61 or more cards in toolbox decks [Archive]

Illissius

12-11-2009, 11:01 PM

I'm not actually sure about this whole 60 cards thing. It seems a bit like the whole efficient markets hypothesis in economics. (Yeah, I just lost 95% of people right there). It seems very elegant and appealing and you can do a good job of explaining away all the potential complications and objections so that it stays valid. But just because it's elegant and appealing doesn't mean it's actually true in 100% of cases.

I mean, the core thesis is that some of your cards are better than others, and you want the highest possible odds of drawing these cards. This is very solid, and most of the time it's probably right. But the theory doesn't actually make any kind of attempt to try and quantify this difference in quality, how much of advantage you're expected to gain, and how and why it actually outweighs all the potential advantages from having more than 60 cards (toolbox cards, land-mana ratio, cards-that-do-one-thing-to-cards-that-do-another-thing ratio, whatever). I mean, maybe there's some cases where it doesn't. Maybe a deck doesn't have such a big difference in power level between its cards. Maybe the power level difference multiplied by the marginal decrease in probability of drawing them from adding another card or two to the deck isn't actually all that significant relative to other considerations. Maybe the fact that cards aren't actually uniformly powerful across matchups and you can't know for certain what matchups you'll face actually matters. But how are we to know? The theory doesn't say. It just assumes.

freakish777

12-12-2009, 01:31 AM

This is very solid, and most of the time it's probably right. But the theory doesn't actually make any kind of attempt to try and quantify this difference in quality, how much of advantage you're expected to gain, and how and why it actually outweighs all the potential advantages from having more than 60 cards

Unfortunately the theory here isn't something that can readily be quantified until Wizards opens up MagicOnline to allow Bots to play games (and then allows you to have access to the statistics of the games played by your Bot, and you're able to run a large enough number of Bots to get a sufficient data set).

I'm going to attempt to provide a method for actually quantifying the quality, it's just that in practice, you would never do it without a lot of computers doing a bunch of work for you.

Let's simplify the equation and say it's Time Spiral Block Constructed instead (in which Patrick Chapin proposed a 66 card Mystical Teachings deck, I don't know if he played it or not, but the reasoning was "you don't actually want to draw Teachings as often as you do with a 60 card deck, otherwise your hand gets stuffed up with cards you can't cast, and you're just tutoring for a bunch of silver bullets anyways, plus the mana being mixed correctly).

If Wizards did open up statistics of games and decks for MagicOnline (you pay a small subscription fee, you get all the data on decks, pairings, results, etc, pretty handy information for the pros, and if they aren't mining this data already to learn how to design better cards in the future, they're retards...), you could query the data for things like:

(first let's set A = "All Decks playing card X", B = "All Decks playing card Y", etc)

1. Percentage of A (how popular is this card).
2. Percentage of matches won by A compared to B, C, etc (how good is this card, where B, C, etc covers all cards in the format).

If you have a lot of data sets from Tournaments, you eventually get some values that estimate "the reality of the situation."

The formula would be something like (this is probably far too simplistic, but it's an attempt):

Card X's Value in the Metagame = Sum (how good is this card against X[i] * how popular is X[i]) (for all cards in the metagame, if you could normalize to 1 that would be even better).

Deck A's Value in the metagame = Sum(value of all cards in A)/Number of all cards in A = "percentage chance your deck has to win against an unknown opponent"

With a quantification of value, you shouldn't even have to do the statistics for probability of drawing X or drawing Y, you'd just say:

Well my Big Game Hunter has total value 0.35 and my 77 card Survival deck (62 main 15 board) has value 0.50, and it would go up if I removed Big Game Hunter...

So SCG will no doubt collect a lot of decklists for their 5k Series. They will also have a lot of data about pairings and results as well. Really, I would like to see SCG do a couple things (since Wizards is slow about doing things, being a large corporation):

1) Allow players to sign up for events electronically (payment, AND deck reg, filling out a web form with your deck reg on it before the event I'm sure would save a whole lot of time on their end manually keying in data later, it also speeds up tournament start time since there's less people to collect money from).

2) Lease the data gained (as opposed to giving select people access to it, it's nice seeing Steve Menendian write articles using your data, and I'm glad someone gets access to it, but why limit it to one person?) I'm sure there's a lot of really interesting things to be mined out of the data...

3) Once you implement #2, tell Wizards to get their fucking act together with MagicOnline data mining.

Maveric78f

12-12-2009, 03:56 AM

Imagine that exhume, entomb and reanimate have so many clone cards (like armageddon being cloned by ravages of war) that you can decide put them in your deck in a lot of copies.

Imagine you're building a mono black deck with no disruption (just the above mentionned cards plus the reanimation targets. In this case you'll definitely want to play a deck with more than 60 cards in order to avoid drawing the reanimation targets.

Another default of decks with more than 60 cards is that it weakens the sb.

grahf

12-12-2009, 04:18 AM

I think it matters more in draft, where you have very little card draw or deck manipulation. There, drawing land when you need some business, any business, will conclusively kill you. And the percentages of drawing what you want will be more greatly reduced by playing extra cards as you're starting with fewer cards to begin with. And also, one can make more conclusive judgments about "this card is less good than these other cards."

Kuma

12-12-2009, 01:51 PM

At some point, I wondered, is it actually a wise desicion to cut down to 60 cards?

Yes. This is Deckbuilding 101.

If I play a 60 card deck, the probabilities of having at least one Survival in the opening hand is about 39,95%.

If I play a 61 card deck, the probabilities of having at least one Survival in the opening hand is about 39,40%.

If I play a 62 card deck, the probabilities of having at least one Survival in the opening hand is about 38,87%.

Why would you want to reduce your chances of drawing Survival at all? It's the engine that makes the deck work. Not only does running extra cards decrease your chances of drawing Survival, it decreases your chances of drawing everything else. Need a land? The 61st card reduces your chances of getting it. Need removal? Unless your 61st card was a Vindicate, or something, it's hurting your odds of getting it.

So by adding 2 cards to the deck the probability of having the important enchantment in your opening hand only decreases by about 1,08%. A value which usually makes no difference in practice.

The average tournament has 9-10 rounds counting Top 8, meaning an unnecessary loss every 10 or 11 tournaments, just for not being disciplined. This doesn't even count the damage from brining down the mean average or drawing that worst card actually directly lowering your percentages. Playing with a 61st card is like playing Russian Roulette. Every match you spin the chamber and have a one in one hundred chance of getting a loss... just because you couldn't cut that last card.

Those may not be the worst odds in the world, but would you play Russian Roulette with a 100-chamber gun in real life? Remember, there is no gain (unless your deck falls under one of the exceptions I'll list, but even then most people just pretend their deck is an exception to rationalize their sin).

Now imagine playing this one in one hundred Russian Roulette over and over. Every match you play, you take another spin and pull the trigger. It will catch up to you. You cannot beat math.

I wonder if this little value is worth the inclusion of more silver bullets, which are usually only good in certain matchups, but can decide games. Cards like Faerie Macabre, Goblin Sharpshooter or even Spore Frog.

When you take all the relative strengths of all the cards in your deck and take the mean average, you arrive at an imaginary line dividing cards that increase your chances of winning in general (Ancestral Recall) and cards that decrease your chances of winning compared to if you drew something else in your deck (Swords to Plowshares). This does not mean Swords to Plowshares is bad or should be cut. What it means is that Ancestral Recall is better in your deck in this metagame, and it is a better draw than Swords to Plowshares in general.

This mean average is how strong your deck is. It reflects the sum total of all contributions towards winning of all the cards in your deck. While it is an abstract idea, we can work with it...

If you add a 61st card to your deck, you are clearly bringing down the mean average. The 61st card cannot possibly be as good as the average. The 60th card is not as good as the average. Most likely, the 40th card isn't as good as the average. The 61st card may help you win games sometimes, but it will contribute to losses more (compared to the other 60) as a whole. Otherwise, it is not your 61st card, another card is and you should cut that one.

Imagine that exhume, entomb and reanimate have so many clone cards (like armageddon being cloned by ravages of war) that you can decide put them in your deck in a lot of copies.

Imagine you're building a mono black deck with no disruption (just the above mentionned cards plus the reanimation targets.

Why would you build that deck? It would be awful.

In this case you'll definitely want to play a deck with more than 60 cards in order to avoid drawing the reanimation targets.

No, no, no.

Not only will the extra cards you run likely be weaker versions of cards in the 60, you're going to hurt your chances of getting enough land. This doesn't even work as a theoretical.

It's not like drawing your reanimation targets is a huge problem in reanimator anyway. It's only a problem if you draw all of them, or you're not running any discard outlets, i.e. Putrid Imp or Cabal Therapy.

@everyone advocating running more than 60 cards:

60 card decks are the established norm for Magic. If you think 61 is somehow better, you need to provide a list of cards that is not improved by cutting it down to 60, and all the necessary proof required.

Read the Patrick Chapin article linked earlier in the thread. He addresses the issue very well and says it all better than I can.

from Cairo

12-12-2009, 01:57 PM

That's very true, and it's a fact clearly reflected in most Gifts-based decks, which opt for far too many bullets in an effort to power up Gifts Ungiven. With that said, however, I'm in favour of going up to 61 cards in a deck with a toolbox-type engine (e.g. Survival, Intuition) if the toolbox options are a relatively small number of cards and/or complement one another closely enough to be considered multiples of that card. One example of the latter scenario would be to use Shriekmaw as a fifth piece of removal. Taking this case, let's say that Shriekmaw was card #61, and the rest of the deck is very tight. Shriekmaw provides a little versatility for your removal (since it's also a beater, and a Bridge-remover), and it also complements your Survival engine (easily fetched removal) and, in addition, it plays into your deck's other mechanics (Genesis, Volrath's Stronghold, etc.). In a case like that, I'd say that the 61st card gives more than a fair account of itself.

Otherwise, I'd probably be against it.

I follow your point and the way you defended it has merit. In your example I'd agree that Shriekmaw does fit well with the other components of the deck and is a worthy inclusion.

But I think it's wrong to look at 1 card as the "61st," in a 61 card deck every card is the 61st. While Shriekmaw may be absolutely defensible, there are 60 other cards that need to be pulling that weight as well.

In some cases cutting a card that is almost always seen as an x4 down to an x3 could make sense if you're running other spells that fill a similar role. Or maybe one's decided that 21 land and 39 spells is leaving them short on land when trying to curve out, and they attempt to solve this by adding the extra "61st" card of a 22nd land to smooth the mana base. Something like this could be addressed through a different angle by replacing some fetchlands with permanent mana sources, so that the deck is less likely to be thinning land drops and will draw into it's 21 land more frequently. Or by cutting a couple clunkier spells and the 22nd land for a couple cantrips, I mean it would all depend on the deck, but there's alot of options to consider when trying to optimize.

Goaswerfraiejen

12-12-2009, 02:40 PM

But I think it's wrong to look at 1 card as the "61st," in a 61 card deck every card is the 61st. While Shriekmaw may be absolutely defensible, there are 60 other cards that need to be pulling that weight as well.

Touché. That's why I felt the need to specify, for the sake of our thought-experiment, that the rest of the deck was also very tight. But I guess that's dangerously close to begging the question.

Ultimately, as you say, it's something that will have to be determined on a case-by-case basis. The specific scenario I pointed to is not one that I can easily see occurring outside of a deck with a Survival or Intuition engine, however, so that might narrow the list down some. Although, come to think of it, it might also work for Lands.dec.

Rico Suave

12-12-2009, 06:38 PM

You can't beat math. 61 cards will lose you games, and matches, if you play over a long period of time (and honestly it's not even *that* long for it to come up).

But I think it's wrong to look at 1 card as the "61st," in a 61 card deck every card is the 61st.

No.

There is *always* a weakest card in the deck, just like there is always a strongest card in the deck. It may be difficult for us to decide which is the weakest, but it is there. Even if a number of cards end up being metagame decisions, there is still a weakest card.

FieryBalrog

12-12-2009, 06:51 PM

That's not necessarily true. Especially in decks that aim to cover different situations, the "weakest card" is not always the same.

Honestly the rigid 60 card limit is more like received wisdom than any actually theoretically solid knowledge. There just isn't enough objective data to make it conclusive. Not to mention many pros do run at least 61 card decks at times. If cutting from 61 to 60 means lower your land count to an unacceptable degree, or cutting a 4-of to a 3-of (which has a much bigger impact on the odds of drawing that card than cutting from 61 to 60 has on drawing any card), then I can see going for a 61 card deck.

undone

12-12-2009, 09:46 PM

You can't beat math. 61 cards will lose you games, and matches, if you play over a long period of time (and honestly it's not even *that* long for it to come up).

No.

There is *always* a weakest card in the deck, just like there is always a strongest card in the deck. It may be difficult for us to decide which is the weakest, but it is there. Even if a number of cards end up being metagame decisions, there is still a weakest card.

Thats is not true. If the weakest card in matchup A exists, and the weakest card in matchup B exists but the torny will consist of something like 30% deck A and 30% deck B with 40 % other and the card your deciding has similar functions in the other matchups than going to a 61st card is not a terrible idea its simply a minimaly risky one. Silver bullets are awsome when they are relivant, the thing is silver bullets are MUCH MUCH more broken when you have brainstorm and can play something like enlightened tutor with 2-3 tutor targets + counterbalances and tops which you would run anyway.

In survival I would say its boarderline, if the list contains any 4 ofs other than Survival, goyf or mana dorks than its probably better to find another card to cut.

grahf

12-12-2009, 11:01 PM

This is getting a little off topic but...

Is a deck with 20 Swamps and 40 Relentless Rats functionally any different or better than 100 Swamps and 200 Relentless Rats? (besides the shuffling mechanics)

Pat Chapin seemed to think so:

What are you playing? Seriously, 36 Relentless Rats and 24 Swamps? Even in a Relentless Rats deck, you only want to play 60. You aren't decking anyone, and the more cards you play, the more inconsistent your mana.
If the ratios are exactly the same, I don't see how the percentages would be any different. But statistics was never my strong point.

Finally, there is a deck where more than 60 cards is correct, but it's definitely not a toolbox deck: 40 Black Lotuses and 40 Wheel of Fortunes. In that "deck," you need a bigger library than your opponent to outlast them.

Nihil Credo

12-12-2009, 11:17 PM

This is getting a little off topic but...

Is a deck with 20 Swamps and 40 Relentless Rats functionally any different or better than 100 Swamps and 200 Relentless Rats? (besides the shuffling mechanics)

If the ratios are exactly the same, I don't see how the percentages would be any different. But statistics was never my strong point.

A bigger library means a bigger variance in your draws, which is something undesirable.

Look at it this way: if your deck is made of 3 lands and 4 spells, you're guaranteed to have a 3 land/4 spell opening hand 100% of the time. If it is made of 30 lands and 40 spells, you won't always get 3/4: a part of that 100% will become 0/7, 1/6, ... , 6/1, 7/0 draws instead. If you have 300 lands and 400 spells, the 'weird' draws will become even more likely, and so on.

4eak

12-13-2009, 01:42 AM

Statistics isn't my strong point either (had to graduate early to provide for my family, so I wasn't able to take advanced classes on the topic [I'm a philosopher, not a mathematician], although I've read on the topic). Forgive my concrete approach (http://pastie.org/741066) to the issue. From a million hands per test:

20/40 -- Land/Spell
0 Land, 7 Spells = 4.83%
1 Land, 6 Spells = 19.9%
2 Land, 5 Spells = 32.4%
3 Land, 4 Spells = 26.9%
4 Land, 3 Spells = 12.4%
5 Land, 2 Spells = 3.11%
6 Land, 1 Spells = 0.412%
7 Land, 0 Spells = 0.0189%
Total= 99.9%

40/80 -- Land/Spell
0 Land, 7 Spells = 5.32%
1 Land, 6 Spells = 20.2%
2 Land, 5 Spells = 31.6%
3 Land, 4 Spells = 26.2%
4 Land, 3 Spells = 12.6%
5 Land, 2 Spells = 3.49%
6 Land, 1 Spells = 0.526%
7 Land, 0 Spells = 0.0309%
Total= 99.9%

80/160 -- Land/Spell
0 Land, 7 Spells = 5.59%
1 Land, 6 Spells = 20.3%
2 Land, 5 Spells = 31.0%
3 Land, 4 Spells = 26.0%
4 Land, 3 Spells = 12.8%
5 Land, 2 Spells = 3.71%
6 Land, 1 Spells = 0.570%
7 Land, 0 Spells = 0.0388%
Total= 100%

160/320 -- Land/Spell
0 Land, 7 Spells = 5.76%
1 Land, 6 Spells = 20.4%
2 Land, 5 Spells = 31.0%
3 Land, 4 Spells = 25.8%
4 Land, 3 Spells = 12.7%
5 Land, 2 Spells = 3.77%
6 Land, 1 Spells = 0.601%
7 Land, 0 Spells = 0.0419%
Total= 100%

Because I suck at programming, I'm impatient, and I don't have a supercomputer, I dropped down to 100,000 hands for this:

1600/3200 -- Land/Spell
0 Land, 7 Spells = 5.86%
1 Land, 6 Spells = 20.5%
2 Land, 5 Spells = 30.7%
3 Land, 4 Spells = 25.7%
4 Land, 3 Spells = 12.8%
5 Land, 2 Spells = 3.81%
6 Land, 1 Spells = 0.619%
7 Land, 0 Spells = 0.0450%
Total= 100%

More educated folk may need to correct me. Perhaps I am missing something vital.

The argument is that by increasing the quantity of cards in the deck, but not changing the ratio of functions in the deck, there will be a higher chance to draw hands which are different from the average hand.

If these numbers are correct (I'm only showing the results of the program), and I'm interpreting them correctly, then it doesn't seem that the quantity of the deck has a very significant impact on the chance to draw our "golden hand" in the context of the difference between 60 to 70 card decks. Ratio seems to do the majority of the work.

From my perspective, even including the odds of seeing your 'MVP' cards, it is certainly possible to build 61 card decks (and even higher) which are actually the best way to build the deck according given the proportion of the functions.

peace,
4eak

SpencerForHire

12-13-2009, 01:53 AM

I think the simple fact of the matter is, there is a difference between drawing a mediocre card for your deck and a good card.

It is expected in deck building that some cards are better than others for a given purpose and it is your duty to tune it down to the best 60 "somethings" you can.

4eak

12-13-2009, 02:19 AM

Surely you can see that it remains at least possible that some decks exist purely on ratio. Land + Relentless Rats, etc.

Even if there was a way to generate "value" for each card in a deck (a very difficult topic to explore), it still would be possible that certain 61+ card decks would be the optimal.

I don't feel I'm qualified to know the value differences between 61+ and 60 cards (and I doubt you are either to be honest). A silver-bullet toolbox deck is unique enough that it could easily fit into the category of decks which benefit from running more than 60 cards though.

I still think, for now, it's a "60-card myth" which we've simply been taught from our early magic days. For most decks, which have such clear MVP's like Goyf, 60 is obvious. But, decks which have more balanced values between the different cards of a deck are less subject to the 60-card rule.

peace,
4eak

Rico Suave

12-13-2009, 04:57 AM

Before addressing specific responses, I challenge anybody in this thread to post a list that is optimal with 61 cards, and I will show you a way to improve it by cutting down to 60.

That's not necessarily true. Especially in decks that aim to cover different situations, the "weakest card" is not always the same.

No it is not always the same, but there IS a weakest card. As such, you should cut it.

Honestly the rigid 60 card limit is more like received wisdom than any actually theoretically solid knowledge. There just isn't enough objective data to make it conclusive. Not to mention many pros do run at least 61 card decks at times. If cutting from 61 to 60 means lower your land count to an unacceptable degree, or cutting a 4-of to a 3-of (which has a much bigger impact on the odds of drawing that card than cutting from 61 to 60 has on drawing any card), then I can see going for a 61 card deck.

There is plenty of data to support it. Have you not seen the math in this very thread? Read Pat Chapin's article, as it has plenty of objective data.

Pros do not run 61 cards. They have in the past, many many years ago, but today? No. Even if you run around and find one list from a recent PT top 8 with 61 cards, I will show you dozens upon dozens of lists that run 60 cards. Just because one pro among 50 might do it doesn't make it right.

If you have trouble with your mana, you can run cards like Ponder which function as something like .3 in regards to your mana count. This is addressed in Chapin's article...

Thats is not true. If the weakest card in matchup A exists, and the weakest card in matchup B exists but the torny will consist of something like 30% deck A and 30% deck B with 40 % other and the card your deciding has similar functions in the other matchups than going to a 61st card is not a terrible idea its simply a minimaly risky one. Silver bullets are awsome when they are relivant, the thing is silver bullets are MUCH MUCH more broken when you have brainstorm and can play something like enlightened tutor with 2-3 tutor targets + counterbalances and tops which you would run anyway.

In survival I would say its boarderline, if the list contains any 4 ofs other than Survival, goyf or mana dorks than its probably better to find another card to cut.

Not every card is a silver bullet. If you are deciding between 2 silver bullets, and can't afford to cut either one, then you need to investigate the other 59 cards because there is always a card that can be cut.

4eak

12-13-2009, 05:19 AM

@ Rico Suave

I challenge anybody in this thread to post a list that is optimal with 61 cards, and I will show you a way to improve it by cutting down to 60.

Just how do you think you can "prove it"?

No it is not always the same, but there IS a weakest card. As such, you should cut it.

Before you go off on someone for "not reading the content in this very thread", please recognize the hypothetical situations already presented in this thread which counter your above assumption.

X Mountains and Y Lightning Bolts.dec may actually be optimal at more than 60 cards. (Relentless Rats, whatever)

If you want to argue about optimality outside a vacuum, and inside a specific metagame, the problem becomes so much more complex. Believe me, I would know, I actually try to measure things like this.

Again, you are assuming the above claim. There's plenty of discussion on it, but directly measuring the value of cards is unbelievably difficult to do. I've yet to see such a full mathematical treatment. Genetic algorithms/neural networks for this problem (the easiest solution which isn't biased upon human results and too small a sample size) don't even exist.

Pros do not run 61 cards. They have in the past, many many years ago, but today? No. Even if you run around and find one list from a recent PT top 8 with 61 cards, I will show you dozens upon dozens of lists that run 60 cards. Just because one pro among 50 might do it doesn't make it right.

It doesn't make it wrong either. You should also acknowledge that nobody has said that it would be COMMON for the optimal number to be 61 or more. Even if it is 1 in 50, the point would still be made. There could be exceptions. You have your experience, and some very good reasons to believe that the vast majority of decks are optimal at 60 cards, but you don't proof that there are no exceptions to this rule.

peace,
4eak

FieryBalrog

12-13-2009, 06:40 AM

Before addressing specific responses, I challenge anybody in this thread to post a list that is optimal with 61 cards, and I will show you a way to improve it by cutting down to 60.

How do you know it will be an improvement? There's a lot of value-loaded words in this thread that I don't think have earned their keep.

No it is not always the same, but there IS a weakest card. As such, you should cut it.
See above.

There is plenty of data to support it. Have you not seen the math in this very thread? Read Pat Chapin's article, as it has plenty of objective data.

That's not data, or at least its not relevant data. By data I mean objective tournament performances with control groups. You would need to conduct some rigorous tests to determine superiority when even Chapin himself indicates that in a best case scenario, the improvement gained by going from 61 to 60 cards is on the order of 0.07%.

Illissius

12-13-2009, 08:30 AM

I still think, for now, it's a "60-card myth" which we've simply been taught from our early magic days. For most decks, which have such clear MVP's like Goyf, 60 is obvious. But, decks which have more balanced values between the different cards

I wouldn't call it the "60-card myth". I'd call it the "60-card rule of thumb". In other words, unless you have a very good reason not to, use 60 cards. One good reason (paraphrased from some article) is that if you really don't know which card to cut and it's the morning of the tournament, you might be better off taking the small but guaranteed hit in consistency from playing a 61 card deck over the chance that you cut the wrong card and do even more harm.

Edit: That said, let me go back to the question of ratios. Land/spell ratio is only one part of the question. Let's say you're playing a control deck, and let's say it has 61 cards in it: 12 draw spells, 12 counterspells, 12 removal spells, 2 win conditions, and 23 lands. What do you cut? The "there is always a worst card" theory doesn't only ignore the variance in card power in the context of different matchups -- which you can explain away with "just average it out over the expected metagame" -- it ignores the variance in the context of different game states. Most of the time, you need a combination of cards which do different things to win a matchup, and furthermore that combination is going to be different in different matchups. Maybe the best card in your deck is Force of Will. So you want the best possible odds of drawing Force of Will. But you sure as hell aren't going to win games if your deck is nothing but Force of Will. To win games, you will need some combination of lands, countermagic, spot removal, mass removal, library manipulation, card draw, and win conditions. And once you break it down like this the change in ratios from adding or removing a card is much more pronounced than just looking at the land-spell ratio, where both are 20+ or 30+ and the difference is so miniscule you maybe can't even fine tune it to the point where you can be sure that the 61 card ratio is better than the 60 card one. So maybe you need at least 3 mass removal spells in your deck to win the midgame against aggro, so you can't cut one of those. And you want at least 6 spot removal spells to reliably survive the early game, so you can't cut any of those either. And you want at least 5 card drawing spells for the late game against the control mirror. And 7 fast counterspells against combo. And you need at least 23 lands. And so on. And once you get down to this level, the difference between 4 and 3 is much greater than the difference between 24 and 23. If you cut one of the cards you "can't cut" to get down to 60, one or more of your matchups is going to suffer. The difference, in perspective, is probably going to be small. But so is the difference in the probability of drawing your best cards between a 60 and a 61 card deck! How can you really say for sure whether the one really small number is bigger than the other really small number?

This is what I mean when I say the 60 card rule is at best a rule of thumb. The idea behind it is compelling, and it's probably going to be right most of the time. But there's all kinds of potential stumbling blocks to the theory which it doesn't even try to address, and the very fact that it's dealing with such miniscule differences makes it very hard to say anything for certain.

Rico Suave

12-13-2009, 09:01 AM

[B][SIZE="3"]Before you go off on someone for "not reading the content in this very thread", please recognize the hypothetical situations already presented in this thread which counter your above assumption.

I don't care about hypothetical situations. I care about actual situations that apply to the game we play. As such, I asked someone to post a list that is optimal at 61 cards. So far, there has been nothing of the sort.

Your post is like "well, IF there was a guy who could run the 40 yard dash in 2.2 seconds then this situation would happen in the game of soccer..." but who cares? It doesn't exist.

Does the toolbox deck have 4 Tarmogoyfs and 4 Survival of the Fittest? Then it needs to be 60 cards for best performance.

It doesn't make it wrong either. You should also acknowledge that nobody has said that it would be COMMON for the optimal number to be 61 or more. Even if it is 1 in 50, the point would still be made. There could be exceptions. You have your experience, and some very good reasons to believe that the vast majority of decks are optimal at 60 cards, but you don't proof that there are no exceptions to this rule.

You would make a lot of credit for your argument by providing even just one example of a deck that is optimal at 61 cards. Just one.

Until then, there are no exceptions to this rule.

Illissius

12-13-2009, 09:14 AM

Rico_Suave: See, we're coming at this from different directions.

We're saying the theory can't be proven, and we can't be sure it's always true.

You're saying it can't be disproven, at least not until someone shows you a specific example.

The two aren't mutually exclusive.

Edit: And the whole idea that if we show you a 61 card deck you're going to correctly pinpoint the right card to cut is preposterous. I could look up decklists from a Pro Tour, find one that's 61 cards, and paste it here. How could you possibly know what kind of metagame the deck's pilot expected? (And whether that expectation was correct is an entirely different question from whether the deck was correct given the expectation). Have you playtested the deck for weeks so that you know the all of the ins and outs of it and how many of which cards you need when to beat each of the expected matchups? No, you haven't. There's no way you're going to have a better idea of which card to cut than the guy who played it for every single 61 card deck from every single high level event ever.

4eak

12-13-2009, 09:48 AM

@ Illissius

Come now, you can finish my quote off...

... of a deck are less subject to the 60-card rule.

I certainly think of it as a 'rule of thumb' in my own deckbuilding. I hope my posts were clear about that. The vast majority of cases, I would suggest a 60-card deck maximum. In fact, I doubt you could find a post of mine which suggests a deck above 60 cards (Minus Battle of Wits.dec). I consider the possibility of an exception pretty rare, although silver-bullet toolbox decks seem the most reasonable place for it to occur.

I called it a myth simply because so many people accept it blindly. You obviously, Illissius, do not accept it blindly.

@ Rico Suave

I don't care about hypothetical situations. I care about actual situations that apply to the game we play. As such, I asked someone to post a list that is optimal at 61 cards. So far, there has been nothing of the sort.

You've not only misunderstood my argument, but you continue to make an outrageous claim that you can optimize 61+ card decks into 60 card decks in all possible cases. You are out of your league. You don't realize you can't even prove your own belief. I could show you a 61 card decklist, and you have no mathematical means to even begin the discussion of the decks optimality or suboptimality, and you have no way to compare the alternatives.

You would make a lot of credit for your argument by providing even just one example of a deck that is optimal at 61 cards. Just one.

Until then, there are no exceptions to this rule.

You would actually have a valid argument if you could say something other than:

I believe X, and since you haven't provided an exception to X, then X must be true.

And, again, it would be nice if you supported your claim that you can optimize any 61+ card deck (which I assume you mean for any format and any metagame as well). Your claim presupposes that you can calculate card value and deck optimality. Please, show me.

My main point is that 61-card decks aren't necessarily suboptimal. We don't have proof for it. The proof is very difficult to produce. Understanding card value and deck optimality is exceedingly difficult. Few here can claim to know beyond a doubt that their deck was truly optimal (even if all of us have said it without thinking).

Testing for the optimality of a single deck type (Burn, for instance) requires atleast millions of games played PER decklist (of which there are many possible decklists) to realistically have a basis for comparison among its variants, even in a vacuum. Playing actual magic against a specific metagame is infeasible to compute for us mortals.

No deck has ever been played enough games (assuming all games were recorded) to say that 'the optimal' build has been reached. You'd be lucky to have 10,000 games with any particular list of cards, and even that is far too small a sample size to see the value discrepancies between a 60 and a 61 card deck or between one card or another.

Here (http://pastie.org/741278) is an example Burn program which might help illustrate how truly small a difference in value we are arguing about. (If you are wondering, yes, I've been working on a robust Burn and Belcher optimizer) Burn is the simplest deck in the game; and for the sake of simplicity, I'm going to argue that we can understand its win-condition status in terms of dealing 20 or more damage.

I set it to "Shock" as the burn spell, and I mulliganed until I had a mountain in hand or only 3 cards left (after brief testing, that appeared to be a decent mull rule). At a million hands per decklist:

Mountain / Shock --- Average Turn on which you Win (Deal 20 damage)
09 / 51 --- 7.9806
10 / 50 --- 7.8711
11 / 49 --- 7.8261
12 / 48 --- 7.8358
13 / 47 --- 7.8832
14 / 46 --- 7.9769

These are really very small differences. Small enough that no human could validly "feel" the difference between the decks in practice. Here's what I got for 1,000 hands:

Mountain / Shock --- Average Turn on which you Win
09 / 51 --- 7.930
10 / 50 --- 7.905
11 / 49 --- 7.839
12 / 48 --- 7.755
13 / 47 --- 7.845
14 / 46 --- 7.962

Before you move on, please compare the numbers. Do you see the problem? From 1,000 hands of testing per decklist, one could assume that 12/48 was the optimal decklist; one could even believe they have "proof" for it. They would be wrong though. Now, I just chose a single set of tests. I'm sure if I run the "1,000 hands" many more times that I could find a set which showcased this issue even more. I don't think I need to though.

You all see how easy it is for humans with even the best anecdotal evidence (that's 7,000 games I just showed you), in as controlled experiment as possible, with no metagame, and least amount of variables to control, etc. would still be quite incapable of proving the optimality of a decklist. You may or may not find optimality by chance with only 1,000 games per decklist (in this case, we didn't). Even if you did find optimality with just 1,000 hands per decklist, that still wouldn't be proof.

Proof is your problem. You can't generate it. Your magic career is largely educated guesswork; so be careful about the claims you make. Do you see why you can't prove to anyone that a 61-card list is truly suboptimal? Even this simple thought experiment is impossible for either of us to prove or know through our own play experience and anecdotal evidence. Imagine how much more complex the discussion of optimality becomes in the context of a wider card pool format, more complex functionality in the tested archetype, the human choice element, flaws in the tournament structure, and the specific metagame in which we wish to play.

A silver-bullet deck, which is practically the opposite of Burn.dec in terms of the ratio of functions (and the variance of each functions value per match) and reliance upon card manipulation, could very well be the deck which breaks the 60-card maximum rule of thumb. Card values in silver-bullet lists have such a huge range, as they are truly metagame defined values. It isn't unreasonable to think that one could produce a metagame in which a silver-bullet deck would prefer the 1-card larger toolbox.

(Just to be clear, this suggests that even huge sets of tournament data is FAR from being proof of optimality; in fact, tournament data is even less reliable with so many variables. You could take every game ever played, and you still wouldn't not know what was the "best deck" or who was the "best player".)

peace,
4eak

Elfrago

12-13-2009, 10:19 AM

Without entering in the 61 vs 60 debate, based on my experience, your toolbox should be trimmed to the minimun, because, due to their own nature, the cards in a toolbox shine in one particular situation and suck the others, often being dead draws.
So, revisit your toolbox, I'm sure you will find something to cut.

Illissius

12-13-2009, 11:02 AM

Come now, you can finish my quote off...

Sorry, that was just a mispaste. I didn't even notice. I know where you're coming from. I just thought "60-card myth" was a bad choice of words, that's all. (You could call it the "60-cards-are-always-correct-in-each-and-every-situation myth", but that one doesn't roll off the tongue quite so easily.)

Giles

12-13-2009, 11:58 AM

A bigger library means a bigger variance in your draws, which is something undesirable.

If we are doing a spell-to-land count and the only thing we care about is if we have a spell or a land. Then it really does not matter how big the deck is. It really is just a Poisson ratio.

Aside:
When magic players dip into the world of statistics, they know nothing. It is not always simple as Thing/Total, and that is true statistic.
What is being dealing with is a binomial probability,expected values, confidence intervals and testing the statistic.

So for the 4eak's brute force method, I would not calculate it like that at all. I would do a binomial probability mass function. Which will give you a better statistic ( Link (note use the pfm to do the test.) (http://en.wikipedia.org/wiki/Binomial_distribution))
Then I would find the expected value (E(x)=np), then calculate the standard deviation with a sample size larger than 50. Then I would do a Z test to make sure that we are conclusive that it effects something.

SpencerForHire

12-13-2009, 02:29 PM

Even if there was a way to generate "value" for each card in a deck (a very difficult topic to explore), it still would be possible that certain 61+ card decks would be the optimal. This is merely hypothetical conjecture from you, like Rico Suave said you have yet to give an example and I have plenty of 60 card list examples that have won tournaments in every format.

I don't feel I'm qualified to know the value differences between 61+ and 60 cards (and I doubt you are either to be honest). A silver-bullet toolbox deck is unique enough that it could easily fit into the category of decks which benefit from running more than 60 cards though.I'm much more qualified than you given the fact that I have taken the time to read numerous mathematical threads, articles and even in the past done some napkin math myself to calculate exactly why 60 is better than 61. You are giving reasons why I am wrong based on your own beliefs which are not backed up by any empirical data to prove yourself correct.

I still think, for now, it's a "60-card myth" which we've simply been taught from our early magic days. For most decks, which have such clear MVP's like Goyf, 60 is obvious. But, decks which have more balanced values between the different cards of a deck are less subject to the 60-card rule.It is the 60 card fact based on the mathematics of consistency. If I wish to see card X in number Y of opening hands I have to run Z copies. The math changes for every card over 60 you play. If your argument is then: "well the percent change between 1 to 2 cards is only a fraction of a point" then you are playing with the consistency level of your deck over all games played. Sure it may not matter in one game, but in every tournament you play where you miss top 8 by one round because you just didn't see that one card you needed, well then you can thank yourself for believing the world is flat because no one ever gave you sufficient evidence to believe otherwise.

Edit: And if your argument is then: the 61st card (which I cut) would have won me that matchup; I would suggest you may need to build a better deck or a better sideboard.

paK0

12-13-2009, 02:44 PM

SpencerForHire

12-13-2009, 03:14 PM

The main problem I have with this thing is that everyone is approaching it very mathematical. You cannot mathematically define the powerlevel of a deck, it's just not possible. The only argument weather a deck is good or not is succes, and since there are hardly any 61-decks out there I think giving the right answear is not really possible.

http://www.starcitygames.com/magic/fundamentals/17670_Reflecting_Ruel_The_61Card_Debate.html

I disagree. You are looking at this in a very "package deal" way. The power level of a deck has nothing to do with optimization and reduction of randomization as much as possible.

Name any deck of which you may be thinking. Now tell me; if that deck was 0.5% more consistent due to an optimization of a card choice or say, bringing it to 60 cards from 61. Would you rather play that version in a tournament or the 0.5% less consistent version?

hjalte

12-13-2009, 03:47 PM

This is merely hypothetical conjecture from you, like Rico Suave said you have yet to give an example and I have plenty of 60 card list examples that have won tournaments in every format.

I'm much more qualified than you given the fact that I have taken the time to read numerous mathematical threads, articles and even in the past done some napkin math myself to calculate exactly why 60 is better than 61. You are giving reasons why I am wrong based on your own beliefs which are not backed up by any empirical data to prove yourself correct.

It is the 60 card fact based on the mathematics of consistency. If I wish to see card X in number Y of opening hands I have to run Z copies. The math changes for every card over 60 you play. If your argument is then: "well the percent change between 1 to 2 cards is only a fraction of a point" then you are playing with the consistency level of your deck over all games played. Sure it may not matter in one game, but in every tournament you play where you miss top 8 by one round because you just didn't see that one card you needed, well then you can thank yourself for believing the world is flat because no one ever gave you sufficient evidence to believe otherwise.

Edit: And if your argument is then: the 61st card (which I cut) would have won me that matchup; I would suggest you may need to build a better deck or a better sideboard.

If you want to do this mathematically, it doesn't matter how many 60-card lists you have, until you have shown, that every single 61-card list, can become more consistent/better, by removing one of the cards.
The problem arises when you, to make a mathematical theory, have to evaluate the power level of every card in the deck. Then you have to take into account the expected/known meta, and compare the cards in your deck, against that meta, and against the overall game plan of the deck. This evaluation of the power level of the cards will certainly be subjective, which will lose you any scientific backup, you wanted to achieve. Even if it wasn't subjective, the problem would be much to complex to model anyway.

So the only way you can possible figure out, if a 60-card list is better than a 61 card list is, if you can obviously point out the weakest card in the list, but that might not be so easy, if you haven't played the deck yourself. Or you can test the hell out of the list, but as 4eak pointed out, it will be close to impossible to test a deck enough to tell, with statistical backup, which card is the weakest.

paK0

12-13-2009, 03:59 PM

I disagree. You are looking at this in a very "package deal" way. The power level of a deck has nothing to do with optimization and reduction of randomization as much as possible.

Name any deck of which you may be thinking. Now tell me; if that deck was 0.5% more consistent due to an optimization of a card choice or say, bringing it to 60 cards from 61. Would you rather play that version in a tournament or the 0.5% less consistent version?

If the other version is 0.5% more powerful I might give it a shot. Its a given that we have a shitload of inconsistend Decks running around here. Lets look at Dragon Stompy, it had a time where it was really succesful and a force to be reconed with. However it has always been inconsistent. It trades that for additional power.

Same thing is here, I'm not an expert player nor a math genius, but decks like that or the existence of Dredge with and without LED show me that it can be a good deal to trade consistency for power. And thats what you do by adding the 61st card. Hard to give examples, maybe adding a Relic to a finely tuned Dreadstill list.

Having a greater toolbox is most helpful when you don't know what you will face. If you breakdown a metagame chances are you don't need 61 cards. But if you do not know what to face you can slam the Relic in there, and maybe win just 1 game on the back of it. In the other games you just have to make free two mana and it will cycle. Ill happily trade the 0,5% if I know I get a reasonable shot at winning g1 against any gravebased Deck that might show up.

Elfrago

12-13-2009, 04:02 PM

SpencerForHire

12-13-2009, 06:49 PM

And lets say that you win all of those 0,5% of matches. Now, the card you're adding MIGHT win you more than 0,5% of matches, and thus running it would still be correct.

Then that card should have been in the main over another card. That comes back to optimal deck building.

If the other version is 0.5% more powerful I might give it a shot. Its a given that we have a shitload of inconsistend Decks running around here. Lets look at Dragon Stompy, it had a time where it was really succesful and a force to be reconed with. However it has always been inconsistent. It trades that for additional power.

I would argue that the most powerful factor of a deck is it's consistency.

Same thing is here, I'm not an expert player nor a math genius, but decks like that or the existence of Dredge with and without LED show me that it can be a good deal to trade consistency for power. And thats what you do by adding the 61st card. Hard to give examples, maybe adding a Relic to a finely tuned Dreadstill list.

You won't find any LED Dredge players advocating 61 cards in any large amount. Before you go hit up DeckCheck.net, I am aware there are a few 62 card lists; there will always be outliers.

If you want to do this mathematically, it doesn't matter how many 60-card lists you have, until you have shown, that every single 61-card list, can become more consistent/better, by removing one of the cards. By that same logic you are failing to find information to prove your point.

Look it comes down to tried and true. Until you can find new data that shows a specific reasoning or technology making 61 cards better, there is no proof. There are countless articles on the subject, all arguments always advocate the math being too minute. If you like playing the odds because your one card is so powerful that it will just win games but doesn't fit into your 60 for god knows what reason then best of luck to you.

Maveric78f

12-13-2009, 07:03 PM

I'm writing from a mobile phone, so please forgive me in advance for the short response. I won't elaborate all my statements but I really cannot let this interesting debate without reacting.

First of, Illisius and 4eak made brilliant posts. Maybe the shock/mountain deck is better with a n/61 ratio than with a n'/60 ratio. Just proving so, would demonstrate that 60 cards is not necessarily optimal, even if in practice on a real metagame (not the one restricted to the shock spell). Another argument in favor for 60+ cards is that sometimes, you simply don't know what to cut and the most risk-free choice, given your limited knowledge, is not to cut anything.

Second-of, Nihil is absolutely true about his variance statement. I don't have a clue of what you meant Giles, except maybe: "you all suck at statistics, and I'll prove by claiming something wrong. Most statistic theory are just tools to evaluate (exactly or approximately) complex event probabilities. Basic knowledge can still prove things when it's simple.

Humphrey

12-13-2009, 09:28 PM

As others mentioned. IF all cards could be rated on a Powerscale, then it might be possible to use statistics - but thats not how Magic works.
You need a combination and you need it at the right time.

I dont think, there is a huge difference in running 60/61 cards, exept some decks like combo where you might draw through the whole deck or you absolutely depended on a specific card in your starting hands - like FoW/Daze

Sometimes I use the 61st card as a metagamecard and add it above the ususal list.
In Goblins i run the 23rd land there, because i want my small toolbox.

Its more about luck than about math to win a match.

Cire

12-13-2009, 09:45 PM

wouldn't all 4eak have to do to prove that a 61 card deck could in some situations be better than a 60 card deck is to run a million plus hands with shock deck, and then if it turns out that the optimal variation of land/shock (61 cards) > land/shock (60) cards, then 61 cards could in theory be better at least once over 60 cards?

Did that make sense? basically 4eak run your million hands with every configuration of land/mountain with 60 cards and then do the same for every configuration of land/mountain with 61 cards; then compare the most optimal (fastest) configuration....

Rico Suave

12-14-2009, 12:36 AM

I am going to write this in a very simple way so it does not get confused.

Mountain/Shock deck is irrelevant. You are only allowed to run 4 of any particular card (except basics) in any given deck. In those decks, there are cards that are simply better than other cards, much like Lightning Bolt is simply better than Shock.

Playing a 61st card means you are less likely to draw what matters. There are no exceptions to this, much like how a Survival deck can't possibly run a 61st card that is better than Survival.

The proof for 60 cards being optimal is in Pat Chapin's article, it is supported by math, and it is supported by common sense. I don't need to do millions upon millions of sample tests to know that walking across the street can be dangerous because I can get hit by a car, and similarly I don't need to conduct millions upon millions of sample tests to know that running a 61 card decklist means I'm less likely to draw my best cards.

You are getting caught up in the math and theoretical nonsense that you are failing to see that running 61 cards will lose you games because you did not optimize your decklist.

Does it matter *how* much of an improvement it is going from 61 to 60 cards? No. All that matters is that it is an improvement.

Jak

12-14-2009, 01:27 AM

4eak

12-14-2009, 01:35 AM

@ Giles

I would be grateful if you would show the math. Of course, this variance is only one small problem of the larger one: calculating card and deck values. I'm unaware of any extensive mathematical solution to problems of this nature (Chess, for example, is still "unsolved" in this sense). Show me what you've got, I'd appreciate it.

@ Cire

The point of the exercise wasn't to find an exception; the point was to show how nobody here could possibly claim they can calculate optimality for all decklists. I've given proof of a lack of accuracy in our own testing and tournament data which, even in the most simplistic and quantifiable decks (decks for which it is easiest to calculate value), prevents anyone here from knowing beyond a shadow of a doubt that the 60-card rule has no exceptions. Unfortunately, because of that lack of accuracy and resources, I don't think I'm in a position to prove an exception exists either! I'm arguing that we are technologically incapable and too ignorant to know the answer (that doesn't mean we shouldn't keep searching and looking for the answer).

I did look for exceptions on the offchance I might find one. I only ran a few tests, but I didn't find significant exceptions. Several 61 and 62 card lists were equal to the best 60 card lists (as far I could tell). It's possible that some possible burn list would benefit from running more than 60 cards, but I don't have the "microscope" to prove it either way.

The problem is that I'm forced to look at the 4th and 5th decimal place, which frankly, isn't accurate enough at a million hands per decklist. If you run the same test several times, you'll see that the very last digits can vary. It's clear that my brute-forcing isn't accurate enough (although it's hella better than nothing!). The brute force technique needs to be re-written in a faster language, perhaps C, and run several orders of magnitude larger to give us any semblance of the accuracy that we need.

As I said before, I think the exceptions, if any, would be rare. If the exceptions are rare, then testing might simply be infeasible for an individual. There is a large space of decks that would need to be tested.

Finding an exception to the 60-card rule may not be found at 61 cards. Perhaps the perfect ratio is found at 62, 63, 65, or more cards. The population of decklists, even for something as simply as X Burn cards and Y mountains (as simple as you get), grows out of control.

Furthermore, it's possible there are no exceptions for Shocks/Mountains.dec or for LightningBolt.dec, and so on. This would not prove the 60-card rule to be true though. If we widen our format to include more than just one type of burn spell and mountains (obviously, we have to!), then we'd have an exponentially larger set of decks to test. We'd then also need to test all Shock + Lightning Bolt.dec's, and so on. The trick is that there are too many combinations of decks for us to go through.

Beyond burn decks, we don't even have a solid ground on which to evaluate deck optimality, card values, or create valid comparisons between two decks in specific metagames. I have some of the best data to be found on the subject, constanty updated/crawled and replicated databases of morphling.de and deckcheck, my personal matchup testing history (fairly extensive), and, some fairly unique brute-force techniques to measure things that simply couldn't be measured by hand. These are still drops in a bucket. (I think the creator (http://en.wikipedia.org/wiki/Richard_Garfield) of Magic probably liked this sort of thing.) And, that's okay. For now, we should be willing to admit ignorance and the possibility that our assumptions could be wrong.

Admittedly, this isn't an easy topic to wrap your mind around. Here's a quick analogy of what I'm saying:

Imagine an unquantifiably large kitchen floor (all possible decks) tiled with an uncountable number of tiles (each tile representing a deck).

My buddy over here told me that "he knows" for a fact (as in, he somehow can prove) which tiles have the least quantity of bacteria on them (he can magically know the optimality values of all the decklists). He told me that he knew one particular row of tiles, row 60 (those decklists with 60 cards exactly in them), from which he could select tiles which had the fewest bacteria on them (were the most optimal) of all the tiles in the kitchen.

I turned to my buddy, and said: "How do you know that?", and he couldn't me an answer.

My buddy doesn't even have a microscope. He just looks with his naked eye at the tiles and knows this to be true. I point out to him that it can be difficult to know if a tile is free of bacteria just by using his naked eye, but he didn't even realize what I had said.

Not only was my buddy incapable of observing all the tiles in the kitchen (many of which were obviously bacteria-ridden, but some we couldn't tell), and he had no system by which to make a comparison, but he lacked the tools to keenly observe even a single tile in that kitchen.

Like the difficulty in measuring those tiles, we face a similar (if not even more complex) problem in quantifying magic decks. Pimpin' ain't easy.

@ Rico Suave

Mountain/Shock deck is irrelevant.

Buddy, everyone knows that the mountain/shock deck doesn't exist. Not only that, we know that even if it were possible (and actually, it is almost possible when checking over the set of legal cards), it still wouldn't be anywhere near the best deck in Legacy because there are so many cards which are better.

These facts do not detract from my argument against your claim that all 61+ card decks can be optimized into a 60 card deck.

You've failed to understand what I'm doing for you. You are claiming that your 60-card rule holds in all formats right? In this case, we had a severely restricted format. The mountain/shock deck is relevant to your claim. I simplified the format in this example, simply to showcase how impossibly difficult it would be for you to support your thesis. Even in the best conditions, a burn.format, you can't justify your claim that 61+ card decks are always suboptimal. If there is no way for either of us to justify your claim in a simple vacuum, then we will also fail (to an even greater extent) to justify such your claim in the more complex and chaotic formats. We don't have the tools to measure this problem, not even if we cut the problem down to a simple form.

Playing a 61st card means you are less likely to draw what matters.

Again, you clearly have no idea how difficult it is to support the claim that there is always a "best" card in a deck in a given metagame. It is quite possible there are decks which have more balanced values between cards. In fact, they don't have to even be equal. If they were close enough in value, the value of having the proper ratio of each function might outweigh the benefits of going down to 60 to see the handful of cards which are only barely better than the rest in your deck.

You are getting caught up in the math and theoretical nonsense that you are failing to see that running 61 cards will lose you games because you did not optimize your decklist.

I can only provide you with so much support. This sounds like you don't believe the game is subject to the laws of mathematics and that your 60-card rule is true without need of more rigorous proof.

Hear me out:

I am not saying the 60-card rule is not true.
I am saying neither of us are capable of knowing whether or not that rule is true at this point in time.
For now, we can only claim that it is possible for the 60-card rule to be either true or false, and that's the best we can do for now.

The point of contention is "what is knowable", which we clearly disagree about, probably because you aren't reading carefully enough.

peace,
4eak

Rico Suave

12-14-2009, 02:34 AM

What about Gabriel Nassif's 5 color control deck? In a deck like that you care more about the land/spell ratio than having a higher percentage of drawing a Cruel Ultimatum or something. In decks with redundancy like that, I would much rather have the right land/spell ratio where you can only get that by having more than 60 cards.

There are ways to achieve a tight mana:spell ratio without 61 (or more) cards.

Deck manipulation in particular will smooth draws. If 20 land is too few and 21 is too many, cards like Brainstorm will make 20 land just right.

Buddy, everyone knows that the mountain/shock deck doesn't exist. Not only that, we know that even if it were possible (and actually, it is almost possible when checking over the set of legal cards), it still wouldn't be anywhere near the best deck in Legacy because there are so many cards which are better.

This is the entire point. You said it yourself, there are so many cards which are better.

When you build a deck in Legacy, you want to maximize your chances of drawing the cards which are better. You do this by playing 60 cards instead of 61.

It is that simple.

Again, you clearly have no idea how difficult it is to support the claim that there is always a "best" card in a deck in a given metagame.

A 10 year old can recognize that Survival of the Fittest is the best card in a Survival deck, or that Tarmogoyf is the best card in deck X.

I can only provide you with so much support. This sounds like you don't believe the game is subject to laws of mathematics and that your 60-card rule is true without need of more rigorous proof.

The laws of mathematics prove that you are more likely to draw at least 1 of 4 Tarmogoyfs if you run 60 cards instead of 61.

The laws of math are EXACTLY what I believe.

Hear me out:

* I am not saying the 60-card rule is not true.
* I am saying neither of us are capable of knowing whether or not that rule is true at this point in time.
* For now, we can only claim that it is possible for the 60-card rule to be either or false, and that's the best we can do for now.

The point of contention is "what is knowable", which we clearly disagree about, probably because you aren't reading carefully enough.

No, you are not capable of knowing it's true. It is painfully obvious to be true for anybody with common sense.

Like I said before, if you can provide just one example of a deck that is optimal at 61 cards then be my guest. Until then, the rule remains.

Made up formats, fantasy worlds, and hypothetical scenarios don't exist. You can go on and on about how there might be a possibility of a chance that a deck could exist that perhaps would be better off as 61...blah blah. It doesn't exist.

I can claim that in a made-up scenario a man can bench press an elephant. But guess what? It doesn't really exist, and until someone actually does bench press an elephant, then the rule remains that nobody on this planet can do it.

Do you see how silly your argument is? Please, just stop. You can argue for the sake of arguing, because that is all you are doing, but I will not be a part of this anymore.

Jak

12-14-2009, 03:31 AM

There are ways to achieve a tight mana:spell ratio without 61 (or more) cards.

Deck manipulation in particular will smooth draws. If 20 land is too few and 21 is too many, cards like Brainstorm will make 20 land just right.

Seriously? What you said makes no sense. For one, Brainstorm isn't in type two, which is what the example was. Secondly, you just proposed adding Brainstorm in place of a land. How does that change anything?

Sometimes you can't just remove sweepers, counters, draw, targetted removal, etc to play the right number of lands. This was Nassif's dilemma.

The list for example. I am sure you are just going to post what card to cut but can you really? You have way too many factors to just decide what card to cut. The deck won. It was 61 cards.

2 Cascade Bluffs
2 Exotic Orchard
3 Island
1 Mystic Gate
4 Reflecting Pool
4 Sunken Ruins
2 Vivid Crag
4 Vivid Creek
3 Vivid Marsh
2 Vivid Meadow

3 Broodmate Dragon
4 Mulldrifter
3 Plumeveil
3 Wall of Reverence

4 Broken Ambitions
1 Celestial Purge
2 Cruel Ultimatum
4 Cryptic Command
4 Esper Charm
1 Pithing Needle
1 Terror
4 Volcanic Fallout

Sideboard
1 Celestial Purge
2 Infest
2 Negate
1 Remove Soul
4 Scepter of Fugue
1 Wispmare
2 Wrath of God
2 Wydwen, the Biting Gale

Maveric78f

12-14-2009, 04:30 AM

Generally, I hate making personnal attack, but I think that you're all losing your time arguing with Rico Suave. I already tried to do it in 2 threads unsuccessfully. The first thread was the dream halls' thread. And the second was the thread about the uncountable tokens provided by the doubling season, followed footsteps opalescence combo. In this very last one, I mathematically proved he was stating bullshits. He never understood it, nor admitted it.

The mountain/shock format is really full of insight since it shows what can happen in a magic format, that is computable.

Rico Suave

12-14-2009, 04:30 AM

Seriously? What you said makes no sense. For one, Brainstorm isn't in type two, which is what the example was. Secondly, you just proposed adding Brainstorm in place of a land. How does that change anything?

Sometimes you can't just remove sweepers, counters, draw, targetted removal, etc to play the right number of lands. This was Nassif's dilemma.

I said cards like Brainstorm.

I agree that sweepers and targetted removal are pretty important for a control deck, but you'd be surprised how easy it is to remove a counter. Draw can be very important, but frequently that is also easily put on the chopping block.

The list for example. I am sure you are just going to post what card to cut but can you really? You have way too many factors to just decide what card to cut. The deck won. It was 61 cards.

2 Cascade Bluffs
2 Exotic Orchard
3 Island
1 Mystic Gate
4 Reflecting Pool
4 Sunken Ruins
2 Vivid Crag
4 Vivid Creek
3 Vivid Marsh
2 Vivid Meadow

3 Broodmate Dragon
4 Mulldrifter
3 Plumeveil
3 Wall of Reverence

4 Broken Ambitions
1 Celestial Purge
2 Cruel Ultimatum
4 Cryptic Command
4 Esper Charm
1 Pithing Needle
1 Terror
4 Volcanic Fallout

Sideboard
1 Celestial Purge
2 Infest
2 Negate
1 Remove Soul
4 Scepter of Fugue
1 Wispmare
2 Wrath of God
2 Wydwen, the Biting Gale

I was referring to a Legacy list. After all, we are on Legacy boards.

I don't have nearly enough experience with recent Standard or the metagame at that event to make an educated decision.

What is the Pithing Needle for?

Maveric78f

12-14-2009, 04:35 AM

What about dragon stompy as a deck being better with 61 or 62 cards in order to be able to burn for 10 with Arc Slogger?

hjalte

12-14-2009, 04:47 AM

I'm beeing a bit pedantic, I know, but:

The laws of math are EXACTLY what I believe.
The laws of math tell you, that if you can't prove something, it's only an assumption, therefore:

Like I said before, if you can provide just one example of a deck that is optimal at 61 cards then be my guest. Until then, the rule remains.
You can't possibly prove, that the 60-card rule always holds mathematically, therefore it's an assumption, that it is true.

I can claim that in a made-up scenario a man can bench press an elephant. But guess what? It doesn't really exist, and until someone actually does bench press an elephant, then the rule remains that nobody on this planet can do it.
How would this help your argument? If you want to use the laws of mathematics, it is almost always nescessary to make an idealized world. By simplifying your problem, you will get some deeper knowledge of the problem, which might not be obvious, when you are examining the full problem.
Let's look at a simple example from physics. You throw a ball, with a certain speed in a certain angle. We calculate the trajectory of the ball, using simple newtonian mechanics, in the gravitational field of the earth. This will, in many circumstances, be a good assumption, and it will certainly give you quite a lot of information on the balls flight through the air.

Your argument is, that we can't learn anything, if we make these idealized assumptions. We need to take into account, that we have drag from the wind, that the earth isn't a uniform sphere, that the suns gravitational field is also pulling in the ball. Hell, we should even look at local fluctuations in the atmospheres density. This problem is unsolvable, to any meaningful extent.
But nonetheless, our simplified model explains a lot of the behaviour of the ball, so why can't we use this model?

Do you see how silly your argument is? Please, just stop. You can argue for the sake of arguing, because that is all you are doing, but I will not be a part of this anymore.
4eak did exactly that. He made our problem the simplest possible, so it could be handeled by a computer with relative ease. That approach made it possible to know some mechanisms in the game, which might not be visible, if we analyzed the full problem with all cards in magic. I wouldn't call that arguing for the sake of arguing.

4eak

12-14-2009, 04:48 AM

@ Rico Suave

If you are tired of debating, then stop debating. There's no reason to get angry. Your argument is not improved by your use of ad hominem, which is about as uncouth as violating Godwin's Law (http://en.wikipedia.org/wiki/Godwin's_law). You seem averse to discussing theory, hypotheticals, and thought experiments; and, unfortunately, this thread's very topic is itself quite theoretical.

The laws of math are EXACTLY what I believe.

Good, then we have some common ground to talk about the issue. I'm calling your 60-card rule the semi-equivalent of Goldbach's Conjecture (http://mathworld.wolfram.com/GoldbachConjecture.html). You say you "know" the answer; and for now, I'm saying you can't. You're claiming to know something which is beyond our current ability to know.

I'll try to fit this into the analogy (because perhaps that will not be as emotionally charged in your eyes). We can agree that the vast majority of the kitchen tiles are largely green and filled with bacteria, and that we can safely assume they aren't as optimal as a distinct group of tiles starting at row 60. However, in that small set of kitchen space, we are without the necessary tools (a microscope, etc.) to perfectly evaluate the differences in such a way for us to make the claim you've made.

As Illissius put it, we both agree to at least a 'rule of thumb'. Extending the rule of thumb to a proven maxim of fact is quite a bit more difficult. The problem of measurement and quantifying value will continue to haunt this problem.

Your argument (http://en.wikipedia.org/wiki/Negative_proof) so far rests upon the notion that we haven't provided a counterexample. I've tried to show you that it isn't very easy to find a counterexample in the same way that it isn't very easy to prove your claim to be true, and more importantly, that it isn't necessary to find a counterexample to rebut your claim to finding proof for the 60-card conjecture because neither of us are capable of pinpointing deck optimality for all decks in all metagames, which is the basis upon which one could justify an argument for or against the 60-card rule.

If it makes you feel any better, I hedge my bets upon that 60-card conjecture as well. Even in toolbox decks, I do my best to goto 60 (write a python script and digup my posts, you'll see I only post 60-card lists). I consider sticking to the rule of thumb to be a risk worth taking (one which is heavily in my favor), but not fact which is provably known.

peace,
4eak

Rico Suave

12-14-2009, 05:11 AM

No, my argument is that there are 60 cards which are the best fits, and that anything which would fit into card slot 61 is simply weaker than the other 60 cards.

My argument is that it's better to have the highest possible chance to draw your best cards instead of having a lower chance. Why would you ever want to draw that 61st card if you could just cut it and draw a stronger card?

Theoretical examples *can* lead to practical conclusions, like in physics, but in this discussion there has been no practical conclusion. There has just been talk of possibilities and it has led to no concrete examples.

Even with the Nassif case above, he himself admitted that he was rushed into making the deck (he constructed his SB the morning of the event) and that he would change maindeck slots.

What about dragon stompy as a deck being better with 61 or 62 cards in order to be able to burn for 10 with Arc Slogger?

Dragon Stompy's strength isn't burning with Arc Slogger, but rather getting Magus/Moon/Trinisphere in the opener with acceleration.

4eak

12-14-2009, 05:31 AM

@ Rico Suave

my argument is that there are 60 cards which are the best fits, and that anything which would fit into card slot 61 is simply weaker than the other 60 cards.

To be clear, that is the same conjecture to which I was referring. That argument can be paraphrased as: no 61+ card deck is optimized in a format that allows a 60-card deck. Your argument in defense of "knowing it to be a fact" was fallacious.

Theoretical examples *can* lead to practical conclusions, like in physics, but in this discussion there has been no practical conclusion. There has just been talk of possibilities and it has led to no concrete examples.

I think the theoretical examples have explained the boundaries of our provable knowledge about the concrete world. I didn't tell you the answer. I told you the boundaries to such an answer given our current circumstances, which are broader than you claim.

Moreover, I think future progression on the issue will continue down the path I showed you. If more concrete results are to be obtained, it will be through neural networks, genetic algorithms, bots, and grinding it out on a computer. For now, I don't see how it is feasible for us to solve this by hand.

Dragon Stompy's strength isn't burning with Arc Slogger, but rather getting Magus/Moon/Trinisphere in the opener with acceleration.

To be fair, Stompy has two things it must accomplish:

Drop Disruption
Drop a relevant Fatty.

It will usually lose if it can't do both.

Slogger is often the most relevant beater in the deck, in part because of the control elements and alpha-strike threat it offers. Improving the best beater in a deck is relevant.

Is it more relevant than the maximizing the chances to see other important cards and the consistency gains of playing 60 cards? I very highly doubt it, but again, it remains possible that 61+ could be optimal in some metagames.

peace,
4eak

Maveric78f

12-14-2009, 05:56 AM

I also don't have a clue about whether DS is better @60+ cards but another argument in favor of 60+ is tha fact that all cards are highly redundant, 20 accelerant, 16 locks, 16 kills or so, and it has no card on which it relies that much. Trinisphere, chalice, magus or blood moon are not rankable in terms of performance. If you could play more than 4, I'm quite sure you wouldn't because they are complementary and useless in multiple.

Illissius

12-14-2009, 07:31 AM

Theoretical examples *can* lead to practical conclusions, like in physics, but in this discussion there has been no practical conclusion. There has just been talk of possibilities and it has led to no concrete examples.

We are arguing past each other. In basically every single concrete case I endorse playing a 60 card deck. I can't think of any specific exception. Same as you.

Theoretical possibilities are all we were talking about. I am saying the present state of magic theory doesn't preclude the possibility of an optimal 61 card deck existing. The theory of 60 card superiority is not rigorous enough to exclude this possibility. Obviously you can't come up with a strict mathematical model of Magic and all of the interactions in it (and in the metagame), which is what you would need to prove it definitively, but even the simplified model the theory uses is not sophisticated enough to be satisfactory. It's a bunch of handwaving. Very convincing handwaving, to be sure. But the whole thing is more or less unknowable.

Even if we do accept that it's theoretically possible for optimal 61 card decks to exist, because of the very handwaviness of it, it's still basically impossible for us to tell with any certainty whether any specific deck is one of them. Therefore, when in doubt -- and you are always in doubt -- it's best to play 60 cards. The practical consequences of this theorizing are effectively nil. You could say it's pointless. But we play a game of cards with Dragons on them, so that is relative.

4eak

12-14-2009, 08:05 AM

The conclusion from Illissius' post (which I agree with) for those who aren't interested in pure magic theory:

Handwaving dragons theoretically play 61 card SurvivalBurn decks on pointless concrete kitchen tiles

It was an interesting thread. I learned how pointless my record keeping has been in many ways. Looking at the numbers was pretty shocking. I'll keep working on some optimizers for real decks. Belcher and Burn are prime candidates. Anyone else interested in writing some code?

peace,
4eak

my eyes!---frogboy

Rico Suave

12-14-2009, 08:37 AM

Going beyond the math, there is something else to consider:

Personal experience.

My personal experience has shown that there is a lot more to a 61 card deck than the ratio of mana:spell and the slightly reduced chance of drawing your best cards. While these factors are more than enough in their own right, there is much to be said that theoretical possibilities do not encompass.

Every single game that 61st card shows up, it is a factor. Every single game you draw that 61st card, you have to acknowledge that all of your subsequent draws are a full turn later than what you would have gotten if you just ran 60 cards.

Back when I tested 61 cards I would keep track of what my draws were after the 61st card. Far too many games I would draw a Cursed Scroll a turn too late. Far too many times I would draw a Wasteland a turn late, and oh hey my opponent used that opportunity to ramp up his mana and drop a bomb. Far too many games I would lose with a control deck to an aggro deck, and then flip over the top card of my library to reveal the sweeper I needed to stabilize. But hey, I had the 61st card!

Guess how many times the 61st card would win a game for me? Once or twice in hundreds of games. In other words it lost a hell of a lot more games just by being in the deck than it ever contributed to winning.

The main argument in this case is that a deck like Survival would benefit tremendously from having a silver bullet. Ironically, I found that the decks with heavy card/deck manipulation were negatively affected by the 61st card more than decks without manipulation. Why? Think about it, when you see more cards you are more likely to see the 61st card, and as a result the rest of your draws are now skewed.

In conclusion, the 61st card of a deck has a lot more impact than decimal percentages would lead you to believe, and no amount of theorycrafting will make up for raw experience learned from an enormous amount of time building and playing decks. This is not concrete evidence, but to anyone with a lot of experience it is self-evident.

Eventually it will catch up to you one way or another.

One last thing:

Drop Disruption
Drop a relevant Fatty.

It will usually lose if it can't do both.

Slogger is often the most relevant beater in the deck, in part because of the control elements and alpha-strike threat it offers. Improving the best beater in a deck is relevant.

Going up to 61 cards is not improving it. Drawing Arc-Slogger a turn late, simply because you drew that 61st card first, will deny you 4 damage. =\

4eak

12-14-2009, 08:52 AM

@ Rico Suave

Your personal experience, as well as mine, is irrelevant (http://www.mtgthesource.com/forums/showpost.php?p=409226&postcount=29) in this discussion. I wish that weren't the case. The scope of the question, however, forces this into non-experienced based solutions.

One person can pickup a deck, play 1,000 games with it, and another person can pickup the exact same deck, play 1,000 games with it, and they are just too likely to have different experiences and results. They can draw essential "rule of thumb" claims about the deck, but questions concerning something as small and difficult to quantify and measure as the value and optimality of a single card (in a non-toolbox setting) are beyond the scope of their experiences to validly answer.

Experience is important and having that flashlight in the dark is obviously very useful. It provides context and let's you separate out the wheat from the chaff on more obvious issues. Experience and intuition will help you find good "rules of thumb", but as far as the actual answer to this particular problem, no one will ever be experienced enough by playing magic itself to find it.

You still aren't catching some of the complexities of this issue, but whatever. That's fine; even if you understood the reality of this issue, it would not amount to any difference in how you play magic at this point in time. (Perhaps it would change how you study magic though)

Fun argument, I know. As I said, it's theory. Although, plenty of the things we now know about the game were at one point in time 'just theory' =). Give it time.

peace,
4eak

paK0

12-14-2009, 12:35 PM

Going up to 61 cards is not improving it. Drawing Arc-Slogger a turn late, simply because you drew that 61st card first, will deny you 4 damage. =\

Well, maybe the card is the acclerant that you needed to play him in the first place.

If you just add random cards it is obvious that they won't show results.

tivadar

12-14-2009, 12:58 PM

Ok, let me start by saying that this thread really can't "prove" anything. Yes, mathematics shows us that your probability of drawing a "better" card in a 61 card deck is lower than 60. What this *doesn't* consider is how cards interact with one another, or, for that matter, the game of Magic at all.

My *opinion* is this. Start by saying that obviously (to most people) a 70 card deck is non-optimal. I'd even argue that 65 cards is also non-optimal. For me personally, I'm either playing 60 or 61 cards. Why do I even allow 61 cards? Because, like it or not, there are games that run to someone being decked. And there are people playing Solidarity out there. Yes, the odds of you being helped by having 61 cards vs. 60 is fairly miniscule, but then again, so are the increases in probability of drawing a certain card.

I'd be willing to allow up to 63 cards. Why 63? Because I feel after that, that you can remove a full set of 4 cards from the deck and probably increase the power level, or meet some ratio you're trying to achieve in some other way (lands, to creatures, to removal, for instance). However, I really feel that *either* 60 or 61 are optimal. But once again, that's just my opinion.

Elfrago

12-14-2009, 01:41 PM

Then that card should have been in the main over another card. That comes back to optimal deck building.

No, it isn't always true.
A simple example is when every card in my 60 allows me to win 0,7% of my matches, the 61 card wins me an addition 0,6% of my matches while playing 60 cards allows me to win 0,5% more of my matches. Looks weird indeed, but maybe the added flexibility of the additional card allows me to win more matches than the slighltly optimization offered by 60.

MattH

12-14-2009, 02:58 PM

I challenge anybody in this thread to post a list that is optimal with 61 cards, and I will show you a way to improve it by cutting down to 60.

Question: does the deck have to be good, or merely a legal decklist?

I'm assuming that if someone posts a vintage-legal Boros deck that you aren't planning on responding with, "Cut 61 cards, add a 60-card Tezzeret deck." Claiming that you can find a better decklist is not the same as claiming that you can find a better decklist which is a strict subset of the original. As quoted above, your challenge is the latter.

.
.
.

I would like to untie two issues being discussed in this thread which some people have conflated.
Statement One: Every 61 card deck can be improved by cutting one card.
Question Two: Assume Statement One is true. Choose a card to cut. Can it be proved that the chosen card was the correct one to cut?

These are NOT the same question, and should not be treated as such. Also, it seems like some people (looking at you, Rico Suave, although you're far from alone on this) need a refresher on the definition of proof. "A convincing demonstration" is not proof. 'Evidence' is not proof.

Kuma

12-14-2009, 03:02 PM

"60 cards is optimal" is a theory in the way that gravity is a theory. We can't prove that 60 card lists are always optimal, just like we can't prove "gravity" is what's keeping us from floating off the planet. However, both ideas are the best and most realistic approximations of reality we can come up with right now. As such, the burden of proof is on those who say 61 cards is optimal in Deck X.

I believe that we can all agree on the following statements:

Some cards in your deck are better in more situations than others, i.e. stronger. Force of Will is generally better than Spell Snare, etc.

You want to draw the strongest cards in your deck as often as possible. In a Survival deck, this means drawing Survival of the Fittest. In Goblins --- Goblin Lackey, In CounterTop --- Counterbalance, Tarmogoyf, or Force of Will.

If you don't need the strongest card in your deck, then you need to draw the strongest card for the situation. This could be drawing a Trygon Predator against Stax, or a Tormod's Crypt against Ichorid.

Running a 60 card deck maximizes your chances of drawing both your strongest cards, both for most specific situations and overall.

The difference between a 60 card deck and a 61 card deck is minimal, but real.

------------------------------------

Given the above statements, how can it be beneficial to run 61 cards? Yes, it doesn't make much of a difference if you run 61 cards instead of 60, but why would you want to dilute the quality of your draws at all? What benefit do you gain from reducing the power of your average draw? Sure, if you run a Faerie Macabre in your Survival deck as the 61st card because there is a lot of Loam and Ichorid in your meta, you're improving your chances of winning. But by running 61 cards, you're reducing your chances of drawing that awesome Faerie Macabre. You could make the matchup even better by going to 60 cards. Maybe cut that fourth Thoughtseize or Cabal Therapy that you're always sideboarding out.

Every deck I've played has always had a weakest card or couple of cards. Do you find yourself siding out a certain card or cards more often than the rest of your deck? Consider cutting one and going to 60 cards. Obviously the card isn't needed in the first place.

A very excellent Patrick Chapin article was linked early in the thread and ignored. I would like to see some in the 61 card camp issue a rebuttal to the article's points. Here it is again:

http://www.starcitygames.com/magic/standard/12478_61_Cards_Magic_Russian_Roulette.html

I don't really care if 12 Mountain/48 Shock.dec is optimal at 61 cards, because real Magic decks don't look like that. Sure, it's an optimal 61 card list, but how is that relevant to Legacy?

DISCLAIMER: I'm not saying it's impossible in Legacy for a 61 card deck to be optimal in theory. I'm just saying that we have nothing to suggest that it is, and everything we know suggests that 60 cards is optimal in most, if not all situations.

Illissius

12-14-2009, 10:23 PM

"60 cards is optimal" is a theory in the way that gravity is a theory. We can't prove that 60 card lists are always optimal, just like we can't prove "gravity" is what's keeping us from floating off the planet. However, both ideas are the best and most realistic approximations of reality we can come up with right now.

That's not the point. The problem isn't that it's only a theory, the problem is that the theory isn't powerful enough. To take the gravity analogy, what we have with 60 card superiority at this point isn't general relativity, it's not even Newton, it's more like saying "things fall down because look around! things fall down", while trying very hard to ignore helium balloons. Nobody is going to doubt the fact that things fall down. That doesn't mean everyone has to be in perfect satisfaction with the theory.

(As an aside, while in the physical sciences it's not possible to prove anything, only to disprove, the same isn't true of mathematics and logic. You have plenty of proofs in math. And because Magic is a strictly artificial/logical construct rather than a physical phenomenon I'd say it falls into the latter camp. But again, this is besides the point, because I don't even expect a strictly rigorous mathematical proof here -- that would be foolish and more or less impossible.)

As such, the burden of proof is on those who say 61 cards is optimal in Deck X.
Indeed. However, the burden of proof is also on those who insist that 60 cards is optimal in 100.000000% of cases with no possible exceptions whatsoever.

DISCLAIMER: I'm not saying it's impossible in Legacy for a 61 card deck to be optimal in theory. I'm just saying that we have nothing to suggest that it is, and everything we know suggests that 60 cards is optimal in most, if not all situations.

Great! Then we're in perfect agreement and have nothing to argue about.

All I'm really for, I guess, is for people to have a greater awareness of how things work rather than just relying blindly on dogma. To the best of our knowledge, playing 60 cards is best, but the best of our knowledge isn't terribly advanced.

Rico Suave

12-14-2009, 10:56 PM

Question: does the deck have to be good, or merely a legal decklist?

I'm assuming that if someone posts a vintage-legal Boros deck that you aren't planning on responding with, "Cut 61 cards, add a 60-card Tezzeret deck." Claiming that you can find a better decklist is not the same as claiming that you can find a better decklist which is a strict subset of the original. As quoted above, your challenge is the latter.

.
.
.

I would like to untie two issues being discussed in this thread which some people have conflated.
Statement One: Every 61 card deck can be improved by cutting one card.
Question Two: Assume Statement One is true. Choose a card to cut. Can it be proved that the chosen card was the correct one to cut?

These are NOT the same question, and should not be treated as such. Also, it seems like some people (looking at you, Rico Suave, although you're far from alone on this) need a refresher on the definition of proof. "A convincing demonstration" is not proof. 'Evidence' is not proof.

Of course, the correct card to cut depends on a lot of factors. Nevertheless, the deck can be improved by cutting a card (the correct card).

Is this true beyond the shadow of a doubt, with no possibility now or in the future that 61 cards may perhaps be optimal? No. It's quite possible a mechanic can be printed that will change things. One has already been printed in the form of Battle of Wits.

However, 60 cards being optimal is a fact. A rule. Facts and rules have been known to change in time. However, a fact by definition is known through experience and observation, much like scientists collect facts through the same methods.

As such, I don't feel wrong calling it a fact or a rule.

Rule of thumb, however, is very rough and approximate, and I disagree with the color in that term because there is a lot more to support 60 cards than a rough unscientific method.

You still aren't catching some of the complexities of this issue, but whatever. That's fine; even if you understood the reality of this issue, it would not amount to any difference in how you play magic at this point in time. (Perhaps it would change how you study magic though)

I understand everything you are saying.

You claim that it cannot be proven that 60 cards is always optimal, but I am saying it is irrelevant when building a deck because we cannot ever prove that 61 cards is optimal either. It's just a silly argument.

We can, however, concede to the overwhelming evidence in favor of 60 cards and apply a little common sense.

4eak

12-14-2009, 10:56 PM

@ MattH

does the deck have to be good, or merely a legal decklist?

I'm assuming that if someone posts a vintage-legal Boros deck that you aren't planning on responding with, "Cut 61 cards, add a 60-card Tezzeret deck." Claiming that you can find a better decklist is not the same as claiming that you can find a better decklist which is a strict subset of the original. As quoted above, your challenge is the latter.

I think its quite possible to consider both pretty easily. And, for clarification, Rico's series of arguments could certainly include the former. One of his problems with my shock/mountain example was the poor quality of cards chosen for the deck in (what he presumed to be) a format with much better cards. The idea was "why consider shock.dec when we have Tarmogoyf?" and so on.

This quote strikes me as questioning what we mean by the word "optimal". I should have been more specific a while back about it.

Optimal = Highest chance to win given the format's cardpool and the targeted metagame.

You'll probably think this is a severe definition in some ways. Also, please note that this allows for possible "ties" between 2 or more decks sharing the top spot. I'll be brief in my discussion of the terms.

You can artificially limit cardpools. If you only have certain cards to work with, then that is a limit on your cardpool. If you are only interested in playing Landstill or Survival as your pet deck, then you'll have a limited cardpool. Of course, we must distinguish your cardpool and the cardpool of your metagame (as your opponent's may not have the same artificial limits). One might argue that the metagame shapes the cardpool, but at this point, I'm not going to explain what that isn't true enough to abolish the distinction between these two things.

There is the mathematically perfect metagame (which we'll never arrive at), the average world metagame (which our DTB tries to emulate, and what most people call "Legacy"), and what I usually call "specific" metagames which are basically local to any particular tournament ("specific" despite the fact that all of these are theoretically specifiable, not just the local tournament). Metagame information will also include odd things like tournament structure (they can vary), and perhaps other game mechanics. It is important to note that defining metagames is necessary to define deck optimality--there is no way to discuss a deck's optimality without reference to the other expected decks and players in whatever environment they are to be compared and tested.

You appear to dislike your former claim. I'm guessing it would amount to disliking some aspects of how one would define optimal. In that case, I'd initially want to ask then, if "optimal" is so restrictive in meaning, then why would you even want to consider developing tier 2 decks?

I see 3 reasons to develop the so called "tier 2" decks:

It is possible that the current "tier 1" isn't the optimal deck in a format, and perhaps a currently recognized "tier 2 deck" could be optimized into what is actually the optimal deck in a format/metagame.
A player has artificially imposed a cardpool limit (for whatever personal/financial/"for the challenge" reason you could think of)
What is considered a "tier 2" deck, which may in fact be suboptimal in the perfectly mathematical and the average world metagames, may actually be optimal in more specific metagames.

Optimality is not easy to calculate. Defining the cardpools and metagames are preconditions to testing for optimality, and that ain't easy. Even if you could create a simple vacuum (as I did in the very restrictive burn.format), you'll still see how difficult it can be to truly distinguish optimality.

I would like to untie two issues being discussed in this thread which some people have conflated.
Statement One: Every 61 card deck can be improved by cutting one card.
Question Two: Assume Statement One is true. Choose a card to cut. Can it be proved that the chosen card was the correct one to cut?

These are NOT the same question, and should not be treated as such.

Instead of just begging the question (http://en.wikipedia.org/wiki/Begging_the_question) in statement one, automagically limiting the scope of this discussion, let's restate that as an open question. "Can every 61+ card deck can be improved by cutting one card?"

I don't think this explication is true to the spirit of the discussion though. I think we want to know if any 61+ card decks can possibly be called optimal (in whatever cardpool and metagame).

Yes, some people have simplified the 60-card rule debate into "improvement through the cutting of one card". That is not, however, the overall point in question. The "cutting one card" concept is a gross oversimplification of the optimization process which would be used to move from 61 to 60 cards.

To give a brief example, a 61-card toolbox deck might run:

1x Circle of Protection: Red
1x Circle of Protection: Green

It might be the case that the correct optimization of the deck within the context of the cardpool and metagame would not amount to removing just 1 of these CoPs, but perhaps the the replacement of both of these cards with:

1x Story Circle

Even this is far from showcasing the complexity of optimizing. Clearly, we aren't 'just' talking about cutting one card though, even if that is how it was phrased. I think its difficult to talk about the issue (Ill's "handwaving" perhaps). This is, after all, a less-than-perfectly explored topic without a great deal of shared terminology. Surely you will admit that the larger question in the discussion, "Are all 61+ card decks supoptimal?", to be the heart of the matter.

Question one: "Are all 61+ card decks supoptimal?"

Before anyone can honestly say they know the question, before we even try to answer that question,, we need to ask whether or not it is physically possible for us to answer that question. The second question I've tried to answer then is:

"Can it be proven that all 61+ card decks are suboptimal?"

If the answer to this second question is "No", then we will need to answer question one with "I don't know". That's a lot of metagames and cardpools to consider.

Now, you remind us of what "proof" means. I'm willing to discuss epistemology with you, but I doubt you have the patience or interest in it. Let's be charitable. If we have very good reason to believe that the answer to the second question is "No", then likewise we have a very good reason to answer question one with "I know don't".

Because we lack the mathematics and computation power, we inevitably must answer question one with "I don't know".

We may even want to qualify that and say, "In the vast majority of cases, 61+ card decks are very likely suboptimal." However, we will still need to leave room for exception.

@ Rico Suave

You claim that it cannot be proven that 60 cards is always optimal, but I am saying it is irrelevant when building a deck because we cannot ever prove that 61 cards is optimal either. It's just a silly argument.

As I said, for someone who isn't concerned about magic theory or progress, this is good enough. This isn't a silly argument for anyone who might believe that progress is possible. One day enough progress could be made that this discussion will be relevant to how you build decks. The route I'm taking (computerizing the game) has a long-standing tradition in general game theory of "solving" games and optimizing strategy.

peace,
4eak

nodahero

12-14-2009, 11:34 PM

I will admit I have not read this thread in its entirety but I think there is a very obvious example to prove that 61 card decks can be better than a 60 card deck mathematicaly at least.

Let us assume for simplicity sake that magic does not limit us to 4 copies of a card. thus if your deck was 20 Mountains and 40 Lightning Bolts each card in your deck would deal on average 2 points per card ((40 bolts X 3 damage per bolt)/60 cards). Then lets say you add a 61 card thus increasing the average damage per card to 2.02, a small increase per card but an increase none the less. On the other hand one can argue to merely cut a mountain instead of going to 61 to make the average damage per card 2.05, better yet correct? Sort of.

In the above example the 41 bolt deck with 60 cards deals more damage per card on average but runs less mana then the 40 card bolt deck with a 61 card deck. The other aspect of this then becomes the change in the odds of drawing the needed land for your spells. With the 61 card, 20 mountain deck your odds are about 32.79% of drawing a land as opposed to 31.67%. While this change is not very noticeable on such a small scale imagine this on a larger scale to see the eventual magnitude because a 1.1% change does not SEEM large on paper.

While I think most people would cut the 20th Mountain that may not always be right. It seems much better in this example since Bolt only costs one but imagine using some really bad burn spell that costs like 3 mana... Eventually you will begin to see that there are so many variables to account for that such a determination of wether or not a 61st card is better than a simple 60 is nearly impossible. To help explain the 3 mana burn spell argument note that land 1,2,4,5,7, and 8 do nothing. The only lands that count are multiples of 3 so until a multiple of 3 land is in play those lands are dead draws. You would want to hit the exact right mix to maximize your odds of winning.

Illissius

12-14-2009, 11:39 PM

That's basically the example 4eak was using, I think, only he did it with Shocks.

Clearly the best way to optimize a 41 Shock, 20 Mountain deck is to replace them with 40 Lightning Bolt, 20 Mountain. QED.

4eak

12-14-2009, 11:51 PM

@ nodahero

I see where you are coming from. I'd love to see a purely mathematical demonstration of the value of lightning bolt in that deck. Unfortunately, the problem may be too complex for us to feasibly apply a purely mathematical treatment. Brute-forcing on a computer might be necessary.

As Nihil explained, variance from running more cards is a factor (which I mistakenly assumed to be somewhat insignificant until I ran the simulation).

Mulligan rules make it even more difficult to consider.

Conditions and choices in the game are pretty modular, and "averaging" is difficult. Each card in the deck isn't 1/3 part Mountain and 2/3 part Bolt, and that really makes the deck play out differently.

The best way I know to realistically understand the value of those cards, at this point, is to grind it out on a computer.

I will admit I have not read this thread in its entirety

You probably should read the thread then.

@ Illissius

I actually tested Lightning Bolts and 4 damage for 1cc (Uberbolts?) to see what would happen. I didn't post the results though.

It was useful, I just forgot to talk about it because I was busy frying other fish.

Having looked at the differences in Shock/Bolt/Uberbolt, for example, I know that as the damage of your 1cc burn card rises (moving from Shock to Bolt, etc), the more mountains you will want to run (which seems obvious in hindsight). It wouldn't be difficult to imagine a 30 Mountain, 30 1cc for 20 damage Burn Spells showing this to be true. That was interesting to me because I sometimes mistakenly think of "curving out" without regard to the value of the spells I'm casting (assuming the same mana costs).

There are implications to this. One might, for example, make a formula which analyzes the average CC of cards in the deck and try to create the appropriate mana-base for it. But, clearly, that wouldn't be effective. The values of land are in complete feedback with the values of the other cards in the deck; it was nice to see "proof" of that so clearly before my eyes.

In fact, this also led to a change in the mulligan-rules. And, even if I didn't adjust the mulligan rules, because I should/could run more land in higher damage burn decks, as the damage increased on my burn spells, and the numbers of mountains played increased, the less i needed to mulligan, which only made each card all the more valuable on average (because I was more likely to open with a keepable 5, 6 or 7 hand and win even earlier).

peace,
4eak

MattH

12-15-2009, 01:53 AM

Of course, the correct card to cut depends on a lot of factors. Nevertheless, the deck can be improved by cutting a card (the correct card).

Is this true beyond the shadow of a doubt, with no possibility now or in the future that 61 cards may perhaps be optimal? No. It's quite possible a mechanic can be printed that will change things. One has already been printed in the form of Battle of Wits.

However, 60 cards being optimal is a fact. A rule. Facts and rules have been known to change in time. However, a fact by definition is known through experience and observation, much like scientists collect facts through the same methods.

As such, I don't feel wrong calling it a fact or a rule.

Rule of thumb, however, is very rough and approximate, and I disagree with the color in that term because there is a lot more to support 60 cards than a rough unscientific method.

You made a specific challenge, one that I would like to answer. You said, "I challenge anybody in this thread to post a list that is optimal with 61 cards, and I will show you a way to improve it by cutting down to 60." Now, I love a good challenge. But my answer hinges on two things you've left vague: what you mean by "optimal" and what you mean by "show". I would like you to define these before I give my answer so there's no weaseling out of it later with, "But that's not what I meant!"

1. Does the deck need to be good, or just a legal decklist? As stated, your challenge says that you are simply going to make a single cut to a decklist I provide, but you include that fuzzy word, 'optimal', which looks like a secret escape pod for you. I don't want to come up with a 61-card Survival deck only to see you try to get out of it by saying, "Survival isn't a good deck anymore, therefore it isn't optimal. Cut the Survivals and utility guys for Brainstorms and Counterbalances and FoW and then it's optimal."

To avoid this, are you willing to claim that you can improve any 61-card list by finding a 60-card proper subset of that list? Your original boast sounds like this, but I want to be sure.

2. I also don't want to waste time coming up with a 61-list only to have your definition of "show" turn out to mean, "I will show you that 60 cards is better by cutting a card and then claiming that the deck is obviously better because it's now 60 cards." That would be circular reasoning.

However, you don't have to show that the card you're cutting is the best possible cut; it's certainly possible that there are several possible cuts to a 61-list, any of which would be an improvement! You only have to find one such, but you DO have to show, without a doubt, that your cut has made the deck stronger.

Rico Suave

12-15-2009, 02:33 AM

You made a specific challenge, one that I would like to answer. You said, "I challenge anybody in this thread to post a list that is optimal with 61 cards, and I will show you a way to improve it by cutting down to 60." Now, I love a good challenge. But my answer hinges on two things you've left vague: what you mean by "optimal" and what you mean by "show". I would like you to define these before I give my answer so there's no weaseling out of it later with, "But that's not what I meant!"

1. Does the deck need to be good, or just a legal decklist? As stated, your challenge says that you are simply going to make a single cut to a decklist I provide, but you include that fuzzy word, 'optimal', which looks like a secret escape pod for you. I don't want to come up with a 61-card Survival deck only to see you try to get out of it by saying, "Survival isn't a good deck anymore, therefore it isn't optimal. Cut the Survivals and utility guys for Brainstorms and Counterbalances and FoW and then it's optimal."

To avoid this, are you willing to claim that you can improve any 61-card list by finding a 60-card proper subset of that list? Your original boast sounds like this, but I want to be sure.

I'm not looking to tell people to play a different deck entirely. For the sake of this, let's just go with something that could reasonably win a match in a Legacy tournament. Survival has a legitimate chance to win the entire tournament, let alone a single match, and is actually a great example considering the original question was asking about 61 cards in a toolbox deck.

I used the word optimal, meaning that the person who presented it feels that there is no way to improve the list at 61 cards, or maybe they just aren't sure.

With the list, the following would help:
1) An expected metagame
2) A SB
3) Any concerns about mana, specific choices, or any other important information.

I will mention maybe 1-5 possibilities for a card that could be cut, why they can be safely cut, and how it impacts the rest of the deck and its strength in certain matches. If other issues come up, like mana issues, I may recommend 2-5 changes max.

2. I also don't want to waste time coming up with a 61-list only to have your definition of "show" turn out to mean, "I will show you that 60 cards is better by cutting a card and then claiming that the deck is obviously better because it's now 60 cards." That would be circular reasoning.

However, you don't have to show that the card you're cutting is the best possible cut; it's certainly possible that there are several possible cuts to a 61-list, any of which would be an improvement! You only have to find one such, but you DO have to show, without a doubt, that your cut has made the deck stronger.

I would certainly explain why. In fact it might become a very lengthy reasoning because it would involve how the deck would play out against several decks.

@4eak

As I said, for someone who isn't concerned about magic theory

Do not assume I am not concerned about magic theory.

You are discussing math and computer theory, which is different.

4eak

12-15-2009, 03:15 AM

@ Rico Suave

Do not assume I am not concerned about magic theory. You are discussing math and computer theory, which is different.

Perhaps you are concerned with magic theory to a small extent, but by and large, you have me convinced otherwise. Just because a theory doesn't have immediate impact on your choices doesn't make it silly (or pointless). You seem to think the argument I've provided you (which uses a computer and is theoretical) is not useful because it doesn't directly impact your deckbuilding at this very moment. As I've said, sometimes it takes a while for a theory to be cultivated and developed or for a breakthrough to occur which is applicable. If I thought you were more fully concerned about the topic, you wouldn't continue to hold your position.

It looks like you haven't quite received the message yet because you keep walking into the trap (for example, MattH is setting one up for you) in this thread and boasting about a capacity we know you don't have.

Lastly, you realize that mathematics and computers are languages and observation tools in which we can objectively consider this topic, right? They are quite appropriate to magic theory discussions.

peace,
4eak

Maveric78f

12-15-2009, 06:39 AM

About computer simulations, I built one to know which one between Ponder and Serum Visions was the best to have 2 lands by turn 2 and less than 4 lands by turn 10. I was considering that I was playing 4 braintorms, 8 fetches, 10 other lands (all fetchable). Even in this simple case, I had to cope with situations where it was not clear what the best strategy was (2 BS + Ponder in hand, what do you play first?, do you mulligan with a hand with ). My solution to this was first to ask these questions on forums, but the debates it created were definitely not productive. Then I planned to use reinforcement learning (which is somehow my PhD thesis subject), but, since ponder was obviously better, I've never done it.

I'd be very interested to deal with this kind of stuff later.

Ps: even the shock/mountain metagame is very interesting in the way that you can't ignore your right to mulligan and this mulligan decision makes this a reinforcement learning problem too.
PPs: Mulligan choices is probably the first topic on which simulation might help.

tivadar

12-15-2009, 08:29 AM

However, 60 cards being optimal is a fact. A rule. Facts and rules have been known to change in time. However, a fact by definition is known through experience and observation, much like scientists collect facts through the same methods.
...
You claim that it cannot be proven that 60 cards is always optimal, but I am saying it is irrelevant when building a deck because we cannot ever prove that 61 cards is optimal either. It's just a silly argument.

This is just... wow... 60 cards being optimal (all of the time) is not a fact/rule/proven. Something that is known through experience and observation (without the rigour of proof) is a theorem. That's all there is to it. Anybody who says otherwise is just plain wrong. Do I really need to explain this? In order to prove that 60 cards is optimal, you need to show it's optimal for all possible decks. Good luck with that...

Anyways, can we stop stating things as fact? This thread is all people giving there opinion, and that's it. I will agree that 60 cards is better than 61 cards in a majority of cases (though I won't dare to say I could prove that either).

P.S. As percentages show us, in order to show a difference in 0.5% probability, one would need to play a minimum of 200 games. Actually, it's more than this due to random fluctuation. In addition, since the chance of winning over any one deck varies, you'd have to play all these games against the same deck (to have a controlled experiment). Then you'd have to do that against every deck in the field.

P.P.S. No automated deck builder/game player will *ever* be able to prove 60 cards is better than 61, not even with any reasonable error margin. Do you know the compute time for this!? So let's not even suggest that.

Maveric78f

12-15-2009, 09:07 AM

P.P.S. No automated deck builder/game player will *ever* be able to prove 60 cards is better than 61, not even with any reasonable error margin. Do you know the compute time for this!? So let's not even suggest that.

Computer time? 10 minutes at most for the simple shock/mountain problem. Computer time is not the problem, developer's time or player/game models is the issue.

tivadar

12-15-2009, 09:11 AM

I more meant the compute time to test a deck against "every deck" in the field. Not to mention trying to determine what the best 60 vs. 61 card list is itself. The number of possibilities are endless. But you're correct, building an oracle would probably be even harder.

Humphrey

12-15-2009, 09:54 AM

I thought about that CoP example and it shows the problem with the value of a card. While in some matchups CoP:Red has a value of 100 (against burn) its useless against other decks (value 0 against Merfolk)

Therefore you should only pack cards in your deck which have the highest average value. Now you can come to the point that you should pack those cards as often as allowed in your deck (4ofs)
If you think this further, mathematically you shouldnt build a deck with cards you only want to have 1,2 or 3 times.

In the end you shouldnt play toolbox-decks at all, because statistically they are less consistent than piles of quadruples (aka Canadian ***** f.e.)

MattH

12-15-2009, 12:50 PM

For the sake of this, let's just go with something that could reasonably win a match in a Legacy tournament.

Ahh there you go, hedging your bet! (And still leaving yourself outs, with that fuzzy word, "reasonably".) That was not part of the original challenge. ;D

How about we start small. I claim that the following deck is optimal:

26 Swamp
35 Relentless Rats

and by "optimal" I mean that removing either a swamp or a rat will result in a worse deck. I claim that if you remove a rat, you will draw too many lands, and if you remove a land, you will not draw enough land. According to you, one of those two claims MUST be wrong, and the other right. Find out which one, and prove that you chose the right one.

This should be super-easy, right? There's only two possible choices, so you've got 50% odds if you just blindly guess! Tell me, which card should be cut from this? And remember, you do have to justify why the cut is better than not making the cut without resorting to "it must be better because now it's 60".

This is just the easiest example. I have others. P.S. This challenge is open to anyone who cares to try it!

Rico Suave

12-15-2009, 08:34 PM

@ Rico Suave

It looks like you haven't quite received the message yet because you keep walking into the trap (for example, MattH is setting one up for you) in this thread and boasting about a capacity we know you don't have.

I have been on friendly terms with Matt for years. He has seen every Legacy list I've played since the B/R list was revamped to be more than just a shadow of T1.

I would hardly call it a trap.

Ahh there you go, hedging your bet! (And still leaving yourself outs, with that fuzzy word, "reasonably".) That was not part of the original challenge. ;D

How about we start small. I claim that the following deck is optimal:

26 Swamp
35 Relentless Rats

and by "optimal" I mean that removing either a swamp or a rat will result in a worse deck. I claim that if you remove a rat, you will draw too many lands, and if you remove a land, you will not draw enough land. According to you, one of those two claims MUST be wrong, and the other right. Find out which one, and prove that you chose the right one.

This should be super-easy, right? There's only two possible choices, so you've got 50% odds if you just blindly guess! Tell me, which card should be cut from this? And remember, you do have to justify why the cut is better than not making the cut without resorting to "it must be better because now it's 60".

This is just the easiest example. I have others. P.S. This challenge is open to anyone who cares to try it!

I was hoping you would have gone through the efforts to make a Legacy Battle of Wits deck, as I didn't rule that out either. :D

Let's assume this deck could reasonably win a Legacy match-up.

The only change to be made is to change it from 61 cards to 300+ cards. Now your match-up against High Tide is nearly impossible to lose, and to boot you can achieve a more precise mana:spell ratio.

None of this takes away from the power of the deck or its strategy or how it plays out either.

For the sake of the argument, yes, Relentless Rats and Battle of Wits are 2 exceptions to the rule. This was also pretty evident in Pat Chapin's article too.

Sevryn

12-15-2009, 09:55 PM

Ahh there you go, hedging your bet! (And still leaving yourself outs, with that fuzzy word, "reasonably".) That was not part of the original challenge. ;D

How about we start small. I claim that the following deck is optimal:

26 Swamp
35 Relentless Rats

and by "optimal" I mean that removing either a swamp or a rat will result in a worse deck. I claim that if you remove a rat, you will draw too many lands, and if you remove a land, you will not draw enough land. According to you, one of those two claims MUST be wrong, and the other right. Find out which one, and prove that you chose the right one.

This should be super-easy, right? There's only two possible choices, so you've got 50% odds if you just blindly guess! Tell me, which card should be cut from this? And remember, you do have to justify why the cut is better than not making the cut without resorting to "it must be better because now it's 60".

This is just the easiest example. I have others. P.S. This challenge is open to anyone who cares to try it!

Remove both a swamp AND a rat, then add one street wraith. Or a cycling land.

That'd be choice C, I reckon.

4eak

12-15-2009, 11:09 PM

@ Humphrey

I thought about that CoP example and it shows the problem with the value of a card. While in some matchups CoP:Red has a value of 100 (against burn) its useless against other decks (value 0 against Merfolk)

First, having a dead card against one deck in a metagame does not make a card unworthy against the rest of the metagame, nor does it make it a suboptimal choice in the deck. It is quite possible that having that silverbullet is so valuable against certain matchups in the specific metagame that it is worth the loss in value in the other matchups where the silverbullet is not useful. For example, someone running a Tormod's Crypt in their Trinketmage-something.dec in a Loam and Dredge heavy metagame will find that crypt dead against Goblins, but overall might find the Crypt to be a valuable addition on average.

The point of the exercise wasn't to show the optimality of CoP though. In fact, I assumed that you would understand why Story Circle could often replace the CoPs. The idea is that optimization of 61-card decks is much more complex than simply "dropping a card" from the deck.

Therefore you should only pack cards in your deck which have the highest average value. Now you can come to the point that you should pack those cards as often as allowed in your deck (4ofs)
If you think this further, mathematically you shouldnt build a deck with cards you only want to have 1,2 or 3 times.

In the end you shouldnt play toolbox-decks at all, because statistically they are less consistent than piles of quadruples (aka Canadian ***** f.e.)

This is also a gross oversimplification. Not only is it not mathematially shown, but even intuition and study of current Legacy decks and metagames will disagree with you.

Your Tempo Thresh example itself runs singletons! Wipe away, Rushing River, Island, Forest, some sideboard singletons, etc.

You didn't think long enough about it. Tool-boxing isn't necessarily about finding control cards -- sometimes your silver-bullet is a raw win-condition (or whatever). Storm decks are a counterexample. Tutoring is toolboxing. I've Mystical'd for Wipe Away and Duress just as I've mystical'd for AdN, IGG, and Tendrils. You might argue that this isn't a toolbox because the deck doesn't revolve entirely around it, but that is a slippery slope. I can't name a toolbox deck that doesn't also play cards that don't revolve completely around the tutoring process.

Card quality through direct tutors or Brainstorm is really complex. Redundancy can be very important, but there is more to consider in this problem. Go ahead and explain some of the problems you have with running singletons and toolboxing, but perhaps you should reconsider your harsher claims against these strategies.

@ MattH

P.S. This challenge is open to anyone who cares to try it!

The challenge's context is that our cardpool is limited to swamps and relentless rats only, and our opponents' cardpools are the entire Legacy cardpool, and this is against the world metagame?

I don't need to show what is the optimal relentless rats build, just that cutting one certain card would make the deck more optimal, right?

What if I can't prove it? Would you take a good guess with some evidence? =)

@ Rico Suave

I have been on friendly terms with Matt for years.

Don't rule out the possibility that a friend is setting a trap for you.

Many people who have taken time out of their day to debate with you, sometimes even setting intellectual traps for you, might care for you or actually want you to know the truth.

peace,
4eak

Master Shake

12-15-2009, 11:58 PM

So... wait. Let me get this right:

Someone is trying to prove that a 61st card in a toolbox deck is sub-par by using an example of 35 Relentless Rats and 26 swamps?

Where is the toolbox in that?

The point seems to have been vastly overlooked. The question is, "Although it affects your chances of drawing a specific card by a fraction of a percentage point; is it worth it to include an extra card into your deck that wins the game in some match-ups knowing that you have at least 4 cards in your deck that can get it at any point?"

The question is not, "Can having an extra Relentless Rat in my deck help me against the field." No one is talking about adding a dead card to the deck, you're thinking about this in theoretical terms and not practical terms, no one is looking at adding a maindecked copy of Triviadar's Crusade to a deck filled with Worldly Tutor and Eladarmari's Call. People are asking if its a good idea to play something like Gaddock Teeg as a 61st card in Survival or Elf Survival because it shuts down a number of cards that they do not want to see from 40%ish of the decks out there game one.

Furthermore, when you use an organic shuffle you are never going to get a truly randomized deck. This is so important for this discussion because you're throwing around percentages that apply to a mathematically randomized deck. People have shuffling patterns and these patterns present themselves in unrandomized decks and patterns that can be observed. So the argument of "You have .7% less chance of drawing your enabler if you play a 61st card" is thrown out the window. This cannot even be applied to MWS or MTGO for reasons that have been stated and discussed at great length that I will not get into here.

As far as optimization is concerned, Toolbox decks basically can never be optimized. They have tentative and relative slots that reflect tastes anticipations and metagames, that is the entire advantage of playing a toolbox deck. The Chapin article which has been cited at least twice now; he issues no statement about a toolbox deck, which once again this is the topic of the thread. Part of the reason that toolbox strategies are not covered is because standard nor extended have had access to that kind of strategy on a tiered level in years.

There are lists of Landstill which I am convinced are optimized at 61 cards. Many Landstill players play 61 cards, and there is likely no thread in the Proven or DTB forums where people discuss card advantage and optimization more.

As far as the challenge that any deck can be optimized by removing one card of Rico Suave's choice: If this choice is made from a competitive list, that is taste not optimization; despite how much thought he puts into it.

Phoenix Ignition

12-16-2009, 12:21 AM

I don't disagree with almost all of your post, save this statement:

The question is, "Although it affects your chances of drawing a specific card by a fraction of a percentage point; is it worth it to include an extra card into your deck that wins the game in some match-ups knowing that you have at least 4 cards in your deck that can get it at any point?"

The double question here should be, "if there exists a card in your deck such that its specific addition increases your win percentages greatly against a few matchups and slightly decreases the win percentage against every other matchup, can we not find a card in the original 60 that could be taken out to strengthen the deck overall?"

There is a definite advantage, using your example, including Gaddock Teeg in a toolbox deck, but how many Gaddock Teeg-style silver bullets are you using? A 61st card may not completely throw off your draw percentages or land-spell ratio or any of the above, but having, say, 5 silver bullets in a deck compared to 7 can dramatically reduce your overall win percentage. Yes hitting the card may flat out win against certain decks, but having too many of those more or less dead cards in the main deck loses plenty of games.

I've been straddling that line with my Survival and Zur decks a lot recently, and you can definitely feel how clunky it can become.

Rico Suave

12-16-2009, 12:43 AM

The point seems to have been vastly overlooked. The question is, "Although it affects your chances of drawing a specific card by a fraction of a percentage point; is it worth it to include an extra card into your deck that wins the game in some match-ups knowing that you have at least 4 cards in your deck that can get it at any point?"

That is in fact not the question.

If the "61st" card you add is a dramatic improvement to the deck, then you need to investigate the other 60 cards and determine what the real 61st card is.

Nobody is saying that adding Teeg or some such thing is a bad idea. We're saying that there is something else in that deck which is a bad idea.

Master Shake

12-16-2009, 01:31 AM

I would now like to offer a proof that a 61st or 62nd card in a deck can mean its optimized using a truly absurd over simplification

First, we must establish the principle of what the best deck would be: Clearly all the best cards are banned or we are restricted from playing enough of them to make them truely powerful. If these rules were not in place The best deck would be some mix of these cards:

25 Simian Spirit Guide
35 Surging Flame

Now, the numbers can be discussed, but say that you were to walk into a tournament with this deck. it wins on your opponent's first upkeep every game, except if you are in the mirror, then whoever attempts to go off first will lose. And the fact that the combo requires 3 cards to go off ensures that a player could never go off three times in a single turn without 9 cards in hand.

Now, you could include a card such as Spellbook or Reliquary Tower to maintain a lage hand size but this card can only hurt your ability to combo and as such is awful in everything but the mirror, so you could simply run a 61st card. This will ensure that on the play or on the draw, you will always have more cards in your library than your opponent, thus giving you statistically a better chance of having lethal surging flames open to you.

So, while you may walk into the venue with the deck at an optimized 60 cards, you can [and probably will] lose to someone with a 70 card deck.

I don't know if I did a good job of explaining how this example works and this still has nothing to do with a tool-box approach, but it does prove that there are times where having only 60 cards in your deck can be a liability. If this example translates to real magic or not is not an assumption that I feel qualified to make. However, I will assert that it holds at least as much relevance as the Relentless Rats deck.

Phoenix Ignition

12-16-2009, 01:36 AM

4eak

12-16-2009, 01:48 AM

@ MattH

How about we start small. I claim that the following deck is optimal:

26 Swamp
35 Relentless Rats

We have very different definitions of optimal, and of course, I don't think I can prove optimality, but adjusting for those issues, and for the fun it, I'll assume the challenge to be this: provide evidence for why it would improve the deck to remove one card or the other. Again, this isn't a true optimality test in my eyes, and I think you've not given enough parameters (what is my metagame?) for anything of the sort, but it's fun anyways.

It didn't take much to modify the script (http://pastie.org/745253) for it. Summoning sickness accounted. Lands (if any) played first, Rats (if any) played, calculate damage of rats, swing with the rats that can, check for 20 damage, and start another turn off with a draw, etc. The most work was just thinking about a better set of mulligan rules. For each decklist, you need mulligan rules for a 7-card hand, rules for 6-card hands, 5-card hands, and so on. A fun problem to setup. (help Maveric78f!)

I set to mull when I didn't have 2 Land/1 Rat at 7 or 6-carders, mull when I didn't have 2 land at 5, and 1 land at 4, keeping all 3's. Very brief testing showed it to be decent, but I have no idea what the rules should look like. Again, this needs a lot of work.

This test, of course, doesn't take into account the metagame, my current opponent or his deck, any defense, control elements, blocking, hell...this is straight goldfishing. This is merely one lens (better than nothing) to consider a much more complex problem.

I'm on my wife's netbook here at work, so I'm only doing 100,000 hands per run, for the sake of time.

26 / 35
mull count --- 14150 (lazily includes multiple mulligans, such as 7 to 6 to 5, etc.)
The average turn you win --- 6.20037

25 / 35
mull count --- 15665
The average turn you win --- 6.21983

26 / 34
mull count --- 13168
The average turn you win --- 6.18495

So far, just in goldfishing damage (and perhaps even highest average toughness on the board per turn), cutting a rat might be the correct direction. While it wasn't part of the challenge, I thought it would be interesting test the goldfish kill rate by continuing to modify the ratio. Continuing the trend:

28 / 32
mull count --- 9135
The average turn you win --- 6.12848

30 / 30
mull count --- 6530
The average turn you win --- 6.09629

35 / 25
mull count --- 3823
The average turn you win --- 6.10909

40 / 20
mull count --- 5698
The average turn you win --- 6.28126

It could certainly be the case that the mull rules are incorrect, which would skew the results. Still interesting /shrug.

peace,
4eak

Master Shake

12-16-2009, 03:46 AM

Wow, why don't we use an even less helpful analogy and say "I have a 60 basic land deck, obviously if I was playing the mirror I would rather have 70 basic lands."

You didn't respond to anything either of us just said.

You're right, its was more like a part two of my original post, but people had responded so I think it warranted a new reply. And yes, my example does not apply to real magic in the exact same way that the Relentless rats example does not apply to real magic, not even in mathematical theory.

Maveric78f

12-16-2009, 04:47 AM

(help Maveric78f!)
claclaclap claclaclap hiiiiiii (\me on my white horse)

In the following, I consider you've fixed your swamp/rat deck (S swamps, R rats are constants).

In order to compute the optimal mulligan policy, you need to first define an objective function for each game you play. You partially do it when you say that your objective is to goldfish the fastest possible. As you computed your evaluation of each rat/land ratio, you basically define that killing in 5 turns (just noticed it's not possible, but anyway, you got my point) then killing in 8 turns is as good as killing in 6 turns then killing in 7 turns. But in a mirror where rats can't block (to simplify the problem), it's probably not that simple. You can't basically average those performance, because even if the objective function is only function of the kill turn, it is not necessarilly linear. Anyway, let's assume it's linear for the sake of simplicity and we define it as follows:
O(t) = 100-t
Where O(t) is the objective function (the bigger it is the better the objective has been reached) and t is the number of turns to kill.

Each game you play from a starting hand with X swamps and Y rats (we'll call it X-Y later) will provide you with a randomised reward r calculated thanks to the objective function.

For every X-Y hand, you need to compute as accurately as possible the average of rewards you obtain on large numbers of runs, in order to compute the reward expectation for a X-Y hand.

Once you've got these numbers with your simulations, I should be able to compute the mulligan policy and thus the overall reward expectation for a game for the swamp/rat split you've considered.

Then we can do it again for another swamp/rat deck.

Ps: I'm quite sure your mull rules are incorrect.

Maveric78f

12-16-2009, 09:47 AM

I did some home work to day. I made myself a simulator for the mountain/lightning bolt (or shocks) problem and I computed the optimal mulligan strategies.

I used 10^6 runs for each X-Y couple, resulting into a precision approximately of 1/1000 turn, which is more than enough. The numbers are given with more precision because I did not want to hand-edit the automatically generated numbers.

I'll try to give the results here, but they may not appear well. Here we see the detailed results of 15 mountains, 60 lightning bolts deck on the draw. (ahah I actually bugged this one I played with 15 mountains and 60 lightning bolts instead of 15+45)

.......................0 Land..1 Land..2 Lands..3 Lands..4 Lands..5 Lands..6 Lands..7 Lands
0 cards in hand: 10.97933
1 cards in hand: 10.32619 8.6062
2 cards in hand: 9.85735 7.66289 8.50415
3 cards in hand: 9.44258 6.78216 7.3104 8.38371
4 cards in hand: 9.07986 6.03483 6.09754 7.20203 8.25967
5 cards in hand: 9.03304 5.50793 4.89888 6.02701 7.11136 8.15386
6 cards in hand: 8.82419 5.26999 4.20487 4.81369 5.92343 6.99877 8.02635
7 cards in hand: 8.78163 5.23695 4.02711 4.18834 4.75521 5.83343 6.88926 7.91529

The mulligan policy is as follows with the expected reward on the right
KEEP 10.97933
KEEP KEEP 9.982192
KEEP KEEP KEEP 9.09443
MULL KEEP KEEP KEEP 8.19047
MULL KEEP KEEP KEEP MULL 7.29479
MULL KEEP KEEP KEEP KEEP MULL 6.52890
MULL KEEP KEEP KEEP KEEP MULL MULL 5.85700
MULL KEEP KEEP KEEP KEEP KEEP MULL MULL 5.45042

You can see that with 5 lands and 2 Shocks you'd better to mulligan because the expected kill turn 5.83343 is inferior to the mull to 6 expected kill turn 5.85700.

The expected kill turn of this deck is 5.45042, even if with a god hand of 5 bolts + 2 mountains you can reasonably guarantee you'll kill on turn 4 after playing 7 bolts (you just have to draw 2 bolts in your 4 first turns).

Now I'm going to make the mountain/bolts proportions vary and give the results as soon as I get them.

Maveric78f

12-16-2009, 10:03 AM

In the mountain/bolt 60 cards deck:
nbLands = 10.......4.918240947564987
nbLands = 11.......4.771720262107798
nbLands = 12.......4.670287238892705
nbLands = 13.......4.602975338599499
nbLands = 14.......4.565034868705364
nbLands = 15.......4.542160885292292
nbLands = 16.......4.516900709495308
nbLands = 17.......4.502407004477004 (3.10^6 runs)
nbLands = 17.......4.509242603102798 (10^7 runs, it's really imprecise...)
nbLands = 18.......4.526854826727408
nbLands = 19.......4.560510644890347
nbLands = 20.......4.611037856304454

In the mountain/bolt 61 cards deck:
nbLands = 10.......4.945788544110351
nbLands = 11.......4.7972723032045215
nbLands = 12.......4.69113490435934
nbLands = 13.......4.617977151416818
nbLands = 14.......4.574533102606943
nbLands = 15.......4.544577807688063
nbLands = 16.......4.521217195170542
nbLands = 17.......4.508391037357003 (3.10^6 runs)
nbLands = 17.......4.511090603710212 (10^7 runs, it's really imprecise...)
nbLands = 18.......4.519467472942914
nbLands = 19.......4.5474408641115165
nbLands = 20.......4.591792490981584

I published these results too fast (these numbers were done without enough data). I still don't know if they are enough. Please wait for a comment.

Running @10^7 runs right now. I'm going to make it more (10^8 for the 15-19 range) during the night.

I'll also make the same calculus with shock instead of bolt (just have to change 3 by 2 in my code ;-)).

tivadar

12-16-2009, 10:15 AM

@Maveric78f: First off, this is the right kind of analysis. It's perfectly valid and completely correct. It's nice to see this rigour with posts.

However, what's potentially incorrect with this and a lot of other posts are their base assumptions. This is similar to deriving a proof that 1 = 0 by first assuming that 1 + 1 = 3. The proof is completely correct, your initial assumptions are wrong.

Some things we can't assume:
* A deck with 40 of one card and 20 of another is a valid deck (it's not)
* One deck is always strictly better than another deck independent of the meta
* One card is always strictly better than another independent of other cards that are in your hand (it probably isn't)
* There is always a single worst card that you can swap out of a deck, irrelevant of what you're adding in (similar to above)

Anyways, a lot of you offer "proofs" that you're correct. And while a lot of the math/logic works out, your underlying assumptions just don't seem as strong. I'm all for statistical analysis and examination of this, but in the end, it's going to come down to how people view the game.

P.S. I now want to play RelentlessRats.dec

4eak

12-16-2009, 10:27 AM

A thing of beauty! (Sourcespeak: **Masturbates furiously**, right?)

Your precision is much better. I assume you're working in a much faster language (optimized code perhaps) or with a better computer. 8(10^6) takes a while in Python. As you said though, we can still get a good idea of the answer with even fewer hands played, but the precision is very nice.

As expected, this has to be done on a per decklist basis. It obviously needs to be the frontend to this testing procedure. Only after mull-rules are calculated (removing the guesswork and margin of error in our final outcome) can we run the real test.

This would become absurdly difficult in normal decks (which play more than 2 different types of cards). There are a ton of combinations. We might need to lower the precision in more complex decks.

Imagine how the rest of your choice-engine influences the mulligan decisions test. In burn, obviously it doesn't. Play a land and burn (well, sorta, in a metagame you actually play instants in response to other things etc.) in a vacuum. However, take a deck which is only slightly more complicated, yet still quite linear in many ways, like Belcher, and those mulligan tests start exploding not just on the number of possible hands, but also on the number of ways in which one could play each hand. Sure, you can hardcode a lot of Belcher's obvious choices (and even from intuition and experience give it even more hardcoded choices), but wouldn't you want to know the exceptions (the perfectly optimal choices) to our ghetto choice-engine?

As a sidenote, this seems like a interesting place to consider the overall value of making mulligan decisions, although we'd need more complex decks. I've always wondered how much proper mulliganing adds to your chance to win. Even the difference between just newbie type mulls (no lander mulls, etc) and optimized mull strategies. Obviously, it must be answered on a per deck basis, but a few decks might be a good glimpse. Just from our current example, optimal mulling rules, beyond the no-lander and keep at 3 (which were intuitively obvious), account for ~.4 turns on average (I didn't wait for your comment while I'm saying this, and additional precision will not be necessary to show that this amount is relevant). That is a huge amount!

And, as a second sidenote, I wish there was a generic testing engine whereby I could just input the cards (their rules) manually (although parsing the Oracle might do it as well), set parameters for generating the decklist combinations, maybe add choice-engine rules, setup opponent's, etc. and then let it grind it out. You wouldn't need to write the program over for each deck then. Admittedly, it is too large a project for a noob like me to do.

peace,
4eak

Maveric78f

12-16-2009, 10:36 AM

My precision is false. I thought it was 1/1000 but it looks much worse. Finally, that's why I said to wait for me to conclude. (I use java, so not THAT fast even if it's better than it used to be)

Tivadar> Thx for all. About the limitations, I'm aware of them. I don't know if this work is going to be any use for real magic. But I think it's a valid demonstration to prove that it is possible that a deck with 61 cards is better than one with 60 cards (if my numbers prove it by the way, which I'm not sure anymore, I'm waiting for results in bigger number). It's impossible to say which one and in which metagame.

However, as it's impossible to say in real magic if a 61 cards deck is better than a 60 cards deck, I'd still recommend to shrink your deck until 60, the main reason not being Rico's (you want to get your 4-ofs as much as possible) but Nihil's (Variance goes up with deck size).

4eak

12-16-2009, 10:57 AM

However, as it's impossible to say in real magic if a 61 cards deck is better than a 60 cards deck, I'd still recommend to shrink your deck until 60, the main reason not being Rico's (you want to get your 4-ofs as much as possible) but Nihil's (Variance goes up with deck size).

Nihil's variance is the same problem I've been seeing in duotonous lists. At best, 61-card burn lists appear to only come close to equal to the optimal 60-card lists. I wasn't going to take all night like you are to get more precision though. I'm not surprised a burn list with only 2 different cards might arrive at that, I pretty much stopped looking for an exception (although, it remains a possibility, however insignificant the difference).

Rico's reason has no application in this context. It will matter when we pit Tarmogoyf up against Nimble Mongoose though, etc. And, I think Rico's reasons vastly outweighs the Nihil's variance problem for the majority of decks.

Tool-boxes and more complex decks, particularly within the context of a specific metagame, seem to be the best place for an exception. Nihil's Variance may matter less in decks which have good ways to generate card quality (tutoring, brainstorm, etc.), to the point that the metagame-based value of a 61st card silverbullet might be worth the cost.

peace,
4eak

Maveric78f

12-16-2009, 11:18 AM

Even if Rico's argument is the most widespread in magic world, I don't think it stands in every legacy deck. I can imagine a deck without any 4-of (theoretically imagine, I would not reject a deck because it has no 4-of). I even think that several legacy decks would not want to play a 5th copy of any card they play even if they had the opportunity. Nihil's argument though is unavoidable. And if it works in the moutain/bolt metagame, it may even be the proof that 60-cards decks are always optimal. Toolboxing or cantripping does not change anything IMO. It's only an intuition. I have no proof of this... yet ;-)

4eak

12-16-2009, 11:32 AM

My guess is that the variance problem seems to have the most effect in decks which only 'see' a small portion of all their cards per game. I would think that decks which are more capable of 'seeing' their entire library (Dredging, Tutoring, Chain Ponder/Brainstorm/SDT) will find variance less problematic. The more library you 'see' per game, the less variance there might be.

Consider that Nihil's variance illustration is dealing with moving from a 7-card library to a larger one. This variance isn't just the opening hand (obviously), it really entails all the cards which you see during the game. Burn decks might only see 15-cards, and so that variance is going to be important. A control deck might see 30 cards in the deck, which might make variance less important.

Nihi's variance might impact the end 'wincon' measurement very differently among decks which do and don't have a good deal of card quality/library manipulation. The "variance" cost, in terms of that 'wincon' measurement, in a tool-box deck, or any deck with high card quality generation, might be lower than what we see in the very simple burn deck (in this deck, the variance cost negates the gains of the 'perfect ratio' quite effectively).

Hehe, regardless this is a very tall order. Any ideas how you will measure the values of a tool-box deck?

peace,
4eak

Kuma

12-16-2009, 03:10 PM

That's not the point. The problem isn't that it's only a theory, the problem is that the theory isn't powerful enough.

O RLY?

Indeed. However, the burden of proof is also on those who insist that 60 cards is optimal in 100.000000% of cases with no possible exceptions whatsoever.

Come on.

There are probably more possible 60-card Magic decks than there are atoms in the universe. You said that Magic is an artificial/logical construct, and therefore we should be able to derive proofs for various claims about the game. However, there is much more to Magic than math and logic. Magic is played with two or more people, which means we cannot derive optimization purely from math, because math cannot take the complex actions of another individual into account. Logic is also insufficient, because there is hidden information which muddles any logical claim. One play may be logically correct if card X is in a players hand, but may be logically wrong if card Y is in that players hand.

tl;dr --- Magic is more than math and logic and so you will never be able to write formal proofs deducing anything about the game.

On the other hand, all you have to do to disprove the statement that 60 cards is always more optimal than 61 is to provide one counterexample. This is a much easier thing to accomplish.

The closest we can come to "proof" for 60 cards being optimal is Hypergeometric probabilities of drawing certain cards that are generally held to be better than others. It's not absolute, but it is very strong evidence.

So someone in the 61 card camp needs to find an exception to this rule, i.e. the burden of proof is on them.

All I'm really for, I guess, is for people to have a greater awareness of how things work rather than just relying blindly on dogma.

Statistics isn't dogma.

tivadar

12-16-2009, 03:59 PM

Magic is played with two or more people, which means we cannot derive optimization purely from math, because math cannot take the complex actions of another individual into account. Logic is also insufficient, because there is hidden information which muddles any logical claim. One play may be logically correct if card X is in a players hand, but may be logically wrong if card Y is in that players hand.

Actually this isn't true. One can use minimax game theory in a game with two players. And in fact there is an extension of this theory to games with incomplete information. Obviously with incomplete information, you can only predict what the best move is in most situations.

On the other hand, all you have to do to disprove the statement that 60 cards is always more optimal than 61 is to provide one counterexample. This is a much easier thing to accomplish.

Perhaps "much easier", but is it easy? You're insinuating that showing one deck is better than another is an easy task. It's not. This is evidenced by the fact that we keep going back and forth on the forums trying to decide if merfolk is better than goblins is better than zoo...

All in all, I don't get your point. You argue that Magic isn't defined by math and logic, yet later you say that someone uses a hypergeometric series to predict the probability of drawing a card is higher with 60 vs 61 cards, and that's close to a proof? And how do you intend someone in the 61 card camp to find an exception to that when what they're arguing is that the increased chance of drawing a particular card is not the same as the increased chance of winning a game.

Rico Suave

12-16-2009, 10:05 PM

Carabas

12-17-2009, 01:19 AM

If I have a 61-card deck, all 1-ofs, and each card is an answer to a unique situation, there is one more situation that I can answer than a 60-card deck of the same nature.

Doesn't this mean that for the one situation my deck can answer and your deck cannot, my deck would have a better chance at drawing a card that matters?

MogulKahn

12-17-2009, 02:15 AM

@Carabas

But for the other 60 situations, your 61 card deck is inferior because there is a less chance of drawing what you need.

Phoenix Ignition

12-17-2009, 02:17 AM

@Carabas

But for the other 60 situations, your 61 card deck is inferior because there is a less chance of drawing what you need.

This is a good point, and you can most likely take out 1 of those cards based on knowing that that deck won't show up in your metagame.

4eak

12-17-2009, 02:17 AM

@ Rico Suave

"Playing a 61st card means you are less likely to draw what matters."
That is variance by definition.

Perhaps you've mistaken Nihil's point (read page one again please) and the definition of variance (http://en.wikipedia.org/wiki/Variance). I doubt you understand why Maverick said what he did. Your self-quote is also a very slippery statement. I disagree with what Maverick thinks, but at least I gave some reasons that were pertinent to his argument.

Your argument boils down to the assumptions that there are always 'best' cards in a deck, always 'weakest' cards in the deck, that perfect balance between the values of cards and functions is somehow irrelevant, and that by adding cards past the 60 minimum, you are directly lowering the average card value of the deck.

Here's an exaggerated and simplified example of your argument:

20x Land
4x 10/10 Creatures for 2cc
36x 2/2 Creatures for 2cc

Presume the mana-curve is correct (modify the curve if youwant, but it will still arrive at the same conclusion); just in a vacuum, and not considering a metagame, we are willing to admit that it wouldn't make the deck more optimal to add the 37th 2/2 creature. Even if you added creatures to have the same land ratio, it still would be a mistake. You really want to see the 10/10 creatures as often as possible, and adding the 61st card lowers your chances.

Your argument works in simple situations. Your argument works when you make huge assumptions about card values, some unimportance of balancing functions, and metagames. We've tried to explain that magic becomes much more complicated, and that as you add and balance different functions to a deck (beyond just vanilla beaters), and as you take into consideration the metagame, and as decks grow in complexity, that your argument may hold less weight.

We agree this issue is important to the majority of decks, but 'how important' is also difficult to measure. There isn't a formal mathematical treatment of the issue (and if you think there is, you really have not understood what was said in this thread). This problem is beyond our current testing capacity as well. We don't know either way on this issue. As I said, we can't make a definitive judgement. We aren't going to make all of your assumptions, and we have to admit there could be exceptions.

Please notice that Nihil's variance (which is much different from your argument) is more supportable, and easier to test (even if only by a slight margin), and that we don't need to make your awkward assumptions. While it is beyond our means to evaluate the validity of your argument at this time, it may not be beyond our means to evaluate Nihil's variance. It might also be easier to formally prove that its cost outweighs any benefits from acquiring perfect ratios.

Maverick might think he doesn't need to make your assumptions to show that 61+ cards is suboptimal. He thinks Nihil's variance, which I believe (as you will when you recognize Nihil's point) might be a weaker effect in the end, is enough to prove that 61+ cards is suboptimal. You of all people in this thread should want Maverick to succeed, it would remove the need to make your assumptions, and let us immediately accept your conclusion.

Maverick isn't necessarily unable "to see the forest for the trees". I disagree with Maverick's hypothesis, and I think that Nihil's variance won't have nearly as much impact as your argument in the majority of decks. However, I agree that testing Nihil's variance is the next step, and that we need to first prove (or provide ample evidence) as to whether or not this variance provides too much cost for any benefit that a 'perfect ratio' might bring. You inevitably will assume that the 'perfect ratio' concept is useless, so you've already begged this question. Of course you wouldn't think it is very important to test Nihil's variance, but that's because you've already made your above assumptions.

@ Carabas

If I have a 61-card deck, all 1-ofs, and each card is an answer to a unique situation, there is one more situation that I can answer than a 60-card deck of the same nature.

Admittedly, this is too simple. It is still interesting, and may still bear fruit. Here's how I think of this thought experiment:

You have 62 people (you included) in an infinitely long tournament, all with unique decks, and everyone shares an unlimited cardpool which contains these cards:

"0cc, instant, A deck can have any number of cards with this name, Split second, cannot be countered, you win against opponent X."

Basically, there is an unstoppable silverbullet for everyone in the room (62 total silverbullets, including one with your name on it). You can play as many as you want, but you have to play a 60-card minimum deck, and magic as usual, etc.

What is the optimal deck?

peace,
4eak

MogulKahn

12-17-2009, 02:51 AM

For your silver bullet experiment, if we were to calculate the chance of winning in the opening hand, assuming you put only 1 of each card to avoid the inefficiency in drawing multiples, would be like this:

60 cards: Expected number of wins in opening 7 = (59 C 6)/(60 C 7) x 60/61 = 7/61 [the 60/61 is to account for the fact that you only have 60 silver bullets]

61 cards: E(win in opening 7) = (60 C 6)/(61 C 7) = 7/61

They are the same? or did I mess up?

P.S. This 60 vs 61 card debate seems to hinge on whether all 60 cards are of equal value or not, because different valued cards would obviously make it easier to cut down to 60. I am not sure whether using cards of identical value would scale at all to solving that problem.

Maveric78f

12-17-2009, 03:02 AM

Rico > you're talking about expectation value of getting the "best" card. If I agree that most decks have their best cards, I disagree in general. Nihil is talking about Variance between cards you draw. Raising the number of cards in your deck (for instance by simply doubling your deck cards) make your draws less constrained by the size of your deck and for a same ratio (same expectation value for a single) you'll have more chance to get 0-land or 7-lands hands.

4eak > your last problem is easy to solve. Having a 61-cards deck with every opponent's win card and having a 60-cards deck with every opponent's win card but 1 at random will behave absolutely the same way. You just have to consider that in the 61-cards deck, the last card of your library is the one you did not take in the 60-cards deck. There is a single case where both decks are not equivalent: when you and your opponent have the relevant card placed 61st (or outside of the 60 cards deck). In this case, the 61-cards deck wins. So it's optimal to have a 61-cards deck in this example. It is a good example to understand Nihil's effect too. With a 61-cards deck with only 1-ofs, you'll have 7/61 to kill on first turn. With a 122-cards decks with only 2-ofs, you might have some doublons in your starting hand, making your deck less efficient.

About the tests I've run this night (10^8 runs), it's very close and I'm still not sure at all the precision level is enough. Anyway I give them:

LIGHTNING BOLT problem
nbCards = 60
nbLands = 15.......4.537758419392885
nbLands = 16.......4.515552156218359
nbLands = 17.......4.510812280148946
nbLands = 18.......4.5249891224780425
nbLands = 19.......4.558726831166056
nbCards = 61
nbLands = 15.......4.54445343489939
nbLands = 16.......4.518167184153388
nbLands = 17.......4.510292013557176
nbLands = 18.......4.520272856053504

It did not finish yet. I'll run other threads in parallel (I'm on a multithread machine now).

Edit: Updated with 61/18 and I've launched the threads. Cya in a couple of days with mountain/bolt and mountain/shock data. Conclusion will probably be that 61/17 and 60/17 are too close to decide. Then I'll probably try a bigger order during my hollidays just for these 2 (10^9 or 10^10).

Maveric78f

12-17-2009, 03:04 AM

They are the same? or did I mess up?
You did not mess up, but they are not exactly the same, given that it's better to draw a 61st card to win than to deck yourself.

Edit: another interesting problem.

Card1 %0
sorcery
Card1 cannot be countered.
As an additional cost, discard card2 from your hand, search for card 3 in your hand and in your deck. Put it into play.
A deck can have any number of cards named Card1.

Card2 %100000
sorcery
A deck can have any number of cards named Card2.

Card3 %100000
Artifact
When Card3 enters the battlefield, you win the game. This ability cannot be countered.
I'm quite sure that the deck made of 30 card1, 30 card2, 1 card3 is optimal.

Cire

12-17-2009, 03:16 AM

LIGHTNING BOLT problem
nbCards = 60
nbLands = 15.......4.537758419392885
nbLands = 16.......4.515552156218359
nbLands = 17.......4.510812280148946
nbLands = 18.......4.5249891224780425
nbLands = 19.......4.558726831166056
nbCards = 61
nbLands = 15.......4.54445343489939
nbLands = 16.......4.518167184153388
nbLands = 17.......4.510292013557176

It did not finish yet. I'll run other threads in parallel (I'm on a multithread machine now).

so a 17 land 44 lightning bolt deck has a better kill ratio than a 17 land 43 lightning bolt deck by a marginal .00052026659177 of a turn? that at least shows that in some circumstances, given a very simplistic deck, with no forethought to the metagame, a 61 card deck may be marginally optimal above a 60 card deck ??

Edit:

what happens if you go up to 62+ card?
Edit:

also for the relentless rat example, for the fastest goldfish (turn 6) you only need 3 swamps and 3 relentless rats in this order
1 Swamp
2 Swamp
3 Swamp - Rat
4 Rat (attack for 3)
5 Rat (attack for 8)
6 (attack for 9)
thus it's clear you need much more lands than rats since you need to lay one down for the first 3 turns. Thats just to help if your going to try to find the correct ratio for both 60 cards or 61 cards. i would guess for the 60 card ratio for it too actually be around 33 swamps/ 27 relentless rats... (just a guess lol)

Maveric78f

12-17-2009, 03:34 AM

As I said, it still shows nothing because the difference of what I measured is not relevant. All we can say for the moment is that they are very close even after testing them more than 3 billion times (after testing them 100 million times with every possible starting hand).

Tangle.Wire

12-17-2009, 03:55 AM

I'am actually playing a 65 cards deck (even for tournaments not casual) and without flaming i am able to say everyone who said running more than 60 cards is crap as you don't draw efficient spells is nearly total wrong.

-> You can break your head on maths getting out by how many % you would draw the card (maybe the survival)

But normally every card we add +60 will be stuff that fits into the Decklist, also by playing fetchlands,tutors, additional draw (like ponder,brainstorm,top,standstill) every current legacy deck will be able to find the cards they need.

-> As for the shuffle effect in real life i figured out even here getting any % how often we could draw card XY is senseless.

The really reason i actually choose between 61 or more cards is the changing of the manacurve as adding 5 cards on a basic list won't work as all of those cards will make the complete mana count less.

Its still possible to get +1-+2 cards as their costs are between 0-2 but back to toppic for toolbox decks i wouldn't even think about to play 61, you will never have a situation where you will say "oh damn i am sure if i'd played 60 cards i would have drawn better", by adding more cards its a question of the Decklist and the personal feeling about the consistance.

Also i have always been someone who likes to play Decklists which not totally clone any common tournament list as good players will know which threats we play and on the worst case they even will know the count of the cards.

practical joke

12-17-2009, 04:27 AM

For Toolbox decks you need to make the cut to 60 or 61 in exceptional cases to get it running. If you run 65, you will have a bigger chance to find the answers to the wrong cards because you will have some trouble finding your toolbox starter. For example the chances are less likely for a tezzerator deck to find it's tezzy or trinket mage when they play 65 cards, since they cannot up those slots ( else the deck screws itself). So why play more than needed.
In a 65 cardlist there are always cards that can be lowered, even with ponders and brainstorms, tops and fetches. ( since if you have to use all of that in addition to the 60+ cards decks to run) you will slow yourself down.

I think 60 card is optimal with a 61 card being acceptable. since a 65 card deck is asking for a total screwage once in a while.

Tangle.wire have you had any awesome results with your 65 card deck?
How often does 65cards screw you over in a tournament? ( this should not be considered a flame, but a question)

which cards do you consider to be 61-65 do they have a consequent and significant effect on your deck ( e.a. post a detailed tournament report or something? it might help to explain why a 65 card deck could be acceptable or not).

Rico Suave

12-17-2009, 06:06 AM

Maveric78f

12-17-2009, 06:15 AM

If you can't know after more than a billion runs, then you won't know after your small personal experience.

Saying a deck must be 60-cards to be optimal is not even a theory, it's just a good-use advice.
But saying that a 61-cards deck from which you don't know what to cut would be better as 60-cards when you cut one card is utterly wrong. The main reason is that you don't/can't know what to cut. And sometimes, in doubt, you'd better not cut anything that taking the risk of cutting an important card.

The fact is that we are not playing with optimal decks. Get over it.

kinda

12-17-2009, 07:02 AM

Well, even if you don't know what card to cut in a deck, you will certainly be able to discern cards you do not want cut. And not cutting one of the more questionable choices in your deck decreases the chance of seeing the cards you most certainly do want to see. I do see maveric's point that cutting the wrong silver bullet could be worse then decreasing the chances of seeing a survival, but I believe this argument to be flawed...

Say you were told to build a survival deck with a minimum of 40 cards. Would you have 40 cards in the deck? 41? 60?80? The point being is that 60 cards in a deck is an arbitrary number, and if we were allowed to build smaller decks then they would be much more consistent and powerful. The reason one extra card over the minimum is in contention (whether it be 41 or 61), is due to the diminishing marginal benefit of each card added to your deck. While people are very happy about adding four tarmogoyfs to a deck...whether to run daze or smother is a much grayer area. However I believe everyone would agree that regardless of whether the benefit of daze or smother is actually greater...neither are constantly as beneficial as tarmogoyf and should thus not get in the way of your drawing one.

Forbiddian

12-17-2009, 07:46 AM

Hmm, I made a post somewhere at length about this like a year ago when this was discussed in another 20,000 page thread. I talked at length about this principle:

When you run more cards, there's less regression to the correct land ratio.

By running 31 lands, 31 non-lands (instead of 30/30 split), you take 0.5% more no-land mulligans, and a comparable increase in 7 land hands (or 1 or 6 land hands, etc.). You're less likely to draw 3 and 4 land hands (which, in theory, is what you'd want if you were really running 50% lands).

Just by running more cards in the deck, you're reducing your chance of drawing the correct land distribution.

I used this to argue against someone stating that it might be that 21 land/40 spells is better than 20 land/40 spells and 21 land/39 spells. I pointed out that this was extraordinarily unlikely because there's less regression on a 61 card deck, and there are almost no precise value points near the center of the distribution that are more probable with a 61 card deck.

It's still possible, but any calculation attempting to prove that 61 cards is superior has to deal with the problem that it won't regress as quickly.

Maveric78f

12-17-2009, 08:03 AM

The problem you raise Forbiddian is exactly the one I deal with my simulation. It looks like 17 lands + 43 bolts is as efficient as 17 lands + 44 bolts. But I'm continuing my simulations to find a sensible domination between those decks.

If you cope with strange decks such as the 2 cards combo + 1 silver bullet I proposed in this post (http://mtgthesource.com/forums/showpost.php?p=410500&postcount=108) or the perfect metagamed deck proposed by 4eak at the end of this post (http://mtgthesource.com/forums/showpost.php?p=410493&postcount=105), it's clear that a 61-cards deck is optimal.

Real magic with the 4-of limitations, the metagame MD slots, and the land ratio lies probably between all these extreme cases and we all agree that there is probably the requirement for every deck to be 60-cards if you want it optimal. But this can't be proved nor calculated.

Kuma

12-17-2009, 02:57 PM

Actually this isn't true. One can use minimax game theory in a game with two players. And in fact there is an extension of this theory to games with incomplete information. Obviously with incomplete information, you can only predict what the best move is in most situations.

I agree 100%.

I wasn't trying to say that logic plays no role in the game or that there isn't a correct play for every situation. My point is that you won't be able to write a logical proof, i.e P=Q, Modus Tollens, etc., involving Magic because the game isn't that simple.

Perhaps "much easier", but is it easy? You're insinuating that showing one deck is better than another is an easy task. It's not.

I didn't say it was easy. I said it was easier in that it is possible, whereas going through an astronomical number of decks and proving that they are more optimal than another even larger number of decks is impossible.

Proving that 60 cards is optimal in every case is the same as proving that 61 or more cards is never optimal. And proving a negative is impossible.

All in all, I don't get your point. You argue that Magic isn't defined by math and logic, yet later you say that someone uses a hypergeometric series to predict the probability of drawing a card is higher with 60 vs 61 cards, and that's close to a proof?

You didn't read my post carefully.

I didn't say Magic isn't defined by math and logic. I said it cannot be totally defined by math and logic, i.e. with a proof. Obviously math and logic are a big part of the game.

I didn't say Hypergeometric probabilities were close to a proof. I said it's the closest thing we have to a proof.

And how do you intend someone in the 61 card camp to find an exception to that when what they're arguing is that the increased chance of drawing a particular card is not the same as the increased chance of winning a game.

If drawing your best card doesn't increase your chances of winning a game, I don't know what does.

tivadar

12-17-2009, 04:06 PM

If drawing your best card doesn't increase your chances of winning a game, I don't know what does.

Drawing a land when you have none in play. Drawing a removal spell when you have no others in your hand. Drawing your accumulated knowledge when 3 more are in your graveyard...

My point here is that removing one card not only removes the strength of that one card, but also changes the strength of every other card in the deck. Even if I remove the current worst card, it could lead to a deck who's average strength is weaker overall due to how that card's removal affects the strength of other cards in the deck. That's essentially why it's impossible to say a 61 card deck is always worse than a 60 card deck using just the probability of drawing a card and a card's relative strength independent of the other cards in the deck.

Mind you, the hypergeometric proof is great for offering evidence that it is typically better to play a 60 card deck than a 61 card deck, but is by no means definitive. And I would assume the burden of proof would be on anyone who says that something is *always* better than something else.

Kuma

12-17-2009, 04:19 PM

Drawing a land when you have none in play. Drawing a removal spell when you have no others in your hand. Drawing your accumulated knowledge when 3 more are in your graveyard...

Read between the lines a little. If you have no land in play, a land is likely your "best" card. Unless your 61st card was a land, it's hurting your chances of drawing your "best" card.

Removing one card not only removes the strength of that one card, but also changes the strength of every other card in the deck.

That's entirely possible. Can you give an example of a 61 card deck where cutting any one card greatly reduces the power of all others?

In addition, and probably more importantly, even if I remove the current worst card, it could lead to a deck who's average strength is weaker overall due to how that card's removal affects the strength of other cards in the deck.

Then the card you cut wouldn't really be the "worst" card now would it?

That's essentially why it's impossible to say a 61 card deck is always worse than a 60 card deck using just the probability of drawing a card and a card's relative strength independent of the other cards in the deck.

It's not about drawing a card. It's about drawing the right card for the given situation. Unless the right card for the situation is your weakest overall card in your 61, you're hurting your chances of drawing the right card.

tivadar

12-17-2009, 04:33 PM

Mind you, I'm not arguing that I know a 61 card deck that can't be made better by removing a card. I'm just trying to say that that deck could exists. And that the burden of proof is on the person who says that something is *always* better than something else.

And if I'm allowed to construct fake cards, I could show an example. But that's just an effort in futility, as we're not playing in an imaginary world. In addition, I could most likely construct a deck with real cards that would have this property as well. However, that deck probably wouldn't be competitive, so would it really matter?

In addition, this is all just silly. If we can show that one deck is better than another, then that implies that there is exactly one best deck for each tournament. And what fun would that be?

Let's face it. No one can prove that 60 cards is always better than 61. No one can prove that there's a 61 card deck that can't be made better by cutting a single card. Everyone accepts that 60 cards is typically better than 61. And I think everyone would acknowledge that if you go with a 61 card deck instead of a 60 card deck, your chances of winning go down so little that it might affect you losing an additional game once every 100 or so tournaments. So people, just do what you want, and stop saying you can prove one thing or another!

SpencerForHire

12-17-2009, 04:49 PM

And that the burden of proof is on the person who says that something is *always* better than something else.

That's preposterous. That means if I were to come up with some crackpot rule it is your job to prove me wrong and until you do so I am infallible.

In any scientific argument, it is the burden of the person with the new theory trying to argue against the old theory to prove their point. I don't know why everyone thinks otherwise.

Similarly, I don't understand how people can say that the math behind why 60 would be better is wrong and then go ahead and talk about how the minute mathematical difference is offset by the alleged game-winning 61st card that just can't fit in the first 60.

tivadar

12-17-2009, 05:05 PM

That's preposterous. That means if I were to come up with some crackpot rule it is your job to prove me wrong and until you do so I am infallible.

In any scientific argument, it is the burden of the person with the new theory trying to argue against the old theory to prove their point. I don't know why everyone thinks otherwise.

Ok, since "60 cards is the best" is the old theory, could you show me the evidence to back up that theory? Where's the study comparing 60 card decks to 61 card decks? Where's the proof that it's statistically significant? Point is that there's no "old" and "new" theory. No one knows these things and it's impossible to prove either. This doesn't even get into the fact that people are trying to say 60 card decks are *always* better, which is an even stronger statement. Of course the burden of proof is on the person stating that something is *always* true.

Similarly, I don't understand how people can say that the math behind why 60 would be better is wrong and then go ahead and talk about how the minute mathematical difference is offset by the alleged game-winning 61st card that just can't fit in the first 60.

The proof that 60 is always better than 61 using the probability of drawing a good card is wrong. Its wrong because its based on the fact that if you remove the card with the lowest win percent from your deck, then clearly the average of the remaining cards is higher. What that fails to consider is that when you remove a card, those averages changes. So yes, I'm saying that that logic, as a *proof* is wrong.

I'm agreeing that 60 cards is probably best in almost all circumstances. But I'm also saying that no one can say that 61 cards is strictly wrong.

kinda

12-17-2009, 07:32 PM

Well 40 red morphers and 20 islands is the same as 80 red morphers and 40 islands...but since cards are differentiated in most decks there is a hierarchy. It is definately possible to build a deck that runs better (or at least equal) at more then 60 cards...but the important part here is that you should build decks that run ideally at 60 cards.

I'll say this again...why is the argument between 60 and 61? not 60 and 80? If someone told you that you could make a 40 card minimum deck...would you still run 60/61 or 40/41?

Forbiddian

12-17-2009, 08:14 PM

Ok, since "60 cards is the best" is the old theory, could you show me the evidence to back up that theory? Where's the study comparing 60 card decks to 61 card decks? Where's the proof that it's statistically significant? Point is that there's no "old" and "new" theory. No one knows these things and it's impossible to prove either. This doesn't even get into the fact that people are trying to say 60 card decks are *always* better, which is an even stronger statement. Of course the burden of proof is on the person stating that something is *always* true.

It's demonstrable, in a real way, that large decks, like 70+ cards, or limit N-->infinity decks are worse than 60 card decks, even if they keep the same ratio of cards.

It's also demonstrable that 40 card decks are much better than 60 card decks.

The tests we have are not powerful enough to prove that 61 is inferior per-se, but it seems ludicrous to me to believe that 61 is some magical number. You have to accept the fact that 40 cards are better than 60 cards, and the fact that 60 cards are better than N-->Infinity cards. You'd then be seeking to find evidence that the otherwise decreasing function for all testable values reverses its trend and shoots upward to a maximum at n=61.

Because the trend can and has been demonstrated that smaller decks are better for every tested value, the burden of proof lies on the "61 cards are good" camp to establish why 61 cards, even for a given specific deck, is an exception to the already established rule that fewer cards are better. It doesn't have to do with who used the word "always" or something stupid like that.

One camp says, "The trend probably holds, and 60 cards is always better."

The other camp says, "Somehow you guys are stuck with burden of proof, and you can't meet the bop, therefore 61 cards in my deck is better."

The only evidence with regard to deck quality that's been thrown out is the evidence that fewer cards is better, in general. Among other things, my principle of regression, and the fact that the cards you want to be 4-ofs are seen more often.

I will say that if I somehow believed I found a deck that is better with 61 cards, I would still run 60. The probability that I misevaluated it is much larger than the probability that my deck, for whatever reason, violates the established principle, even given the fact that I evaluated it to be better than the 60 card variant. See the taxicab problem.

By the way, Maveric, why the hell are you doing a monte carlo simulation for something that needs that much precision. Monte carlo is only useful for like poker bots trying to 6 table and only needing +/- 1% accuracy. If you need more accuracy, it's actually much slower to run Monte Carlo. Right now you're trying like 1E8 trials? There are probably at most a few million permutations of just bolts and lands.

Just go through every combination. It's much faster and will actually give you a precisely accurate result instead of a (large) margin for error. Also, once you set up the table of known win turn values, you can actually set your mulligan thresholds precisely instead of guessing.

SpencerForHire

12-17-2009, 09:46 PM

tivadar

12-17-2009, 09:48 PM

Forbiddian

12-17-2009, 10:33 PM

Ok, first, 60 cards is better than 61 most if not all of the time. I don't mean to be "ludicrous" and suggest otherwise. And yes, 61 *is* actually a magic number of sorts. It's the number of cards that's not what people expect you to be running that gives you the best statistical odds to draw what you want. Back in the days of Solidarity reigning, I know this can and did come into play. Many people ran 61/62 card decks to throw the math off for a good Solidarity player. I don't think these people were wrong at all and I know that there are games that they won because of it.

I mean, that's a decent point, that good players were trying to run 61 cards at some point (for a reason other than deck quality -- to trick the opponent) but:

What good solidarity player wouldn't pile shuffle his opponent's deck before the game and count the cards out? Especially if this was a trick pulled by "many" people. Wouldn't good solidarity players know that trick?

A lot of people do that anyway to make sure the opponent has at least 60 (and if it's not exactly 60, you count the sideboard and call a judge, yippie!). If you stick your opponent with just one game loss, you're addicted!

There would have to be a preposterously slim minority of Solidarity players who 1) Care enough to count your graveyard, hand, and cards in play and then subtract (and then divide by 3), 2) Don't care enough to count your deck during shuffling, 3) Spot your library up and think it's faster to count cards in hand/graveyard/RFG/in play than it is just to count the library, 4) it actually happens in a game where they play their storm count EXACTLY to whatever would kill you and then you're able to win with the 1-2 extra cards before decking.

3 is rare, because the lethal storm count would have to be < 10 (and maybe even <7) for him to make a divergent decision to Brainfreeze rather than attempt to stack more storm count, and then it's much faster to count the library at that point than it would be to account for 30 cards in play, graveyard, RFG, etc.

Anyway, I can't imagine that was a profitable move even at the height of Solidarity. You might know a guy who knows a guy who once had it matter where he had a card left and then topdecked the lightning bolt and swung in for lethal.

But I know a guy who knows a guy who lost because he ran 61 cards and the 61st card was shit.

4eak

12-17-2009, 11:43 PM

@ Forbiddian

By the way, Maveric, why the hell are you doing a monte carlo simulation for something that needs that much precision. Monte carlo is only useful for like poker bots trying to 6 table and only needing +/- 1% accuracy. If you need more accuracy, it's actually much slower to run Monte Carlo. Right now you're trying like 1E8 trials? There are probably at most a few million permutations of just bolts and lands.

Just go through every combination. It's much faster and will actually give you a precisely accurate result instead of a (large) margin for error. Also, once you set up the table of known win turn values, you can actually set your mulligan thresholds precisely instead of guessing.

Showing the inaccuracy of our personal testing was a good reason to draw randomly instead of going through the entire search space.

It's also much easier to generate X^Y random lists than computing the non-duplicate permutations (I think they are called Necklaces). Working on the code now (I am noob though, so I'd appreciate the help of any proficient programmer). It isn't feasible to just 'eliminate' duplicate permutations, we actually need to generate the smaller set without resorting to running through all permutations.

peace,
4eak

Maveric78f

12-18-2009, 03:55 AM

@Forbiddian

Did you actually try to compute the number of deck ordering?
It's 60!/(L!*(60-L)!)
Where L is the number of lands. If L=17, then this number approximately equals: 3.87 10^14
And then, you still have to consider all the mulligans states (from 7 cards to 1 card initially drawn).

However you could consider that the game is reasonnably always over at turn T, when the number of drawn cards are 7+T. If you choose T=20. Then the number of orderings is much less, but difficult to count and each of these does not have the same weight in the mean (it's depending on how many combinations, there exists in the remaining 60-T-7 cards). I still believe that Monte Carlo is the best way to tackle the problem so far. May your creativity prove me wrong!

Edit: The best way to tackle the problem is probably to treat each game opening, treat it until kill, and then compute how many combinations in the library are remaining. It looks possible but much more complex (both theoretically and algorithmically) than what I did in a couple of hours of my time.

Edit2: Actually, it's interesting. I'll give it a look. Damn you.

TheInfamousBearAssassin

12-18-2009, 04:20 AM

There are two valid reasons to run a 61-card deck;

1) You're about to enter a tournament, made some last minute changes, and really don't know what else to cut. In this case it's perfectly possible that removing the wrong card will do more damage than running 61 cards.

2) You anticipate getting your opponents to count your deck via discreet means, i.e. pile shuffling and expecting your opponent to do likewise, and you want them to discover the odd number of cards you're running and become pissed off. The optimum number in this case is actually 63, as they might consider 61 or 62 borderline reasonable if they're bad players. People actually get pissed off about this kind of thing, or if they see you running shitty cards like Scrying Sheets or Helldozer or whatever.

On a side note, the odds of drawing your 61st-best card on any given draw in a 61 card deck is 1.63%. That's really not insignificant. To put that in perspective, only 1.58% of the world's population is Mexican. That means that if you consider the distinction between 60 and 61 cards to be immaterial, it's equivalent to wiping the entire country of Mexico off the map.

Take a stand against genocide, kids. Just cut down to 60 fucking cards.

Maveric78f

12-18-2009, 04:24 AM

Algorithm.

For all the 8+7+6+5+4+3+2=35 possible starting hands I've already explicited (knowing you'll never muligan to 0) {
...T=0
...Label 0
...T++
...Draw shock.
...Play your turn.
...If you don't kill, goto Label 0 and place label T.
...Remove Label T.
...Compute kill turn and how many card combinations are remaining in the library. Save it.
...Undo last draw.
...Draw mountain.
...Play your turn.
...If you don't kill, goto Label 0 and place label T.
...Remove Label T.
...Compute kill turn and how many card combinations are remaining in the library. Save it.
...Undo last draw.
...T--
...If T!=0, goto label T.
...Else, compute this starting hand performance expectation by average of the turns T weighted by the number of card combinations remaining in library.
}
Apply mulligan optimisation.
Compute (and enjoy) the exact performance expectation computation.

The algorithm is clunky and it would be much easier with a backtracking program language such as Prolog.

Maveric78f

12-18-2009, 04:31 AM

Take a stand against genocide, kids. Just cut down to 60 fucking cards.
I don't get your rhetoric here. Cutting to 60 is considering that Mexicans are worse than every other population and that the world would be better without them.

But I think that we have actually no way to know that the Mexicans are the right population to cut to have a perfect 6 billions world. I'd rather cut Americans actually, since it would probably lower the magic staple prices. If we don't have anymore enough population to reach the perfect 6 billions world, I'd add some scandanivian chicks. It can't be a bad move.

4eak

12-18-2009, 05:50 AM

My current code is simply trying to generate and count them, no additional layers to it, just for maximum speed, and it is taking forever (I have no idea when it will be finished grinding through).

I asked for more help from some guy (http://webhome.cs.uvic.ca/~ruskey/). He said he might be able help, but wanted more context. We might get some optimized algorithms for generating these permutations (although, it still just might be infeasible).

As a sidenote, I love tacos.

You all should read the previous thread Forbiddian had written in about this topic. His last few paragraphs were hilarious.

peace,
4eak

Forbiddian

12-18-2009, 06:02 AM

@Forbiddian

I forgot that you would have to write your own hand evaluator. But yeah, you figured out a quick shortcut to running every junk permutation.

If you only look at the top 20 cards, there's around a million permutations even with 20+ lands.

The problem is, that each set is not equally likely (i.e. top 20 all bolts is more likely than top 20 all mountains in a deck with 20 mountains, 20 bolts), but there's a work around from poker that the calculators use:

There are essentially 18 different "sets" of 20 cards. Each corresponds to the different numbers of lands, from 0-17 possible.

If you can find the census data for each set, from 0-17 possible, you can simply multiply that result by the probability of having that many lands in the top 20, given the number of lands in the deck and the number of cards in the deck. Excel can do that from a binomial distribution.

This gives you your precise results.

Maveric78f

12-18-2009, 08:20 AM

The truth.

Thx Forbiddian for your inspiration on how to compute an exact expectation. The fact that these numbers are close to the Monte Carlo's experiment makes me confidant about their validity.

10.587769936773933
9.68029233802506 9.333333333333334
8.91651105153967 8.196075918587304 9.177777777777777
8.285715353417906 7.091908554611495 7.909090909090909 9.022222222222222
7.8262947901831525 6.118500687868058 6.627906976744186 7.7727272727272725 8.866666666666667
7.540851354835817 5.3597544979586615 5.333333333333333 6.511627906976744 7.636363636363637 8.71111111111111
7.384065821135743 4.947604073075771 4.422713095055479 5.238095238095238 6.395348837209302 7.5 8.555555555555555
7.294986035052628 4.8464535741727754 4.070699223085461 4.372973119298341 5.142857142857143 6.27906976744186 7.363636363636363 8.4

KEEP 10.587769936773933
KEEP KEEP 9.587769936773933
KEEP KEEP KEEP 8.64767809874312
KEEP KEEP KEEP MULL 7.705489238999487
MULL KEEP KEEP MULL MULL 6.764998174268332
MULL KEEP KEEP KEEP MULL MULL 5.764953959397981
MULL KEEP KEEP KEEP MULL MULL MULL 4.966470533203395
MULL KEEP KEEP KEEP MULL MULL MULL MULL 4.512281162092423
nbLands = 16.......4.512281162092423

10.624294806368624
9.655464211846283 9.545454545454545
8.82664047921042 8.362162576881676 9.386363636363637
8.134672608671687 7.20764700009983 8.093023255813954 9.227272727272727
7.624533066514198 6.176839336607616 6.785714285714286 7.953488372093023 9.068181818181818
7.3055460263007435 5.357598627787307 5.463414634146342 6.666666666666667 7.813953488372093 8.909090909090908
7.134306011763807 4.89246281321753 4.493448637316562 5.365853658536586 6.5476190476190474 7.674418604651163 8.75
7.045739166493884 4.772808312080466 4.089643985315462 4.440122086570478 5.2682926829268295 6.428571428571429 7.534883720930233 8.590909090909092

KEEP 10.624294806368624
KEEP KEEP 9.624294806368624
KEEP KEEP KEEP 8.67782070676439
KEEP KEEP KEEP MULL 7.722486903645516
KEEP KEEP KEEP MULL MULL 6.796786579991297
MULL KEEP KEEP KEEP MULL MULL 5.8195532693910925
MULL KEEP KEEP KEEP MULL MULL MULL 4.995895710988585
MULL KEEP KEEP KEEP MULL MULL MULL MULL 4.508071722338611
nbLands = 17.......4.508071722338611

10.697995442327246
9.668232691696732 9.767441860465116
8.7734498316054 8.53966094629725 9.604651162790697
8.017857465275538 7.336096485199108 8.285714285714286 9.44186046511628
7.453175126760033 6.2483664134607535 6.951219512195122 8.142857142857142 9.279069767441861
7.096478496677107 5.367587632493293 5.6 6.829268292682927 8.0 9.116279069767442
6.9077835608123594 4.846385193753615 4.570875665215288 5.5 6.7073170731707314 7.857142857142857 8.953488372093023
6.818800266267992 4.705405965783324 4.111976346569489 4.513958849141979 5.4 6.585365853658536 7.714285714285714 8.790697674418604

KEEP 10.697995442327246
KEEP KEEP 9.697995442327247
KEEP KEEP KEEP 8.745443846508444
KEEP KEEP KEEP MULL 7.776741940611625
KEEP KEEP KEEP MULL MULL 6.831795621197604
MULL KEEP KEEP KEEP MULL MULL 5.895177859592948
MULL KEEP KEEP KEEP MULL MULL MULL 5.047458620036484
MULL KEEP KEEP KEEP MULL MULL MULL MULL 4.522402653410277
nbLands = 18.......4.522402653410277

10.589052982968674
9.69801095179232 9.282608695652174
8.951833116080435 8.156667726202317 9.130434782608695
8.337771900533733 7.065244005994956 7.866666666666666 8.978260869565217
7.8924002570243585 6.106627462589219 6.590909090909091 7.733333333333333 8.826086956521738
7.616562200330913 5.363593121639736 5.3023255813953485 6.4772727272727275 7.6 8.673913043478262
7.464541082299129 4.965122489889009 4.40632742519535 5.209302325581396 6.363636363636363 7.466666666666667 8.521739130434783
7.376540153232717 4.868725666367176 4.066574628010431 4.358340820604972 5.116279069767442 6.25 7.333333333333333 8.369565217391305

KEEP 10.589052982968674
KEEP KEEP 9.589052982968674
KEEP KEEP KEEP 8.650692744097455
KEEP KEEP KEEP MULL 7.711887642684867
MULL KEEP KEEP MULL MULL 6.758307580713864
MULL KEEP KEEP KEEP MULL MULL 5.7564934442564315
MULL KEEP KEEP KEEP MULL MULL MULL 4.964815881069069
MULL KEEP KEEP KEEP MULL MULL MULL MULL 4.518264612719188
nbLands = 16.......4.518264612719188

10.614115744350432
9.662498847596936 9.488888888888889
8.852535389881561 8.317122127174729 9.333333333333334
8.178943757993856 7.175719153068565 8.045454545454545 9.177777777777777
7.684707615678623 6.160463575557915 6.744186046511628 7.909090909090909 9.022222222222222
7.376978001034605 5.358214590693408 5.428571428571429 6.627906976744186 7.7727272727272725 8.866666666666667
7.211504498296951 4.908503906994473 4.474157125790812 5.333333333333333 6.511627906976744 7.636363636363637 8.71111111111111
7.124205773262377 4.794629130478187 4.084381551362683 4.422713095055479 5.238095238095238 6.395348837209302 7.5 8.555555555555555

KEEP 10.614115744350432
KEEP KEEP 9.614115744350432
KEEP KEEP KEEP 8.669420307807643
KEEP KEEP KEEP MULL 7.717728601273368
KEEP KEEP KEEP MULL MULL 6.797262516344256
MULL KEEP KEEP KEEP MULL MULL 5.806331407398985
MULL KEEP KEEP KEEP MULL MULL MULL 4.988140058857457
MULL KEEP KEEP KEEP MULL MULL MULL MULL 4.509310121328615
nbLands = 17.......4.509310121328615

10.676247713683718
9.664402147741596 9.704545454545455
8.789677486806932 8.488517348330051 9.545454545454545
8.054110440812275 7.298447498187251 8.232558139534884 9.386363636363637
7.5072881603666115 6.2270337765045545 6.904761904761905 8.093023255813954 9.227272727272727
7.163678774056133 5.36455525606469 5.560975609756097 6.785714285714286 7.953488372093023 9.068181818181818
6.981918181123742 4.860627367042461 4.548351438917477 5.463414634146342 6.666666666666667 7.813953488372093 8.909090909090908
6.894446566343289 4.726625222950943 4.105359350642369 4.493448637316562 5.365853658536586 6.5476190476190474 7.674418604651163 8.75

KEEP 10.676247713683718
KEEP KEEP 9.676247713683718
KEEP KEEP KEEP 8.725489477403642
KEEP KEEP KEEP MULL 7.760676354519623
KEEP KEEP KEEP MULL MULL 6.821328802997435
MULL KEEP KEEP KEEP MULL MULL 5.8729483078678655
MULL KEEP KEEP KEEP MULL MULL MULL 5.032340965204721
MULL KEEP KEEP KEEP MULL MULL MULL MULL 4.51812032124002
nbLands = 18.......4.51812032124002

So, in the mountain/bolt problem, 60 cards is the best. And I can't think of a better case since 16/61 and 18/61 have almost the same performance, meaning that 17/61 is close to the best ratio but that it is not enough to justify adding 1 card into the deck.

4eak

12-18-2009, 09:39 AM

Very nice. Check one particular set of decks (Bolt/Mountain) off the list if you want.

For anyone who is a poor interpreter -- this isn't a proof that there is no exception to the 60-card rule. This does not prove the 60-card rule for all of magic; it is only proof (strong evidence if you are extremely picky) that the 60-card rule is true for this deck in particular.

As explained, Nihil's variance has enough influence on this deck that the benefits of moving closer to the perfect ratio do not outweigh the costs incurred through adding another card and increasing the variance factor. Other decks, which see more cards per average game, or have stronger card selection options, might not be affected as strongly by this variance.

Of course, there are a host of other large variables which we haven't even tried to take into account that could have strong impact on this problem.

peace,
4eak

Maveric78f

12-18-2009, 10:16 AM

It leads to a much wider proof: raising the number of cards of your deck to find the best land/active spell ratio (Ive heard this argument earlier) is a bad move.

I know I've not done the necessary to prove this, and I probably can't, but it looks like a direct corrolary to this result.

I'm running the shock/mountain experiment now. It's longer to compute due to the high number of turns before killing (which means that I need to cope with more library orderings). The only problem, I did not implement the end of turn discard rule which was irrelevant with bolt since we killed off 7 active spells but which is relevant with shocks.

I already have the optimal 60 cards ratio:

14.062158485052645
13.300602226595077 12.0
12.63607186950951 10.99422614770399 11.8
12.052457684917202 10.008433835174383 10.653061224489797 11.6
11.566256301220557 9.096368205745073 9.5 10.46938775510204 11.4
11.182975954452091 8.286609181359387 8.340425531914894 9.333333333333334 10.285714285714286 11.2
10.900806751176317 7.61802305200585 7.173913043478261 8.191489361702128 9.166666666666666 10.10204081632653 11.0
10.70465705491726 7.24570821352622 6.378441536266176 7.043478260869565 8.042553191489361 9.0 9.918367346938776 10.8 100.0
KEEP 14.062158485052645
KEEP KEEP 13.062158485052645
KEEP KEEP KEEP 12.110117746980585
KEEP KEEP KEEP KEEP 11.16737523315986
MULL KEEP KEEP KEEP MULL 10.07391880151556
MULL KEEP KEEP KEEP MULL MULL 8.963263959398555
MULL KEEP KEEP KEEP MULL MULL MULL 7.936271749861053
MULL KEEP KEEP KEEP MULL MULL MULL MULL 7.1573665443332875
nbLands = 11.......7.1573665443332875

13.935958223921041
13.108723290105383 12.244897959183673
12.375844130132288 11.190461806233982 12.040816326530612
11.726868492877783 10.15271179950355 10.875 11.83673469387755
11.181678825376304 9.18222627443102 9.702127659574469 10.6875 11.63265306122449
10.749618256943451 8.308668585057468 8.521739130434783 9.53191489361702 10.5 11.428571428571429
10.432534310605298 7.573552917813021 7.333333333333333 8.369565217391305 9.361702127659575 10.3125 11.224489795918368
10.21743210325729 7.138768092207083 6.471800452599565 7.2 8.217391304347826 9.191489361702128 10.125 11.020408163265307 100.0
KEEP 13.935958223921041
KEEP KEEP 12.935958223921041
KEEP KEEP KEEP 11.97760003205142
KEEP KEEP KEEP KEEP 11.026040502733702
MULL KEEP KEEP KEEP MULL 10.03180304691313
MULL KEEP KEEP KEEP MULL MULL 8.95677206462517
MULL KEEP KEEP KEEP MULL MULL MULL 7.927987080722664
MULL KEEP KEEP KEEP MULL MULL MULL MULL 7.1202621450107495
nbLands = 12.......7.1202621450107495

13.892966348705121
13.00165916855973 12.5
12.200555446075915 11.401330822061517 12.291666666666666
11.484576925109666 10.315905459804922 11.106382978723405 12.083333333333334
10.876913459494718 9.290710271529996 9.91304347826087 10.914893617021276 11.875
10.391126331846978 8.355945174464077 8.71111111111111 9.73913043478261 10.72340425531915 11.666666666666666
10.033506630717767 7.555135926371092 7.5 8.555555555555555 9.565217391304348 10.53191489361702 11.458333333333334
9.794322010381139 7.055206746793807 6.575502302720123 7.363636363636363 8.4 9.391304347826088 10.340425531914894 11.25 100.0
KEEP 13.892966348705121
KEEP KEEP 12.892966348705123
KEEP KEEP KEEP 11.928679982761388
KEEP KEEP KEEP MULL 10.967838595627164
KEEP KEEP KEEP KEEP MULL 10.025744553786776
MULL KEEP KEEP KEEP MULL MULL 8.996147028080241
MULL KEEP KEEP KEEP MULL MULL MULL 7.965586462831094
MULL KEEP KEEP KEEP MULL MULL MULL MULL 7.126385700481913
nbLands = 13.......7.126385700481913

13.919494173707967
12.966222742764323 12.76595744680851
12.097743248242693 11.626694865663728 12.553191489361701
11.313908078192206 10.497169098247266 11.347826086956522 12.340425531914894
10.641018597447887 9.420510475774071 10.133333333333333 11.152173913043478 12.127659574468085
10.097091425859462 8.426940677189448 8.909090909090908 9.955555555555556 10.956521739130435 11.914893617021276
9.693543007169566 7.561350837982569 7.674418604651163 8.75 9.777777777777779 10.76086956521739 11.702127659574469
9.42507499046256 6.993984685593185 6.689714376316564 7.534883720930233 8.590909090909092 9.6 10.565217391304348 11.48936170212766 100.0
KEEP 13.919494173707967
KEEP KEEP 12.919494173707967
KEEP KEEP KEEP 11.949771853644375
KEEP KEEP KEEP MULL 10.978983688348542
KEEP KEEP KEEP MULL MULL 10.023340350232825
MULL KEEP KEEP KEEP MULL MULL 9.069051892093484
MULL KEEP KEEP KEEP MULL MULL MULL 8.042230881388335
MULL KEEP KEEP KEEP MULL MULL MULL MULL 7.1714298763074655
nbLands = 14.......7.1714298763074655

Maveric78f

12-18-2009, 11:09 AM

Shock/mountain deck.

14.103197545955226
13.354528457045767 11.96078431372549
12.703504233117267 10.963497483845245 11.764705882352942
12.132959553965328 9.987008832333979 10.62 11.568627450980392
11.658723588772881 9.08555376190147 9.46938775510204 10.44 11.372549019607844
11.285593564112084 8.287488243915668 8.3125 9.306122448979592 10.26 11.176470588235293
11.011101082910184 7.631146218740125 7.148936170212766 8.166666666666666 9.142857142857142 10.08 10.980392156862745
10.819818390665203 7.270070860324469 6.364586604503605 7.0212765957446805 8.020833333333334 8.979591836734693 9.9 10.784313725490197 100.0
KEEP 14.103197545955226
KEEP KEEP 13.103197545955226
KEEP KEEP KEEP 12.152336133996146
KEEP KEEP KEEP KEEP 11.21130243964525
MULL KEEP KEEP KEEP MULL 10.09598082364132
MULL KEEP KEEP KEEP MULL MULL 8.975871217325851
MULL KEEP KEEP KEEP MULL MULL MULL 7.947539786982111
MULL KEEP KEEP KEEP MULL MULL MULL MULL 7.172603789648985
nbLands = 11.......7.172603789648985

13.96078450479939
13.147099077403322 12.2
12.428771805937826 11.153875497016527 12.0
11.794231773989386 10.12534873730319 10.83673469387755 11.8
11.262642504501411 9.165657173456538 9.666666666666666 10.653061224489797 11.6
10.842480620695303 8.304335487902486 8.48936170212766 9.5 10.46938775510204 11.4
10.53470284309352 7.582446944651587 7.304347826086956 8.340425531914894 9.333333333333334 10.285714285714286 11.2
10.325704314737282 7.160552190994315 6.454511764389678 7.173913043478261 8.191489361702128 9.166666666666666 10.10204081632653 11.0 100.0
KEEP 13.96078450479939
KEEP KEEP 12.96078450479939
KEEP KEEP KEEP 12.003669090726012
KEEP KEEP KEEP KEEP 11.053844056260159
MULL KEEP KEEP KEEP MULL 10.039303502385117
MULL KEEP KEEP KEEP MULL MULL 8.95756690575713
MULL KEEP KEEP KEEP MULL MULL MULL 7.929458724356424
MULL KEEP KEEP KEEP MULL MULL MULL MULL 7.127494652510505
nbLands = 12.......7.127494652510505

13.90212855161999
13.024856394894622 12.448979591836734
12.239178376461219 11.358442208672866 12.244897959183673
11.538883655816313 10.282141254225328 11.0625 12.040816326530612
10.94642134995894 9.267871535298632 9.872340425531915 10.875 11.83673469387755
10.474347671601517 8.34581447673409 8.673913043478262 9.702127659574469 10.6875 11.63265306122449
10.12782226414573 7.559154179049331 7.466666666666667 8.521739130434783 9.53191489361702 10.5 11.428571428571429
9.896181847555026 7.073775934663424 6.5543778517696385 7.333333333333333 8.369565217391305 9.361702127659575 10.3125 11.224489795918368 100.0
KEEP 13.90212855161999
KEEP KEEP 12.902128551619988
KEEP KEEP KEEP 11.93910556703631
KEEP KEEP KEEP MULL 10.980451104616048
KEEP KEEP KEEP KEEP MULL 10.040275685360182
MULL KEEP KEEP KEEP MULL MULL 8.991614405384762
MULL KEEP KEEP KEEP MULL MULL MULL 7.95865770504603
MULL KEEP KEEP KEEP MULL MULL MULL MULL 7.125415278295165
nbLands = 13.......7.125415278295165

13.913405820428626
12.974491242116585 12.708333333333334
12.12213936298278 11.57703838148019 12.5
11.35510400675849 10.456541041355017 11.297872340425531 12.291666666666666
10.69895768637167 9.390906920420159 10.08695652173913 11.106382978723405 12.083333333333334
10.170600152356341 8.410459578342486 8.866666666666667 9.91304347826087 10.914893617021276 11.875
9.780079202254562 7.559881858874402 7.636363636363637 8.71111111111111 9.73913043478261 10.72340425531915 11.666666666666666
9.520805699330339 7.008723329956298 6.664333915517355 7.5 8.555555555555555 9.565217391304348 10.53191489361702 11.458333333333334 100.0
KEEP 13.913405820428626
KEEP KEEP 12.913405820428626
KEEP KEEP KEEP 11.944931096392542
KEEP KEEP KEEP MULL 10.976417296708604
KEEP KEEP KEEP MULL MULL 10.024212297592934
MULL KEEP KEEP KEEP MULL MULL 9.0523931404312
MULL KEEP KEEP KEEP MULL MULL MULL 8.024932896214949
MULL KEEP KEEP KEEP MULL MULL MULL MULL 7.16132282854845
nbLands = 14.......7.16132282854845

Dilettante

12-18-2009, 11:30 AM

The thing about a concept of a toolbox deck is... you need to get your hate in a perfect synergistic ratio between useful and useless. I am not saying that it wouldn't work to have more than 60 cards, but you are playing a metagame that is ever shifting every single moment. When playing at a tournament, it is a microcosm where a single player playing something different can change your ratios. So if you show up at a 60 person tournament where you have a psychological profile of every last player, their behaviors, and the probabilities of them drawing particular cards and then suddenly, someone shows up with Nourishing Lich, all that goes out the window. I'd rather stick to 60 cards and let the deck play out.

SpencerForHire

12-18-2009, 01:16 PM

For anyone who is a poor interpreter -- this isn't a proof that there is no exception to the 60-card rule. This does not prove the 60-card rule for all of magic; it is only proof (strong evidence if you are extremely picky) that the 60-card rule is true for this deck in particular.

Alright I'm calling you out. You were the advocate of the Mountain/Shock deck to prove 61 cards but when another deck proves optimal at 60 you say the whole theory of hypothetical examples is only relevant on a deck by deck basis. I call foul.

DrJones

12-18-2009, 02:40 PM

Guys, this problem is so obvious to anybody who has ever done a problem of linear optimization that I can't understand why it is still an issue. Of course that there exist decks that are optimal with more than sixty cards! It's purely a question of fine-tuning.

Let's say you have 4 copies of a card in a 61-card deck and take out one to make a 60-card deck, then your chances of drawing said card is now a 25% less than before. Now, let's define that the deck 'works' as long as the proportions of certain types of cards don't fall below a certain threshold. It might perfectly be than a drop from 4/61 to 3/60 is too much for consistency, while the one from 4/60 to 4/61 is bearable, even if it happens to all cards.

By types of cards I mean a classification according to their function in the deck. For example, let's say a deck is composed by the following:

4 - Tarmogoyf (6.66%)
6 - card draw (10%)
8 - acceleration (13.33%)
10 - disruption (16.66%)
12 - beaters (20%)
20 - lands (33%)

Now, let's assume that testing shows that 33% of land is not enough, but there's simply no space left in the deck for that land, because every time that you take out a beater you fall short of them, same with disruption, acceleration and the others. After some more tests you discover that the deck doesn't 'work' (bad consistency) if you don't follow this recipe:

Tarmogoyf >5%
card draw >9%
acceleration >12%
disruption >15%
beaters >19%
lands >33%

In this situation, it just happens that cutting any card from the 60-card deck is detrimental, while adding a card is a good move that optimizes the deck. Now, this example might look like an exageration, but these kind of problems actually arise when you are fine tuning heavily optimized decks. Of course, it's not as obvious to see in a game like Magic, where one single card can simultaneously fulfill several roles, but this in turn makes it harder to optimize.

4eak

12-20-2009, 10:58 PM

@ SpencerForHire

Alright I'm calling you out. You were the advocate of the Mountain/Shock deck to prove 61 cards

Please point out where I've made this argument. (And, clarify what you mean by "to prove 61 cards")

I certainly realized the implications of simulations, but that wasn't the reason I posted it (http://www.mtgthesource.com/forums/showpost.php?p=409226&postcount=29). I advocated the idea that we're largely ignorant on this topic and that our experience and tournament data is much less meaningful than we previously believed. I tried to make that abundantly clear (http://www.mtgthesource.com/forums/showpost.php?p=409452&postcount=45).

With the expertise of Maverick, we may have the relevant accuracy we need for these particular burn decks, but my argument in general is that we don't have the necessary accuracy for more complex decks. Even current Legacy burn is much more complicated than an idealized 17/43 Mountain/Bolt deck. This sort of testing can steer us in the right direction. and even reveal things we didn't know before, but there is only so much we can reasonably deduce from it.

when another deck proves optimal at 60 you say the whole theory of hypothetical examples is only relevant on a deck by deck basis. I call foul.

If you haven't already, I suggest re-reading the thread. With regards to the 60-card rule, please remember that only one exception would need to be found to show that the 60-card rule isn't true of all of magic. You may want to think of it like this:

Claim:
All X are Y.

Counter:
This particular X is not Y.
Thus, not all X are Y.

A single counterexample is capable of dismantling the "All X are Y" claim.

However, notice how much more difficult it would be to prove "All X are Y" to be true. Likewise, we've considered how difficult it can be prove the 60-card rule to be true. As you said, this must be done on a deck by deck, metagame by metagame, format by format basis.

@ DrJones

I don't think it is that obvious. You've brought up one of the serious concerns in this thread, but it is just one of many factors.

It is more difficult to see any actual benefits from adjusting a deck to be closer to that 'perfect balance' of ratios by adding cards beyond the 60th. I'm not saying it isn't possible, but please realize that Nihil's variance problem works against many of the possible benefits of fine-tuning the ratio of functions.

However, I can say that as we add complexity to decks, metagame variables, and so on, it is certainly possible that this variance will not be as important a variable (although still necessary to calculate, among the many other variables) in determining deck optimality.

peace,
4eak

Humphrey

12-21-2009, 03:23 AM

Ive found a calculator for starting hand. its german but google translate is useful enough for this i guess

http://tinyurl.com/yh8kwt7

Forbiddian

12-21-2009, 03:28 AM

Holy shit, Maveric, you're amazing.

Maveric78f

12-21-2009, 04:23 AM

Holy shit, Maveric, you're amazing.

First time I appreciate a message with shit next to my name ^^.

Humphrey > misthread?

T=Tarmogoyf >5%
CD=card draw >9%
A=acceleration >12%
D=disruption >15%
B=beaters >19%
L=lands >33%
What are those constraints? I don't believe in such hard constraints. There is function of performance of the deck which has a maximum given the following constraint: T+CD+A+D+B+L >= 60
If T, CD, A, D, B and L could take its value on R, then, (outside from deck size issues, such as playing against solidarity or with Battle of Wits) the "variance rule" would demonstrate that the optimum happens when T+CD+A+D+B+L = 60. However these have to be natural numbers.

This "natural numbers" constraint has a big issue: every tutorable element has to be in 1 copy, so that you'll draw it without tutoring it more than once every 5 games. If we except this tutoring effect, all is about proportions as you said. I think that my exemple showed that even in an exemple where the right land proportion is very close to 17/61, it is optimal to play 60 cards. I think that my lightning bolt/mountain study showed that perfect proportion is an insufficient argument to justify playing 61 cards. However, the toolboxing requirements, the fear of cutting the wrong slot, etc... are reasonable additional motivations for playing more than 60 cards.

The thing about a concept of a toolbox deck is... you need to get your hate in a perfect synergistic ratio between useful and useless.
As I said earlier, try to understand how many cards you've drawn to finish your game and then you'll have the probability of actually drawing your toolbox. Usually, it raises until 1/3 of the games, since a game generally lasts 12 turns. It's a lot, that's why a lot of decks play toolbox that can be cycled: rune of protection, relic of progenitus, survival creatures and/or card selection: brainstorm, top.

Let's come back to this problem:

Card1 %0
sorcery
Card1 cannot be countered.
As an additional cost, discard Card2 from your hand, search for Card3 in your hand and in your deck. Put it into play.
A deck can have any number of cards named Card1.

Card2 %100000
sorcery
A deck can have any number of cards named Card2.

Card3 %100000
Artifact
When Card3 enters the battlefield, you win the game. This ability cannot be countered.
A 29/30/1 60-card deck would have 1.208% chance of not comboing first turn.
A 30/30/1 61-card deck would have 1.2055% chance of not comboing first turn.

So, even in this ideal example where the 61-card deck wins, the difference is so small that we can reasonably assume that in a real magic deck where some cards are inherently stronger than others, we would never have proportion justify to play 61 cards.