Remove both a swamp AND a rat, then add one street wraith. Or a cycling land.Quote:
Originally Posted by MattH
That'd be choice C, I reckon.
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Remove both a swamp AND a rat, then add one street wraith. Or a cycling land.Quote:
Originally Posted by MattH
That'd be choice C, I reckon.
@ Humphrey
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.Quote:
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)
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.
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.Quote:
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.)
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
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?Quote:
P.S. This challenge is open to anyone who cares to try it!
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
Don't rule out the possibility that a friend is setting a trap for you.Quote:
I have been on friendly terms with Matt for years.
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
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.
I don't disagree with almost all of your post, save this statement:
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?"Quote:
Originally Posted by Master Shake
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.
That is in fact not the question.Quote:
Originally Posted by Master Shake
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.
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.
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.
@ MattH
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.Quote:
How about we start small. I claim that the following deck is optimal:
26 Swamp
35 Relentless Rats
It didn't take much to modify the script 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
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.Quote:
Originally Posted by Phoenix Ignition
claclaclap claclaclap hiiiiiii (\me on my white horse)Quote:
(help Maveric78f!)
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.
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.
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 ;-)).
@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
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
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).
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).Quote:
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).
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
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 ;-)
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
O RLY?Quote:
Originally Posted by Illissius
Come on.Quote:
Originally Posted by Illissius
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.
Statistics isn't dogma.Quote:
Originally Posted by Illissius
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.Quote:
Originally Posted by Kuma
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...Quote:
Originally Posted by Kuma
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.