You cannot see the forest for the trees.Originally Posted by Maveric78f
To quote an earlier post of mine in this thread:
"Playing a 61st card means you are less likely to draw what matters."
That is variance by definition.
Suddenly, Fluffy realized she wasn't quite like the other bunnies anymore.
-Team R&D-
-noitcelfeR maeT-
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?
@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.
@ Rico Suave
Perhaps you've mistaken Nihil's point (read page one again please) and the definition of 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."Playing a 61st card means you are less likely to draw what matters."
That is variance by definition.
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
Admittedly, this is too simple. It is still interesting, and may still bear fruit. Here's how I think of this thought experiment: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.
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
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.
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).
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.I'm quite sure that the deck made of 30 card1, 30 card2, 1 card3 is optimal.Card3 %100000
Artifact
When Card3 enters the battlefield, you win the game. This ability cannot be countered.
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)
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).
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.
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).
I was saying your articulation of my argument wasn't quite accurate.
As far as variance is concerned, that was already explained in Chapin's article. It just happens to go hand in hand with what I'm saying too.
But you will say "card X in my deck wasn't good at all and was pretty useless in a lot of situations I drew it."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.
Then you'll realize that it's probably not that good of a card.
And without flaming, then eventually you'll become a better player and cut the cards that aren't that good.
Then maybe you will say "oh damn I am sure that I didn't need card X at all, and I'm better off without it."
=)
Suddenly, Fluffy realized she wasn't quite like the other bunnies anymore.
-Team R&D-
-noitcelfeR maeT-
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.
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.
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.
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 or the perfect metagamed deck proposed by 4eak at the end of this post, 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.
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.
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.
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.
If drawing your best card doesn't increase your chances of winning a game, I don't know what does.
Last edited by Kuma; 12-17-2009 at 05:23 PM.
"Casual sex, NO Touching/Licking/Sucking/Groping/Fondling/Riding/Tickling/Binding/Quitters/Italians. Play Fast, Be Polite, Have Fun."
Sure as hell sounds like fun.
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.
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.
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?
Then the card you cut wouldn't really be the "worst" card now would it?
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.
"Casual sex, NO Touching/Licking/Sucking/Groping/Fondling/Riding/Tickling/Binding/Quitters/Italians. Play Fast, Be Polite, Have Fun."
Sure as hell sounds like fun.
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