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!
Originally Posted by tsabo_tavoc
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.
Team Technology - Think it's good? Prove it.
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.
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.
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?
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.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 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.
There are about 3 pages of the statistical reasoning in this thread as to why 60 is the superior number. However, as usual, we ignore what has been put on previous pages because it hurts our argument.
The "to long didn't read" version is: I want to see cards X, Y and Z asap. I run as few other cards as I possibly can that allow me to have the most streamlined game plan. One card changes this by dramatically small amounts but in the end it will affect a game somewhere some how and that is not something we want. The argument against the 61st silver bullet is because it can just as easily have been a card in your maindeck or sideboard if the deck was tuned tighter.
Team Technology - Think it's good? Prove it.
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.
This is what I'm trying to argue against. Somehow this thread devolved from "what, if anything, should I cut from this survival deck" to "a 61 card deck has to be inferior and there's proof". In general, I don't think that 61 card decks are good. But the argument above just don't hold water. A situation may exist where there is a weakest card in your deck and you *shouldn't* cut it.
Anyways Forbiddian, you're right, in terms of burden of proof, if you have a 61 card deck, cut a card. You're 99.999% doing yourself a favor. I guess I'm just cringing at the people who say "well clearly 61 cards is wrong and here's a proof!" We haven't really proved that.
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.
@ Forbiddian
Showing the inaccuracy of our personal testing was a good reason to draw randomly instead of going through the entire search space.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.
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
@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.
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.
For my confessions, they burned me with fire/
And found I was for endurance made
Algorithm.
The algorithm is clunky and it would be much easier with a backtracking program language such as Prolog.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.
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.
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. 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
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.
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.
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.
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
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:
Shock/mountain deck.
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.
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