Game Theory and the Best Deck

[Elric]

With the recent talk on 'best decks'?, I thought it would be worth it to articulate a game theoretic model of thinking about the best deck.  I don't believe that this describes magic in practice, but it's worth thinking about anyway.

Let's assume that everyone knows all possible payoffs (match expected win percentages, or 'win percentage'?) for all possible combinations of decks and players playing those decks.  These payoffs don't depend on whether you are expecting a certain deck or not'”i.e., once you sit down to play a match knowing what deck your opponent is playing doesn't help you (and vice versa). 

Each player's goal is to maximize his expected chance to win the tournament, which translates into maximizing his expected chance to win each round.  Players choose decks simultaneously and generally follow game theory's assumptions (rationality, perfect information, etc.).

It seems pretty clear that the best deck should follow some conception of 'if you saw this tournament metagame and then got to pick your own deck to enter the tournament with, which deck would you pick?'?  However, unless you're assuming that players are identical this doesn't have to be the same answer for every person. 

I'll assume that the payoffs in each matchup don't depend on the players playing the decks, which is the same as all players being identical.  If all players are identical, then you can drop the 'best deck for A'? notation since the best deck for A is the best deck for everyone and it can thus be considered the uniform 'best deck.'? 

If players are identical you know that each player will have an equal chance to win the tournament.  Optimal play for each player is to choose decks according to the same random rule (each player chooses deck 1 with set probability p1 [0,1], deck 2 with set probability p2, etc.).  If you look at all of the decks that a player will choose with a nonzero probability, his chance to win the tournament is the same conditional on choosing any one of those decks.
 
That's just an argument based on each person being identical and Magic being zero-sum (if you're interested in game theory details, it's essentially the Minimax Theorem).  Note that if optimal play is for each player to choose deck 1 with probability 60%, deck 2 with probability 30% and deck 3 with probability 10% that doesn't imply anything about which of these decks is the 'best'?- each deck will have an equal expected win percentage.  If each player's unique best strategy is to play the same deck with probability 1 then you have a "best deck" with no bad matchups: banning should ensue (from now on I'll assume this isn't the case).

Note that this doesn't imply that 60 Forests deck is just as likely to win the tournament as any other deck.  The 60 Forests deck is going to be totally inferior to at least one other deck so you can say with certainty that it won't be played (each player will choose it with zero probability) 

So ex ante you can say that there will be no unique best deck- there will be some number of "equally best decks'? and every deck that isn't one of these decks won't be worth playing.  'Ex ante'? means that you don't have any knowledge about what decks other players will choose.  You only have knowledge about the chance that each other player will choose each deck.  This doesn't mean that after you see the metagame each deck will have equal chances to win the tournament (read on). 

Now let's change that assumption and assume that you can see the tournament's actual metagame resulting from each player's choice of decks ('realized metagame M'?)(the tournament is completely described by its metagame- playing the tournament out adds no new information).  Once you see the metagame you can take every possible deck and calculate its expected matchup win percentage against the field.  This will (almost always) lead to a particular deck being the unique 'best deck given realized metagame M.'?   

NB: It's possible for the 'best deck given realized metagame '? deck to be a deck that no one would ever play before learning the realized metagame (one that each player chooses with probability zero).  Some decks that you'd never play in advance you'd also never play once you see the metagame while other decks that you'd never play in advance might be the best choice for a particular metagame. 

To see an example of this, consider the simple Rock-Paper-Scissors metagame where each deck splits against itself and (Rock beats Scissors 80% of the time, Scissors beats Paper 80% of the time, and Paper beats Rock 80% of the time).  Each player will have a 1/3 chance to bring each of Rock, Paper, and Scissors to the tournament.

Consider adding the strategy 'Crummy Rock'? as a new option.  'Crummy Rock'? is just like Rock but it only beats Scissors 70% of the time.  You can see that Crummy Rock is worse than Rock in every way- it can never be better than Rock but if anyone is playing Scissors it will be worse than Rock (if no one plays scissors it will be equal to Rock).  There is no 'realized metagame M'? for which 'Crummy Rock'? is a unique best deck.   

Now consider adding the strategy 'Rockier Rock'? instead of 'Crummy Rock'?.  'Rockier Rock'? is just like Rock but it beats Scissors 85% of the time and loses to Paper 95% of the time.  If you expect other players to choose Rock, Paper and Scissors with equal probability that there's no reason to pick 'Rockier Rock.'?  Thus, no one will play Rockier Rock.  However, if you looked at a metagame that was 10 Rock, 10 Scissors and 1 Paper then 'Rockier Rock'? would be the unique best deck.

There are all sorts of reasons why this game theoretic analysis won't hold in practice- in particular, players are not identical (and you don't know who will show up for each tournament) and no one knows exact matchup analysis for any matchup, let alone every matchup between every possible deck.  If you want to go against my initial assumptions even more, it's also the case that players are not solely interested in winning and players do not have unlimited access to cards.

That said, this is a pretty simple game theoretic model of the 'best deck.'? 

If you take a look at Grand Inquisitor's thread from last year on the 'incoherent' metagame, here, I'm pretty sure that walkingdude was thinking of the 'best deck'? along the same lines (and that whole thread is a useful resource).  Thoughts? 

[J J P]

The problem is that game theory is a model to determine the behavior of two players. Even for three players it gets to complicated to make useful predictions. Take 60+ players that don't play perfect builds and make play errors. Add the randomness of the game itself and of the initial pairings to that.

A theoretical apporach doesn't help to decide what to play because the problem is far too complicated. The best one can do is to make a guess on the metagame, playtest his deck, playtest it more and hope to be lucky.

[Tobi]

Magic is always random in some way, but that does not mean that theory is useless.

Theory helps to prepare for a tournament as good as possible. Deciding what deck to play, which cards to include and what to sideboard will increase your chances of placing well. And the more time you spend on this, the better your chances will be -before- the tournament.

Random factors as you describe will interfere with your theory, and you will probably not do well, because you have unlucky draws and opponents being lucksacks, but that does in no way eliminate the need to prepare for a tournament on an extended theoretical basis.

[J J P]

I think you are using the term 'theory' in another context than me I tried to point out that the concept of game theory should not be applied to a tournament because trying to calculate anything useful would be way to complicated. I didn't dismiss what you could call theoretical preparation for a tournament.

[AmbivalentDuck]

I think you are using the term 'theory' in another context than me I tried to point out that the concept of game theory should not be applied to a tournament because trying to calculate anything useful would be way to complicated. I didn't dismiss what you could call theoretical preparation for a tournament.

"Too complicated" is never a scientific argument.  Complex Systems is, however, a field of mathematics.

[Klep]

The problem is that game theory is a model to determine the behavior of two players.

No, it's not.  Problems may get complex above 2 players, but that doesn't mean game theory can't model them.  If game theory was only useful when dealing with 2 players, a lot of economists wouldn't have jobs.

@Elric: It actually gets even simpler.  Given a known metagame (and holding all players of equal and sufficient skill to play their decks), the "best deck" is simply the best response to the other players' strategies.

The only reason applying this kind of thinking in practice doesn't work is because the variables are prohibitively complex in order to accomplish the analysis in any reasonable amount of time.  Nonetheless, the knowledge that it can be done means that metagaming is something that is worthwhile to do.

Incidentally, in your example, I would much prefer regular Rock to Rockier Rock.  Given that Scissors vastly outnumbers Paper, I would be perfectly willing to sacrifice 5% in the Paper matchup to gain 15% in the Scissors matchup.  If the numbers of Paper and Scissors were reversed, then Rockier Rock would be the better of the two.

[Elric]

@Elric: It actually gets even simpler.  Given a known metagame (and holding all players of equal and sufficient skill to play their decks), the "best deck" is simply the best response to the other players' strategies.

Incidentally, in your example, I would much prefer regular Rock to Rockier Rock.  Given that Scissors vastly outnumbers Paper, I would be perfectly willing to sacrifice 5% in the Paper matchup to gain 15% in the Scissors matchup.  If the numbers of Paper and Scissors were reversed, then Rockier Rock would be the better of the two.

You're right about the "best response given realized metagame M" part—what I was doing was illustrating how you'd find what that best response is.  Once you see the metagame you don’t care how players picked their decks- you only care about what those deck choices are (the “realized metagame M�).

You can't simply look at the decks in "realized metagame M" to find a "best deck given realized metagame M."  You also can't even look at the decks that ex ante you know will be played with a nonzero probability to find the “best deck given realized metagame M.�  You also have to consider decks that you would play with probability 0 if you had full probabilistic knowledge of the metagame but no knowledge of the “realized metagame.�

Also, unless I typed something wrong, Rockier Rock (compared to Rock) gains 5% in the Scissors matchup and loses 15% in the Paper matchup.  Thus, since there are 10 Scissors and only 1 Paper you want to play Rockier Rock more than Rock in the example given.

[Klep]

You can't simply look at the decks in "realized metagame M" to find a "best deck given realized metagame M."  You also can't even look at the decks that ex ante you know will be played with a nonzero probability to find the “best deck given realized metagame M.â€?  You also have to consider decks that you would play with probability 0 if you had full probabilistic knowledge of the metagame but no knowledge of the “realized metagame.â€?

In practice, no you can't.  In theory, none of that really matters though.  You talk about the meta as though it were an actual realized thing.

Also, unless I typed something wrong, Rockier Rock (compared to Rock) gains 5% in the Scissors matchup and loses 15% in the Paper matchup.  Thus, since there are 10 Scissors and only 1 Paper you want to play Rockier Rock more than Rock in the example given.

Since Rock beats Scissors all of the time and Paper never normally, a version which only beats Scissors 85% of the time but beats Paper 5% of the time is at a disadvantage to normal Rock in a field which has a lot more Scissors than Paper.

[Elric]

You can't simply look at the decks in "realized metagame M" to find a "best deck given realized metagame M."  You also can't even look at the decks that ex ante you know will be played with a nonzero probability to find the “best deck given realized metagame M.�  You also have to consider decks that you would play with probability 0 if you had full probabilistic knowledge of the metagame but no knowledge of the “realized metagame.�

In practice, no you can't.  In theory, none of that really matters though.  You talk about the meta as though it were an actual realized thing.

Any specific set of decks (10 Rock, 20 Scissors, 20 Paper) is an actual realized thing.  Before you see the specific set of decks that result from player decisions, player 1 is making decisions that depend on how other players are making decisions and player 2 through player N are doing the same thing, but no one can see how the other players are making decisions (or the results of those decisions).  That's what the Minimax theorem is for—it tells you the rules that govern optimal decision-making strategies for each player in a situation like this. 

Since Rock beats Scissors all of the time and Paper never normally, a version which only beats Scissors 85% of the time but beats Paper 5% of the time is at a disadvantage to normal Rock in a field which has a lot more Scissors than Paper.

I wasn't having the default be for Rock to beat Scissors 100% of the time:

To see an example of this, consider the simple Rock-Paper-Scissors metagame where each deck splits against itself and (Rock beats Scissors 80% of the time, Scissors beats Paper 80% of the time, and Paper beats Rock 80% of the time).

[Jacob Orlove]

(the tournament is completely described by its metagame- playing the tournament out adds no new information). 

This is the key flaw in the entire analysis. It is simply not true. In fact, the actual execution of the tournament produces far more relevant information than is contained in the entire metagame analysis--pairings, die rolls, opening hands, and similar.

[Klep]

Any specific set of decks (10 Rock, 20 Scissors, 20 Paper) is an actual realized thing.  Before you see the specific set of decks that result from player decisions, player 1 is making decisions that depend on how other players are making decisions and player 2 through player N are doing the same thing, but no one can see how the other players are making decisions (or the results of those decisions).  That's what the Minimax theorem is for—it tells you the rules that govern optimal decision-making strategies for each player in a situation like this. 

There is no guarantee that other players are choosing their decks optimally.  You are only concerned with 1 optimal choice given the choices made by other players, whether the other players chose optimally or not.

I wasn't having the default be for Rock to beat Scissors 100% of the time:

To see an example of this, consider the simple Rock-Paper-Scissors metagame where each deck splits against itself and (Rock beats Scissors 80% of the time, Scissors beats Paper 80% of the time, and Paper beats Rock 80% of the time).

Ah, I missed that.

[warble]

Well, I'd rather play the deck called CS that is tuned to beat this deck it tends to play against in tournaments...CS.  So that would be...rock against rock?  I mean, nobody can tell me maindeck tormod's crypt or lava dart doesn't pwn this matchup, and that is 1 card.  1 CARD!  These tutors, they're making your analysis using game theory quite useless.  Sure, if we wanted to switch decks we "might" stumble upon one that has anywhere between 2 to 5 percent added bonus dependant on the metagame...but that one card we change might have a similar effect.  Saying a deck of 60 cards can't change and adapt to a metagame, and that a completely new deck of 60 cards is necessary is just...plain...wrong...we have a billion homebrews of CS, probably 5 to 10 accepted versions to run, and still nobody knows what the best deck to play in any tournament is.  We guess, and we hedge our bets by using cards we have tried and tested and that have given us the best results.  Sure, if we want to win a larger tournament we may pick a deck with a faster clock, but who's to say that is right?

I'd prefer if these assumptions about the game were not made at all.  Game theory is not only a rigorous model, it is a model that needs a firm set of assumptions to be even marginally correct.  And even then we won't know with more than a good degree of certainty if our decisions have any benefit at all.  Magic, and fun playing magic, doesn't need to be hampered with game theory.  True tuning of a deck requires good decisions card by card, decisions in the sideboard card by card.  Yes, it's great to get a deck from the net, but those IDEAS are not decks, those are inspirations to go out and construct something that isn't that deck, but that is BETTER.

In conclusion, game theory would lead us to stagnant waters.  Be like the river and flow, change cards in every deck you play.  Metagame like a fiend, put terrible cards like abeyance in your deck and suddenly discover it's the crux of the deck, up the squee+spark count in dragon to 8, see what you can truly accomplish.  If we wanted to change decks constantly, we could play net versions of these decks and try to "metagame our souls".  But I'm not going to do that, I'm going to do the best to better the decks that I already chose because I like them, and they're friggin' amazingly broken.  Don't tell the best TPS player in the world to play Workshop Slavery because there's a Dragon/Fish infestation, see what that player can tweak to allow TPS to compete!  I truly believe at the start of every large tournament there are at least 10 players that could wind up winning the thing.  They don't all have the same deck, and hence your argument has a flaw.  Everyone takes a 50/50 deck into the fray, and the winner is not only skilled, but lucky as well (but we knew that already).

[J J P]

The problem is that game theory is a model to determine the behavior of two players.

No, it's not.  Problems may get complex above 2 players, but that doesn't mean game theory can't model them.  If game theory was only useful when dealing with 2 players, a lot of economists wouldn't have jobs.

Well, I believe many people were better of if a lot of economists wouldn't have jobs. But that is something not to be discussed here.

I draw my very limited amount of knowledge from the book Rules of Play (which I totally recommend to people who have interest in games beyond just playing them). Therefor I am a little biased about game theory, I'll quote just a few lines:

Game theory is a mathematical study of decision making. It looks at how people behave in specific circumstances that resemble very simple kinds of games. [...] Although it caused quite a sensation when it was introduced, the promises of game theory were never quite fulfilled, and it has largely fallen out of favor as a methodology within economics. [...]

I mean, you don't even agree on a basic rock, scissors and papers example.

[Klep]

Magic, and fun playing magic, doesn't need to be hampered with game theory. 

When we talk about Magic theory, we aren't primarily talking about having fun playing Magic, we're primarily talking about winning at Magic.  Knowing a best deck exists is very important if your goal is to win.

In conclusion, game theory would lead us to stagnant waters.

No, it wouldn't, because applying game theory to practical play is infeasible.  It merely presents us with a framework in which we can talk about metagaming with regards to Magic theory.  Unless someone out there is doing a dissertation's worth of work on how to practically apply game theory to something as complex as Magic, you have nothing to fear.

@JJP
My studies of game theory having been substantially more involved than yours, I can tell you game theory has applications in a wide number of areas.  There is a research group here, for example, applying it to matters involving software radios.  It is quite a powerful tool, and is a useful perspective for a lot of problems.  What better a place to apply it than with regards to an actual game?  As for the rock, paper, scissors thing, we only disagreed because I missed something he said in the setup of the problem.

[Elric]

(the tournament is completely described by its metagame- playing the tournament out adds no new information). 

This is the key flaw in the entire analysis. It is simply not true. In fact, the actual execution of the tournament produces far more relevant information than is contained in the entire metagame analysis--pairings, die rolls, opening hands, and similar.

Jacob- that’s not a flaw, that’s a feature

Specifically, that’s mainly the result of the assumption that players have perfect information about decks.  Obviously perfect information is a very strong assumption and won’t be true (or even necessarily close to true) in real life.  If it is true, though, then playing the games out won’t give you any additional information because you already know everything that there is to know.

Warble- a CS deck that has 59 cards the same as another CS deck and 1 card different (with the same sideboard) is considered a different deck in this analysis.  Perfect information says that you will know exactly how this 1 card change affects all possible matchups against every possible deck (again, it’s a very strong assumption).  A deck that is completely identical to another deck has a 50% match win percentage, but the same isn't true for decks that are only 1 card apart.  I’m literally using all possible different decks, not just all deck archetypes (“deck� is often used to mean “deck archetype�). Sorry if this wasn’t clear.

Klep- I don’t think the assumption that players' strategies follows the Minimax theorem is that hard to make compared to the other assumptions (like perfect information) that are much, much stronger than what we’d expect in real life.  By comparison, there’s evidence that professional sports players play very close to Minimax so skilled magic players should be able to do something similar.

[Klep]

Klep- I don’t think the assumption that players' strategies follows the Minimax theorem is that hard to make compared to the other assumptions (like perfect information) that are much, much stronger than what we’d expect in real life.  By comparison, there’s evidence that professional sports players play very close to Minimax so skilled magic players should be able to do something similar.

In general, you shouldn't make any assumptions you don't have to for simplicity's sake.  The method of finding the best deck (best response to the other decks) is no different whether other players are picking optimal decks or not, so there is no need to assume that they will.

[Elric]

In general, you shouldn't make any assumptions you don't have to for simplicity's sake.  The method of finding the best deck (best response to the other decks) is no different whether other players are picking optimal decks or not, so there is no need to assume that they will.

When you want to solve for every players’ method of choosing decks at the same time and have that set of methods be a Nash equilibrium (where no player has any incentive to change the way he is choosing decks), you get Minimax (well, mixed strategy equilibrium) play as a result.  This is a central result of game theory.

[Klep]

When you want to solve for every players’ method of choosing decks at the same time and have that set of methods be a Nash equilibrium (where no player has any incentive to change the way he is choosing decks), you get Minimax (well, mixed strategy equilibrium) play as a result.  This is a central result of game theory.

Why do we want to do that?  I thought we were just looking for the best deck, not a Nash Equilibrium.

[Whatever Works]

I have always used a rather unique formula for calculating deck "Ratings" for a predicted metagame.

My rating system formula is:

Your Decks approx game win % vs. _____ deck (based generally on a minimum of 30 games 20 postboard)
Repeat this proccess for all the major decks in the format that you expect to see...

Lets say you did this for 8 decks... You take each of the 8 totals... and multiply them by the predicted % of the expected metagame... Add the total values and divide by the # of the decks to create an efficiency factor...

Say I was to try and find the rating for say Control Slaver... The equation would look like this. (The %'s are 100% made up)

Deck 1 Win 45% Total Estimated Percent of Metagame 25% Total Value = 11.25
Deck 2 Win 70% Total Estimated Percent of Metagame 20% Total Value = 14
Deck 3 Win 80% Total Estimated Percent of Metagame 15% Total Value = 12
Deck 4 Win 30% Total Estimated Percent of Metagame 10% Total Value = 3
Deck 5 Win 40% Total Estimated Percent of Metagame 10% Total Value = 4
Deck 6 Win 50% Total Estimated Percent of Metagame 10% Total Value = 5
Deck 7 Win 55% Total Estimated Percent of Metagame 5% Total Value = 2.75
Deck 8 Win 60% Total Estimated Percent of Metagame 5% Total Value = 3

Deck Overall Value Score = 55

I often use this system to rank decks in the format, but if you are unable to guess the metagame correctly the #'s can heavily vary. Also, just because you guess 100% accurate and a deck you have an 80% matchup that is 50% of field... There is no guarantee that you couldnt have the unfortunate luck of never playing vs. that deck.

Kyle L

[Elric]

When you want to solve for every players’ method of choosing decks at the same time and have that set of methods be a Nash equilibrium (where no player has any incentive to change the way he is choosing decks), you get Minimax (well, mixed strategy equilibrium) play as a result.  This is a central result of game theory.

Why do we want to do that?  I thought we were just looking for the best deck, not a Nash Equilibrium.

When you don't have a Nash equilibrium you also have to make additional assumptions about what everyone's method of choosing decks is.  When you have a Nash equilibrium, you can simply calculate that method from the payoff matrix.  Game theory predicts that rational individuals will play according to mixed strategy (Nash) equilibrium.

If you want you can have a "best deck ex ante given perfect information, identical players, and non-rational players making choices according to a new set of assumptions."

Whatever Works- this is almost exactly what you’d do to find a best deck if you had knowledge of a “realized metagame M� and perfect information about game win percentages (you’d go through this procedure for each deck and pick the one with the highest total).  If you had perfect information you’d pick the deck with the highest overall expected match win percentage (the match win percentage against each deck depends on the average chance you win game 1, the chance you win a sideboarded game when playing first and the chance that you win a sideboarded game when playing second).   

[Klep]

When you don't have a Nash equilibrium you also have to make additional assumptions about what everyone's method of choosing decks is.  When you have a Nash equilibrium, you can simply calculate that method from the payoff matrix.  Game theory predicts that rational individuals will play according to mixed strategy (Nash) equilibrium.

No, actually.  When you aren't looking for a Nash Equilibrium you aren't making any assumptions about what other people play.  You're allowing them complete freedom in chossing a deck, whether they choose optimally or not.  This isn't a mixed strategy game, remember, and the best deck is defined with respect to perfect information (which is to say, a known meta).  Players can't pick a deck that is one thing half the time and another thing the other half of the time.  You pick one deck for a tournament and you have that deck for the entire tournament.   

The key thing is that the best deck is not necessarily part of a Nash Equilibrium.  It is merely the best response to the metagame.  That may result in an NE, and it may not.  If everyone else in the meta is playing Suicide, for example, no deck you choose would result in an NE.  If other players then changed their decks, you might end up in one, but they can't change their decks because the condition we set to define the best deck is that the metagame is fixed.

[Elric]

If you know the metagame as in "know realized metagame M" then you don't need to worry about the assumptions under which your opponents made their deck choices because those choices have already been made.  It doesn't matter one way or the other.  (A "deck vector" that defines a "realized metagame M" is almost certainly not a Nash equilibrium).

That's not the set of assumptions I'm working with throughout, though.  I'm using perfect information to mean that you know all of the entries about the payoff matrix in question, not that you know the “realized metagame M.�  If you want to solve for the best deck before you can see anyone's choices, then assuming rational identical players taking simultaneous actions (and perfect information about the payoff matrix) leads to the conclusion that each player will play identical mixed strategies and choosing any deck that a given player chooses with positive probability gives that player an equal chance to win the tournament as every other deck that he chooses with positive probability. 

This result isn’t particularly surprising given the assumptions.  The set of players' (possibly degenerate) mixed strategies about what deck to play is a Nash equilibrium. 

[Klep]

If you want to solve for the best deck before you can see anyone's choices, then assuming rational identical players taking simultaneous actions (and perfect information about the payoff matrix) leads to the conclusion that each player will play identical mixed strategies and choosing any deck that a given player chooses with positive probability gives that player an equal chance to win the tournament as every other deck that he chooses with positive probability. 

But you can't know this, and so the point is moot.  This is why game theory tends to have problems when dealing with people.  People aren't rational.  They have biases that lead to them acting in ways that aren't in their own best interest, particularly in Magic, where people tend to very stubbornly stick to decks that aren't very good (just look at the Rock's legendary popularity in Extended). 

[Elric]

But you can't know this, and so the point is moot.  This is why game theory tends to have problems when dealing with people.  People aren't rational.  They have biases that lead to them acting in ways that aren't in their own best interest, particularly in Magic, where people tend to very stubbornly stick to decks that aren't very good (just look at the Rock's legendary popularity in Extended). 

People don't have equal skill so that chance to win a matchup depends on who is playing it.  Even if people had equal skill, people don't have complete and total knowledge of the percent chance to win a match playing every possible deck against any other possible deck.  If they did, and they cared only about winning, they probably wouldn't play a lot of decks that game theory say they should play with probability 0.  But wait- maybe people don't solely care about winning... and so on. There are tons of assumptions that I've made that don't actually hold completely.

This is game theory- it isn't a complete and total description of reality.  I mentioned this at the end of the initial post.

[Klep]

I'm just saying you can't use that assumption to pinpoint the best deck, because the best deck is independent of whether or not the other players choose their deck according to their best interests.  It only relies on what decks they ultimately choose.

[forests failed you]

The best deck is usually the one in the hands of the best player in the room.

[AmbivalentDuck]

As an experimental measure with no analytical basis, T2 tends to have clear "best decks:" Rebels, Broodstar Affinity, Ravager Affinity, etc.  These decks pinpoint broken cards/mechanics and exploit them better than other decks in the environment.  A stable metagame then starts developping around these "best decks."

In game theoretical terms, I don't think I'm out of line to suggest that it's a clear case of a Nash Equilibrium.  Every time a new set is added, T2 is shaken up.  Then pros invest amounts of time unequaled in the T1 community to find the best decks over a much more compact series of tournaments with public results.  The metagame often reaches a stable equilibrium thereafter.  Even when idiots bring trash decks, they don't defeat the Nash Equilibrium.  They merely prove its effect by losing. 

I also feel comfortable suggesting that the difference between T2 and T1 is the size of the cardpool and the power level.  It's easier to find an equilibrium when the selection pressure on the deck itself is more pronounced.  In T1, it's very subtle:

The best deck is usually the one in the hands of the best player in the room.

[Whatever Works]

The best deck is usually the one in the hands of the best player in the room.

I feel that way every time I am in the same T1 Tournement as Rich Shay. No matter what the metagame is like... I would take him playing CS over the perfect metagame deck (which unfortunetly is not High Tide... 3 land belcher... UB Fish... or whatever I have ever won with)...

[Klep]

The best deck is usually the one in the hands of the best player in the room.

When that is the case, that is merely a function of the best player being wise enough to pick the best deck.  As I have stated innumerable times, the identity of the best deck is independent of performance.

In game theoretical terms, I don't think I'm out of line to suggest that it's a clear case of a Nash Equilibrium.

This is almost never the case.  For it to be a Nash Equilibrium, you would have to have a situation where no player could improve his performance by changing his deck.  Given the number of people who play, say, Rock in Extended or  Eye of the Storm in Standard, you can not pretend that you have a Nash Equilibrium in those formats. 

[Elric]

The best deck is usually the one in the hands of the best player in the room.

When that is the case, that is merely a function of the best player being wise enough to pick the best deck.  As I have stated innumerable times, the identity of the best deck is independent of performance.

No, these are two different things.  To me, it seems clear that forcefieldyou is saying that the person who has the highest chance to win a Magic tournament is usually the person in the room who is the best at playing Magic and that the main reason for this is his skill.  His skill translates into higher win percentages than the rest of the field would have even if they all played the exact same deck as him- it's not necessarily the act of having the exact right deck based on what other decks are at the tournament.  See Whatever Works’ point about Rich Shay for an example of this. 

If you assume that players are identical, then by definition there is no "best player in the room" in any sense that is independent of deck choice.  In this case, the player who has the deck with the best match win percentage given the rest of the field will have the highest chance to win the tournament.  I'm not sure where the "independent of performance" part enters into any of this.

There’s no indication that this deck is the best possible deck to bring given the metagame.  It just happens to have the highest chance to win the tournament among the decks at the tournament.  I’d call this “the best deck within realized metagame M.�

And Klep- let's call AmbivalentDuck’s proposal "Nash Equilibrium of deck choice strategies with some dumb deck choice strategies added on" and leave it at that.  Obviously, if even a single person plays a deck when he could have played a deck that is in every way better, we’re not in Nash equilibrium.  That doesn’t mean that if I show up with a deck of 60 Forests deck the concept of Nash equilibrium is now completely inappropriate forever.