Game Theory and the Best Deck

[AmbivalentDuck]

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.

That or play skill matters more than deck choice (within reason).  Some of Buehler's card choices in Goat have since been considered suboptimal at best.  At the same time, his play skill is undoubtable and his results are accepted fact.

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. 

*shrug* It's situational.  Notice how I cited Rebels, and Affinity back when it was clearly broken.  Without high levels of selection pressure, the "best" strategy and choices are far from obvious.  Also, players making bad choices don't preclude a Nash Equilibrium.  The existence of the equilibrium is very separate from whether or not it actually occurs.  It's certainly more likely to occur with more selection events and higher selection pressure.

With T1's low selection pressure (on decks) and relatively few public selection events, it's clear why an equilibrium seldom emerges before a new set shakes everything up.  Even if it existed (an equilibrium, Nash or otherwise), why would be expect to see it?  Hand a random n00b Ravager Affinity with Skullclamps and hand a pro any other T8 deck from that era.  I'd feel safe betting on the n00b.  Conversely, pair a n00b playing (insert tier 1 deck) against Smennen playing (insert tier 2 deck)... I'd put my money on Smennen.

EDIT:  Nash Equilibriums aren't useful in biology because they happen, they're useful because they predict the behavior of an evolving system over time.  Similarly, even though player A is a perpetual Sui player who goes to *every* tournament, it doesn't mean that his behavior makes the theoretical Nash Equilibrium any less useful in predicting tournament representation of decks.  Rather, it just means that the tournament itself is not at a Nash Equilibrium.

[Klep]

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.

Then he should say that instead of attempting to corrupt the concept by making it seem like it is something it's not.

I'm not sure where the "independent of performance" part enters into any of this.

Because the definition of the best deck is probabilistic.  It is not guaranteed to win the tournament, it merely has the highest chance of doing so.  How many times do I have to repeat 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.

The latter implies the former.

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. 

No.  Either it is a Nash Equilibrium or it is not.  I will not accept using the term Nash Equilibrium to describe a situation which is generally not even close to one.

That or play skill matters more than deck choice (within reason).  Some of Buehler's card choices in Goat have since been considered suboptimal at best.  At the same time, his play skill is undoubtable and his results are accepted fact.

That may be, but that's irrelevant to the identity of the best deck, so it doesn't matter for the purposes of this discussion.

Also, players making bad choices don't preclude a Nash Equilibrium.

Yes it does, because by definition, for a Nash Equilibrium to exist players must be making their best choices.

[AmbivalentDuck]

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.

No. Either it is a Nash Equilibrium or it is not. I will not accept using the term Nash Equilibrium to describe a situation which is generally not even close to one.

The existence of a Nash Equilibrium is *not* influenced by whether or not a system displays that behavior in practice.  A simple situation to illustrate this: Two players must choose between "red" and "green" fifty times.  If they both choose "red," they get fifty dollars each.  Otherwise, they get nothing.  Player B is a sadistic masochist and always chooses green.  The Nash Equilibrium for the game is clear and clearly existent despite Player B's behavior.  Whether or not the system reaches that equilibrium does not affect the equilibrium's existence.  The same is true of a system where payoffs for various options are pre-defined but not initially disclosed.  Average trials to Nash Equilibrium will correlate directly with the complexity of the payout system and the intelligence of the players.  Average time to equilibrium will inversely correlate directly with trial frequency.

That or play skill matters more than deck choice (within reason). Some of Buehler's card choices in Goat have since been considered suboptimal at best. At the same time, his play skill is undoubtable and his results are accepted fact.

That may be, but that's irrelevant to the identity of the best deck, so it doesn't matter for the purposes of this discussion.

It doesn't affect the identity of the "best deck," just how quickly we can find it.  If we can't find the "best deck" before a new set is released...

Also, players making bad choices don't preclude a Nash Equilibrium.

Yes it does, because by definition, for a Nash Equilibrium to exist players must be making their best choices.

See above.  The Equilibrium exists despite dumbassedness; however, it's unlikely to *occur* in the presence of dumbassedness.

[Klep]

Also, players making bad choices don't preclude a Nash Equilibrium.

Yes it does, because by definition, for a Nash Equilibrium to exist players must be making their best choices.

See above.  The Equilibrium exists despite dumbassedness; however, it's unlikely to *occur* in the presence of dumbassedness.

Poor choice of words.  For players to be in a Nash Equilibrium they must be making their best choices.  Since players so frequently do not make the best possible choice, there is no point in looking for a Nash Equilibrium when attempting to find the best deck.

[PucktheCat]

Does a Nash equilibrium really depend on your opponents making optimum decisions?

I think that some randomness in decision making can be valuable even when your opponent is an idiot.  As long as they show any tendancy to learn whatsoever predictability can be used against you.  Imagine a metagame where there were only four choices:

Sligh
1996 Keeper
Oath (modern version)
Control Slaver (again, modern version)

Obviously Sligh and old, bad Keeper are going to be poor choices and a good player will always choose Oath or CS.  But imagine if all of your opponents were bad, bad players and only chose either Sligh or Keeper.  Which deck do you choose?  It seems likely that Oath will win more in an all Sligh meta and CS will win more in an all Keeper meta (even if the difference is between an 80% chance of winning and a 95% chance).  As long as your opponents are clever enough to pick up on that difference eventually and start to attack your "weakness" (or relative lack of strength) you would be well advised to mix it up a bit in order to avoid becoming predictable.

Is that a Nash equilibrium, or is it simply a closely related phenomenon?

Leo

[Elric]

Ok, Klep let's give the arguments over terminology a rest.  I don't care if your definition of the "best deck" isn't true when other people don't make all of the same assumptions that you do- that's no reason to suddenly find fault with everything they say.  You can also stop repeating the "best deck in a probabilistic sense" part because there's no evidence that people are thinking about it in a non-probabilistic sense.  

Also, the deck that has the highest chance to win the tournament out of all the decks being played at the tournament is not always the deck that is the best possible deck given the metagame.  Think of my 10 Rock decks, 10 Scissors decks, 1 Paper deck example where the best deck to play given that you know you are going to be playing against these 21 decks is "Rockier Rock" but the best deck being played out of these 21 decks is "Rock" (and 10 people are playing it). 

In advance, you may have a 0% chance to play a deck and it still might end up as the unique best possible deck to play once you know the 21 (or however many it is) other decks that you're going to be up against.

Puck- if your opponents were choosing Slight and Keeper when those decks were each strictly inferior to Oath and/or Slaver, then they would have an incentive to change their strategy and it wouldn't be a Nash Eq.  If they for some reason didn't have the option to choose Oath or Slaver, then there might be a NE where your strategy is to play Oath 40% of the time and Slaver 60% of the time and their strategy is to play Slight 70% of the time and Keeper 30% of the time (I made those specific numbers up).

[Klep]

Does a Nash equilibrium really depend on your opponents making optimum decisions?

As a Nash Equilibrium requries that no one be able to perform any better by changing their strategy, yes.

I don't care if your definition of the "best deck" isn't true when other people don't make all of the same assumptions that you do- that's no reason to suddenly find fault with everything they say.

The definition of the best deck I gave is the definition of the best deck.  Any other definition is erroneous.  If you allow a player's playskill or luck to factor in, you are no longer just considering qualities of the deck, you are also considering qualities of the player.  The player is not a part of the deck, and thus qualities inherent to the player cannot be factored in to determining the best deck.  Assumptions must therefore be made in the definition which remove the player from the equation.

Also, the deck that has the highest chance to win the tournament out of all the decks being played at the tournament  is not always the deck that is the best possible deck given the metagame.

There are two usages of the word metagame.  One is the general sense as to what decks are viable in a format and are likely to see play.  The other is the actual decks that are present at a tournament.  The definition of the best deck uses the latter.

In advance, you may have a 0% chance to play a deck and it still might end up as the unique best possible deck to play once you know the 21 (or however many it is) other decks that you're going to be up against.

Absolutely.  This is the essence of why metagaming is hard.  You have to know the future to do it properly.

[PucktheCat]

So, if I read this right, even if there is no mixed-strategy/Nash equilibrium you can still have an incentive for some random decision making.

In other words the requirements for a Nash equilibrium =/= the requirements for multiple "best decks" to exist for a given event.

The practical outcome of all this being that even if my opponents are a bit slow and everyone is playing bad decks I might still want to switch between Welder Stax and non-Welder Stax pretty much with no reason at all, just to keep people on their toes.  Which, if you'll recall, was the point I tried (poorly) to make in the old thread.

Leo

Edited reply to Klep:

Absolutely.  This is the essence of why metagaming is hard.  You have to know the future to do it properly.

You have said this, apparently without irony, several times in this discussion.  I simply don't understand what you are suggesting as a practical strategy?  Divination?  Hacking team message boards?  Posting about a deck to draw attention to it and then playing its foil?  All of these but one are of extremely limited reliability and the other one doesn't really exist.  At some point you reach the limit of what the information available to you can tell you.  Some people might reach that limit sooner than others (I beleive this is the skill you are referring to), but there is always a limit.  Beyond that point, when you are dealing with players you don't know, or a player you do know who plays a number of decks without any discernable pattern, or a player who is trying to gauge the metagame before committing just like you are, the veil descends.  Theoretically, as you have said, after all those players have made their decisions and their lists are revealed there is a "best deck."  But at that time, when your information is limited (as it always is) there is, also theoretically, a best strategy based on that limited information.  That strategy may be the best deck, or some other deck, or an array of decks that should be selected from according to a some random distribution.  And if you use that strategy you are making the correct choice, regardless of whether later, when you have more information, another strategy turns out to be more likely to win the tournament.

[Elric]

If you're removing the player from the equation when finding a "best deck" then you have to accept the fact that when players are not identical (so players do matter in the equation) then the result you get for a “best deck when players are identical� doesn't imply anything about what the “best deck for player A� is.  I'm fine with that, but it bears repeating.

If you’re trying to find the “best deck for player A� (where player A is different from other players) rather than a “best deck when players are identical� then you can’t assume that players are identical.

When you refer to "the metagame" I refer to as "realized metagame M" while what you refer to as the "general sense of what decks can see play" I describe as the match win percentages of all possible decks against each other.

Also, what you refer to as "knowing the future", I refer to as "being lucky"

[Klep]

If you're removing the player from the equation when finding a "best deck" then you have to accept the fact that when players are not identical (so players do matter in the equation) then the result you get for a “best deck when players are identicalâ€? doesn't imply anything about what the “best deck for player Aâ€? is.  I'm fine with that, but it bears repeating.

That's the difference between theory and practical application of theory.

[AmbivalentDuck]

Also, players making bad choices don't preclude a Nash Equilibrium.

Yes it does, because by definition, for a Nash Equilibrium to exist players must be making their best choices.

See above.  The Equilibrium exists despite dumbassedness; however, it's unlikely to *occur* in the presence of dumbassedness.

Poor choice of words.  For players to be in a Nash Equilibrium they must be making their best choices.  Since players so frequently do not make the best possible choice, there is no point in looking for a Nash Equilibrium when attempting to find the best deck.

Poor choice of words.  
-"For players to be in a Nash Equilibrium they must be making their best choices." True

-"Since players so frequently do not make the best possible choice, there is no point in looking for a Nash Equilibrium when attempting to find the best deck." False.  

If at least two players do care about winning and make intelligent decisions, % of top finishes should more closely approximate the Nash Equilibrium over time.  Here as in evolutionary biology, the Nash Equilibrium can predict the long term behavior of a system.  They refer to this as "Evolutionarily Stable Strategies."  Since linking to literature on the subject could do nothing but bog down response rates and move the discussion further from Magic and more toward math, I was hesitant to do it.  But, even just looking at the wikipedia entry might help to focus this:

http://en.wikipedia.org/wiki/Evolutionarily_stable_strategy#Nash_equilibria_and_ESS

There are conditions for this. But much of the time a strict Nash Equilibrium exists, the "meta" will evolve towards it.  Therefore, the Nash Equilibrium can predict the evolution of a metagame well ahead of its actual advancement and is therefore a potent tool when trying to find the "best deck."

[Klep]

-"Since players so frequently do not make the best possible choice, there is no point in looking for a Nash Equilibrium when attempting to find the best deck." False. 

No, this is quite true.  If you know that in the vast majority of situations a significant number of players will not make their best choice of deck, then in the vast majority of situations you will not end up in a Nash Equilibrium regardless of what you choose.  Thus, in looking for the best deck, it is pointless to look for Nash Equilibria, because the vast majority of the time you won't be in one.

If at least two players do care about winning and make intelligent decisions, % of top finishes should more closely approximate the Nash Equilibrium over time.  Here as in evolutionary biology, the Nash Equilibrium can predict the long term behavior of a system.

We aren't interested in the long term behavior of a system.  We're interested in a single tournament.

[PucktheCat]

A Nash equilibrium can exist when the player have different choices right?  Some better, some worse?  In other words in a game of Rock, Paper, Scissors, Rockier Rock where one player is not allowed to choose Rockier Rock the correct strategy might reach an equilibrium point, correct?

If that is correct, then I think this reasoning follows.  Call the player with all four options Player 1 and the limited player Player 2.

From Player 1's perspective it doesn't matter why Player 2 can't play Rockier Rock.  The optimal strategy for Player 1 will not be affected by the reason.  It could be that Player 2's hand can't form the symbol for Rockier Rock due to some tragic accident.  It could be that he is not aware of Rockier Rock's existence as a strategy option.  It could be that he is misinformed about the strategic balance of the game and thinks Rockier Rock is strictly inferior to other strategies.  It could be because he has a prejudice against Rockier Rock and refuses to play it.  From the Player 1's the important thing is that they can't play it.  It makes a lot of difference from Player 2's perspective because if he is choosing to avoid Rockier Rock voluntarily he is making a poor strategy decision, but it makes no difference to Player 1.

Now translate that to Magic.  When Klep refers to the "significant number of players will not make their best choice of deck" he is talking about Player 2s.  For whatever reason (lack of cards, lack of information, goals other than winning the tournament, etc.) these players have limited strategic options that we can ascertain some information about in advance.  The informed, intelligent players that follow the format closely, playtest, and own the cards are in Player 1's position.  Player 1 would be making a mistake if he concluded that because the limitation to all the Player 2s strategie was self-imposed that was his strategy should be different than if the players strategy was limited by the rules of the game.

In other words, to the extent that bad players choices are predictable they are simply playing the game with limited strategic options.  If you go to a tournament with such bad players the resulting metagame may not conform to the technical requirements of a Nash equilibrium, because the bad players aren't making optimal decisions, but you should treat it as if it were, because the effect of their poor decision making is identical to the effect of a perfect opponent with limited strategic options.  Failing to do so, thinking that because of their incompetance there must be some unique ex ante "best deck," would be an error.  You might win, because these are, afterall, bad players, but it would still be an error.

Leo

[Klep]

When Klep refers to the "significant number of players will not make their best choice of deck" he is talking about Player 2s.  For whatever reason (lack of cards, lack of information, goals other than winning the tournament, etc.) these players have limited strategic options that we can ascertain some information about in advance. 

Actually what I'm referring to is players who don't do the work to find out what their best choice is, or just don't care.  They have the same strategy options as other players, but they don't have the knowledge necessary to make the optimal choice.  Thus, you won't end up in a Nash Equilibrium because all players aren't playing optimally.

[PucktheCat]

Why do I, as a player entering a tournament, care whether my opponent chooses to play poorly or has no choice but to play poorly.  What possible difference does it make to my decisions?  If my opponent insists on playing Suicide Black because he is a moron should I choose to play a different deck against him than if I enter a tournament that requires everyone but me to play Suicide Black?

Leo

[Elric]

Absolutely.  This is the essence of why metagaming is hard.  You have to know the future to do it properly.

You have said this, apparently without irony, several times in this discussion.  I simply don't understand what you are suggesting as a practical strategy?  Divination?  Hacking team message boards?  Posting about a deck to draw attention to it and then playing its foil?  All of these but one are of extremely limited reliability and the other one doesn't really exist.  At some point you reach the limit of what the information available to you can tell you.  Some people might reach that limit sooner than others (I beleive this is the skill you are referring to), but there is always a limit.  Beyond that point, when you are dealing with players you don't know, or a player you do know who plays a number of decks without any discernable pattern, or a player who is trying to gauge the metagame before committing just like you are, the veil descends.  Theoretically, as you have said, after all those players have made their decisions and their lists are revealed there is a "best deck."  But at that time, when your information is limited (as it always is) there is, also theoretically, a best strategy based on that limited information.  That strategy may be the best deck, or some other deck, or an array of decks that should be selected from according to a some random distribution.  And if you use that strategy you are making the correct choice, regardless of whether later, when you have more information, another strategy turns out to be more likely to win the tournament.

Well said.  This was my point about seeing the future just "being lucky"- being lucky is the only "method" that I can come up with for seeing the future.

[Mind_under_Matter]

Because Magic is a game of imperfect information, whether it be from deck matchups in initial pairings, opponents hands, opponent's library contents, opponents sideboard contents, an so on, isn't it impossible for players to make optimal plays at all times? And therefore isn't Nash Equilibrium an impossible state to accomplish?

Edit: Also, because of players influencing each other through bluffs and other things, isn't it possible for decks that provide more of such situations to be better options than a deck that would realistically beat Rockier Rock only 45% of the time?

[AmbivalentDuck]

I don't get why you're all so hung up on *being* in a Nash Equilibrium, they almost never happen.  The ESS theory I cited before is very clear, it is even accounts for dumbassedness.  If at least two players make intelligent choices and care about winning the system will more closely approximate the equilibrium as time goes on.  Therefore:

-despite imperfect information
-despite dumbassedness
-despite intentional defiance
-despite the fact that you almost certainly won't end up in a Nash Equilibrium

If one *exists*, it predicts where the meta is moving.  Furthermore, it predicts what you'll see in the top 8 as opposed to the people scrubbing out round 3.  It should be apparent that if biologists can use ESS to predict the outcome of evolution and its rate, it may be possible to predict future T8's as well as particular T8's.  It will, of course, remain impossible to predict if Moron the Sui player maindecks Snow-Covered Swamps.

If your goal is knowing every deck going into a particular tournament, you shouldn't be looking into game theory.  Instead, you should be scouting.

[Klep]

Because Magic is a game of imperfect information, whether it be from deck matchups in initial pairings, opponents hands, opponent's library contents, opponents sideboard contents, an so on, isn't it impossible for players to make optimal plays at all times? And therefore isn't Nash Equilibrium an impossible state to accomplish?

Not impossible, just very unlikely.

If one *exists*, it predicts where the meta is moving.

That may be the case, but it doesn't matter when it comes to identifying the best deck. And since it is entirely possible that a Nash Equilibrium doesn't exist, the utility to be found in looking for one is limited at best.

[PucktheCat]

If one *exists*, it predicts where the meta is moving.

That may be the case, but it doesn't matter when it comes to identifying the best deck.

Now wait.  You have been talking all this time about how the goal of metagaming was to predict the future metagame in order to ascertain the best deck to play.  Now you concede that Nash equilibrium may be a useful tool to predict where the metagame is moving and yet you say "it doesn't matter"?  I would think that a tool that increased your chances of predicting the future metagame would be exactly the sort of thing you are looking for.  Or are you limiting yourself to peaking over peoples shoulders before the tournament and overhearing conversations in the bathroom?

Leo

[Elric]

Puck- if the way that players choose decks (according to some rule in which they have a probability of choosing each of a number of decks) reaches a Nash equilibrium (and all players are identical at playing their decks), then short of peeking over shoulders before the tournament or overhearing conversations in the bathroom, there's no way to tell exactly what deck you should play- you can only say that you should play each of a number of (as far as you can tell) equally good decks with a given probability.

[PucktheCat]

Right, but the makeup of the field as a whole would be predicted by the equilibrum.  Klep seems to be dismissing it for that purpose.  If the tendancy is for the metagame to move towards the equilibrium point (as Klep seems to have conceded) then the makeup of the metagame as a whole would be better predicted by a process that took that tendancy into account than by a process that did not.

Leo

Edit:  Additionally, I would argue that, to the extent that the metagame fails to move towards the equilibrium due to external constraints on player behavior* those external factors act as de facto 'rules of the game' for the purposes of optimal strategy.  The game that results from the combination of the actual rules of the game and these de facto rules** is subject to solution just like any other game, with the possibility of a single ideal strategy or a mixed-strategy result.

*Players that play the same deck every tournament because they just like the deck, players that have goals other than winning, players that simply refuse to learn from their defeats.
**The actual rules are defined by the Comprehensive Rules and the text of the cards.  The de facto rules are defined by players irrational strategy decisions.

[Klep]

Now wait.  You have been talking all this time about how the goal of metagaming was to predict the future metagame in order to ascertain the best deck to play.  Now you concede that Nash equilibrium may be a useful tool to predict where the metagame is moving and yet you say "it doesn't matter"?

Actually I said it might, not that it does.  I haven't really investigated that.  Regardless, the definition of the best deck is with regard to a specific set of decks at a tournament, not some evolving metagame.  Again, this relates to the difference between theory and practical application of theory.  There's a difference between saying "Ok, if there were a tournament with 4 people playing deck A, 5 playing deck B, and 2 playing deck C, then the best deck is deck D," and saying "Well, I'm going to be playing in Rochester, which is probably going to be heavy on the Gifts because it's in the northeast, but there should be a strong contingent of Stax and a few Belchers thanks to JDizzle's evangelism, so I should play deck D."  The difference is that in the first case you have exact knowledge, whereas in the second you are making predictions.

[PucktheCat]

the definition of the best deck is with regard to a specific set of decks at a tournament, not some evolving metagame.

Are we quibbling about definitions here?  Is the issue simply that you want to call a deck that has the best chance to win a tournament in retrospect, regardless of whether there was any reason to beleive that it was the best strategy decision with the information available at the time?  You can certainly insist on that as a semantic matter, but it simply doesn't reflect the reality of the strategy decisions we face.

Lets look at another context.  It is often asserted that in any given gamestate there is a "best play."  Would your definition of that term be based on the best play given available information or the best play if all information were known.  As a concrete example, imagine a matchup of Sligh and Fish (the old school versions).  If the game state reaches a position where each player has two lands, four life, and no other permanents.  You each have no cards in hand.  Fish draws and plays a Standstill.  You (playing Sligh) draw a Fireblast.  If you know they have 2 Misdirection in their deck you can calculate the chances that they will draw one and another blue card off the Standstill, and it is very low (say, 5%).  Would you say that the "best play" in this position is to play the Fireblast or would you wait to see if they drew the Misdirection to answer?  In other words, is your definition of "best play" ex ante or in retrospect?

I would always say that "best strategy" choices are defined in terms of the avaiable information.

Leo

[Klep]

the definition of the best deck is with regard to a specific set of decks at a tournament, not some evolving metagame.

Are we quibbling about definitions here?  Is the issue simply that you want to call a deck that has the best chance to win a tournament in retrospect, regardless of whether there was any reason to beleive that it was the best strategy decision with the information available at the time?  You can certainly insist on that as a semantic matter, but it simply doesn't reflect the reality of the strategy decisions we face.

No, I am telling you what the definition entails.  The best deck is the deck which, given a known set of decks in a tournament and presuming equal and sufficient skill of all players, has the greatest probability of winning the tournament.  If someone uses the term 'best deck' with regards to a metagame, it can be taken to mean  the deck which that person believes would be the best deck in a tournament consisting of decks in proportion to what that person believes to be appropriate.

For example, Flores believes that the best deck in Type 2 is Mono-U, which means that in a tournament consisting of certain percentages of Gifts, B/G control, Greater Good, BDW, other Mono-U, etc. which he believes to be appropriate, he feels Mono-U would be the deck with the greatest probability of winning that tournament.

The identity of the best deck is not something you can only see in hindsight.  It is set as soon as soon as all other players have determined what they will be playing (though you may not know what it is).  In practice, as I have stated a large number of times, it is infeasible for you to know what the best deck for a tournament is, because doing so requires prescience.  However, the knowledge that the best deck exists means that it is worthwhile for you to attempt to make educated guesses about what will be in attendance, and choose your own deck accordingly, as doing so increases your chances of winning by bringing you closer to playing the best deck. 

This process is additionaly complicated by the fact that the theoretical best deck is defined independent of qualities that are not inherent to the deck, such as the skill of the pilot and skill of other players.  In practice, such factors need to be taken into account because we are no longer considering some abstract, theoretical best deck for the tournament, but the best deck for you, as a player, considering that you and your opponents have varying skill levels.

I strongly urge you to make sure you understand the difference between the theory of the best deck and the application of that theory before further commenting.

[AmbivalentDuck]

Given you can't bring Magic X4 to a tournament with an encyclopedic knowledge of how to instantly build the best deck and play it, if we include necessary preparations as a rule, the evolution would matter.  If anyone here is such an amazing player that they don't need any practice with their decks, ever, I bow to them.

Otherwise, the best deck is the most appropriate broken ball of hate that you can construct after completely scouting all other decks.  Since it's an unclear term, "broken ball of hate" refers to as much intelligently selected scouting-targeted hate as can be packed into the core with the highest average win percentage for the particular field. (without significantly diminishing its broken consistency)

[Klep]

Otherwise, the best deck is the most appropriate broken ball of hate that you can construct after completely scouting all other decks.  Since it's an unclear term, "broken ball of hate" refers to as much intelligently selected scouting-targeted hate as can be packed into the core with the highest average win percentage for the particular field. (without significantly diminishing its broken consistency)

How many times do I have to stress the difference between theory and application of theory?  Of course you can't just walk in to a tournament and put together the best deck.  I have repeatedly said as much.  That doesn't, however, change the theoretical definition of the best deck.  Application of the theory can only approximate what the best deck is.  You don't hear scientists say "Well, we can't actually make a perfect sphere so we'll call this thing we made instead a perfect sphere." That wouldn't make any sense.  If we want to take this aspect of the game seriously, we have to accept that there is going to be theory; and if there is theory it is not going to match up perfectly with reality.

[AmbivalentDuck]

The "broken ball of hate," intelligently selected as defined above, is the best theoretical deck.  I'm not trying to imply anything more or anything less about it.

When we go to apply the theory, we have two critical limitations:
-We cannot scout perfectly.
-We cannot build or learn decks quickly enough.

Since I'm assuming that the goal of this thread is to say something useful about T1 and not just spin our mathematical wheels disagreeing on assumptions, it doesn't seem innappropriate to suggest that our goal is to report a good/better way of figuring out what decks to prepare for a tournament given that limited scouting should be enough to choose between them.

Do we have the same goal?  If not (Klep), what is your goal?  Should this thread *do/accomplish* anything?  If so, what?

[PucktheCat]

No, I am telling you what the definition entails.

Ah, I understand.  Logic need not apply.  You didn't answer my second point.  Would you define "best play" along the same lines?

Contrary to your understanding, nothing about Flores stating that a particular deck is the best deck discriminates between your posited definition and my own.  In fact, your own description seems to confuse the two:

The best deck is the deck which, given a known set of decks in a tournament and presuming equal and sufficient skill of all players, has the greatest probability of winning the tournament.

This is what you've been saying all along, this is the retrospective, unlimited information approach to defining the best deck.

If someone uses the term 'best deck' with regards to a metagame, it can be taken to mean the deck which that person believes would be the best deck in a tournament consisting of decks in proportion to what that person believes to be appropriate.

But here you switch to the prospective mode.  If a person is predicting the metagame ("decks in proportion to what that person believes to be appropriate") all of the limited information game theory concepts would fully apply.

In fact, the definition(s) you offered perfectly captures the distinction between the two possible meanings of the best deck.  The retrospective definition involves perfect information and so the best deck will always be precisely identifiable (given enough time).  The prospective definition, however, involves limited information, prediction, divination, and so the "best deck" will be a product of limited information strategy theory and may be a deck that is the same as the eventually revealed "retrospective best deck" it may be another single deck, or it may be some mixed strategy.

The identity of the best deck is not something you can only see in hindsight.  It is set as soon as soon as all other players have determined what they will be playing (though you may not know what it is).

That is nearly always the case in games of limited information (Rock, Paper, Scissors for example).  The cards in a shuffled deck are in a particular order when the game begins, but that doesn't mean it isn't random to the players.

Leo

[Klep]

The best deck is the deck which, given a known set of decks in a tournament and presuming equal and sufficient skill of all players, has the greatest probability of winning the tournament.

This is what you've been saying all along, this is the retrospective, unlimited information approach to defining the best deck.

If you're saying retrospective as in after the tournament concludes, then no, it's not.  If you're saying retrospective as in after players mave their deck choices, then yes, it is.  The identity of the best deck is independent of the deck's performance, so tournament results don't matter.

If someone uses the term 'best deck' with regards to a metagame, it can be taken to mean the deck which that person believes would be the best deck in a tournament consisting of decks in proportion to what that person believes to be appropriate.

But here you switch to the prospective mode.  If a person is predicting the metagame ("decks in proportion to what that person believes to be appropriate") all of the limited information game theory concepts would fully apply.

No, this is exactly the same thing.  In this case the metagame defines a specific tournament in which the decks present are present in proportions according to what the person making the statement feels are appropriate.  In other words, if you think that the metagame consists of Gifts, Slaver, and Stax and you think that they appear in proportions of 40%, 30%, and 30% respectively, then the best deck for that metagame would be the deck that would be best at a tournament that is 40% gifts, 30% Slaver, and 30% Stax.  By stating that deck X is the best deck for a metagame, you are instantiating a theoretical "tournament" in which you feel deck X is the best deck.