Nash Equilibrium for hold'em

[cairpre]

I would like to know how much success there has been in plotting Nash equilibriums for hold'em. Since I am not a mathematician, I'd rather not calculate them myself if the work is already done. It seems to me that by making some reasonable assumptions optimal strategies could be calculated by this method. If the work has been done, where can I get it? If it has not can anyone recommend the proper tools to work with. And finally Is there a good argument why they would not work for a game like hold'em? Thank you

[lucky_mf]

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I would like to know how much success there has been in plotting Nash equilibriums for hold'em. Since I am not a mathematician, I'd rather not calculate them myself if the work is already done. It seems to me that by making some reasonable assumptions optimal strategies could be calculated by this method. If the work has been done, where can I get it? If it has not can anyone recommend the proper tools to work with. And finally Is there a good argument why they would not work for a game like hold'em? Thank you

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Impossible and pointless for a variety of reasons.
(1) It is hard to define the game - How long does it last? who are they players? How many players are there? You can't really pin these things down as they are fluid.
(2) For a NE your strategy is optimal given your opponent's strategy and visa versa. This isn't the case for a variety of reasons.
(3) Even if you had a well defined game - like heads up with the same opponent for 1000 hands - determining an optimal strategies would be very complex - an optimal almost certainly be a mixed strategy (randomizing) and would also depend on your hole cards and your oppenent's actions (whether he raised, checked, or limped). So I think what you would end up with is a set of frequencies for each possible hand (on the flop, turn and river), in given positions, for each possible action of your opponent (on the flop, turn, and river) - in other words the strategies would be very complex (if you could compute them).

Lucky

[cairpre]

Thank you for your reply

Since I am considering a predominantly probabilistic and statistical approach to the game how long the game lasts and who the particular players are is probably not important. Remember it's one long session. However we can categorize certain types of tables and base our strategy on that. For example a 10 person table at the beginning of a tournament probably is less aggressive than the last 4 players on the final table. Optimal strategy is modified to the the texture of the table as we evaluate it

It just seems to me that rather than practicing sitting stone faced or overindulging in attempted tell reading Nash presented a good alternative. I was actually thinking about combining it with a different approach.

When I read Mr. Slansky's example of using a coin flip to offset an opponents superior ability in a game of odds and evens ( Theory of poker chapter 19 -game theory), I thought about how I might apply that to poker. Betting, folding, raising, and checking on a completely random basis seemed like a poor plan. However since mixing one actions is the proper strategy and Nash helps us to determine the right percentage mix of actions, like how often to raise or call with a particular type of hand, it seemed a good idea to put the two together.

So lets say you're playing the standard tight aggressive style recommended in most books "randomizing" that basic strategy at about the same percentage as the determined equilibrium point should yield the better results over the long run. I think as long as we assume rational play on the part of the opponents, and have evaluated the table texture correctly we will consistently make correct (not necessarily winning) plays. This just seems an easy rather stress free way of getting maximum return.

But being a relative newcomer to both poker and probablity theory I am not sure how correct this is.

[lucky_mf]

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Since I am considering a predominantly probabilistic and statistical approach to the game how long the game lasts and who the particular players are is probably not important.

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The length of the game is very important from a game theoretic perspective as are the number of opponents and the rate at which you discount future profits at the expense of profits now. The longer the game last with the same people, and the lower your discount rate, the more optimal strategies for the game will have you moving away from optimal strategies for the hand (i.e. you will be willing to sacrifice some +EV in the current hand for future +EV). Also it should be obvious that the number of players matters a lot – a 3 handed game is much different than a 6-handed game. Furthermore a flop with 3 players in a 6-handed game is much different that a HU pot in the same game. If you could calculate optimal frequencies they would be different for these different situations (and for your holdings + the board).

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Optimal strategy is modified to the the texture of the table as we evaluate it

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Well this isn’t exactly right – because NE assumes all players are playing optimally with respect to the other players’ optimal strategies. So it really doesn’t matter which players are in the game from the perspective of NE – what does matter is the number of players.

Lucky

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It just seems to me that rather than practicing sitting stone faced or overindulging in attempted tell reading Nash presented a good alternative.

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It is tempting to think that game theory has practical applications to poker – it really doesn’t. It might be useful for formalizing poker theory, but that’s about it. Take a 100 hand heads up match – the easiest case scenario. If there is a way to calculate a NE in this game, and if this calculation was practical in the sense that it provided an advantage, there would be bots capable of beating good players. That no such bots exist is telling.

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So lets say you're playing the standard tight aggressive style recommended in most books "randomizing" that basic strategy at about the same percentage as the determined equilibrium point should yield the better results over the long run. I think as long as we assume rational play on the part of the opponents, and have evaluated the table texture correctly we will consistently make correct (not necessarily winning) plays. This just seems an easy rather stress free way of getting maximum return.

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Two big things:
• You can’t calculate these frequencies. They are hole card, board, number of players, and stage in game specific (your optimal strategy during your last hand will be different than your play in the first because in the first establishing a reputation matters – in the last it doesn’t).
• Your opponents are not playing the best response to your strategy (thus no equilibrium). To assume they are is just silly. Thus, even if you figured out a NE for NLE, your NE strategies would not be optimal given the play of your opponents (who are off the equilibrium path).

Rather than trying to find a precise mathematical way to beat the game I would suggest watching carefully how your opponents play and respond to your play. If you do this you can figure out ways to exploit their play. For example, if someone keeps folding to your c-bets – keep doing it; If they float you with nothing – start double barrel bluffing; If they are calling you down with mid pair – make a hand and value bet the crap out of them. If someone is stealing a lot from the CO and BTN – re-raise them from the blinds; If someone bets behind frequently when you check - start checking made hands to them and go for a check raise.

Lucky

[f97tosc]

The multiple stages is the key features which makes holdem intractable. You can write it down as a well-defined mathematical game, but you can't compute solutions because the number of combinations of cards and actions that can pop up are astronomical. And to play optimally preflop, you have to take into account all possible communal cards and actions from the other players.

On the other hand, if you make approximations and simplifying assumptions about various things then some headway can be made. Some special situations, like heads up river play are relatively tractable.

[DWarrior]

Not too familiar with Nash Equilibrium (flunked out before getting there in Econ).

Anyway, the SNGPT tool (for SNGs) has an unexploitable strategy for SNGs when it gets to HU (at which point the blinds are so high that the game becomes push/fold). If you play that strategy, opponent can only counter by playing the same exact strategy, and any deviation of his loses him money. However, it may be more optimal to adjust to his flaws and create an even better strategy.

Also optimal strategies exist for late-game (but still more than 2 players) SNGs when all the players are playing optimally and the game is in push/fold mode.

[jogsxyz]

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Also optimal strategies exist for late-game (but still more than 2 players) SNGs when all the players are playing optimally and the game is in push/fold mode.

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Are you sure someone has proofs for the 3 person game? Or are they just opinions based on monte carlos?