heads up poker no rake (game theory)

[flopdanutz]

is heads up poker with no rake a finite zero sum game?
if possible can u please provide a simple explanation?
if so is there an optimal mixed strategy?
if so can you apply it?
if so will u write a book on it? hehe jk

[Copernicus]

Yes it is a zero sum game, since the sum of the payoffs to each participant is 0. Assuming there are blinds it is also finite under any strategy, since there is a non-zero probability of a player winning consecutive hands that blind out an opponent who's objective is to make the game infinite.

There is no universal optimal strategy, since it is a game of imperfect information, although I would guess that there are good game theoretical approximations to an optimal strategy based on ratios of chip counts to blinds.

It would seem that for a dominant chip leader a strategy of aggressive pre-flop play, combined with post flop betting in reasonable proportions (of the smaller stack) to his probability of beating random hands would yield something close to optimal.

[M.B.E.]

<font color="purple">There is no universal optimal strategy, since it is a game of imperfect information, although I would guess that there are good game theoretical approximations to an optimal strategy based on ratios of chip counts to blinds.</font color>

Actually there is an optimal game-theoretic strategy for heads-up holdem, but it would take existing computers way too long to do the calculation. A group at the University of Alberta recently used some clever techniques to develop an approximation of the optimal game-theoretic strategy, which they call pseudo-optimal. Here are some links:

<font color="green">University of Alberta Computer Poker Research Group (http://www.cs.ualberta.ca/~games/poker/)</font color>

<font color="yellow"> Article in Poker Theory forum (http://twoplustwo.com/forums/showthreaded.php?Cat=&amp;Board=genpok&amp;Number=281492)</font color>

It appears that the game of "headsup preflop holdem" (i.e. where all but the first betting round is eliminated, a.k.a. "roll-out holdem") has been solved by Alex Selby:

Optimal Headsup Preflop Poker (http://www.archduke.demon.co.uk/simplex/index.html)

It's important to understand what is meant by an "optimal game-theoretic strategy". It is a strategy that cannot be beaten by any counter-strategy. However, against any given opponent the optimal strategy might not win as much as another strategy tailored to exploit the opponent's particular weaknesses.

[Copernicus]

I'm very familiar with the Alberta work. In fact a lot of work I have done paralleled theirs, and actually took the Poki concept several steps further when they sidetracked into the heads-up model. Aaron Davidson was most helpful in discussing a conceptual framework for that next generation.

Unfortunately I don't have the computer skills or funding to turn the work into a "bot" (or more to the point an automated but off-line "advisor", since bots are violations of the TOS of almost all sites). The limitations are not in processing since the results arent necessarily optimal, but rather in I/O of the real time situation. Tested against real hand histories after the fact, however, it learns and predicts other players actions/hands very well. Its impossible to turn that into a theoretical earnings rate, however, since the opponents actions might have been different based on the actions of the "advisor". It is also a monumental data mining task that would benefit from a "team" of players contributing hand histories to be most effective.

I agree with your definition of "optimal", and that is why I commented that there is no optimal game theory solution, since there will always be a counter-strategy for any given fixed strategy. That is why the best poker players will eventually break very good but rigid (although complex) players. (I distinguish game theory solutions (fixed) from AI (learning/genetic) solutions.)

[M.B.E.]

"<font color="purple">I agree with your definition of 'optimal', and that is why I commented that there is no optimal game theory solution, since there will always be a counter-strategy for any given fixed strategy.</font color>"

That may be true if there are three or more players, but not in a heads-up game. In a heads-up poker game, an optimal strategy (as we have defined it) does exist. In fact, the optimal strategy for "preflop holdem" has been calculated (http://www.archduke.demon.co.uk/simplex/index.html). If this calculation is correct (and I have no reason to think it isn't), then there is no counter-strategy that will beat it.

[Copernicus]

If you add "or minimize losses" to your definition of optimal strategy then there theoretically strategies that are close to optimal. However, even the strategy for pre-flop hold em is a mixed strategy. As soon a strategy becomes mixed it can no longer be absolutely optimal under all conditions, even under the looser definition.

One interesting result is that if the deal never moves, then there is a provable positional advantage to acting second, which can theoretically be extended to acting last in a multi-player game. That condition exists (the deal never moves) in "dealers choice" home games where only one player happens to choose (and deal) hold em.

[M.B.E.]

<font color="purple">If you add "or minimize losses" to your definition of optimal strategy then there theoretically strategies that are close to optimal. However, even the strategy for pre-flop hold em is a mixed strategy. As soon a strategy becomes mixed it can no longer be absolutely optimal under all conditions, even under the looser definition.</font>

I'm not sure what you mean by this.

Yes the optimal strategy for headsup holdem is a mixed strategy, not a pure one. But it is still optimal, in the sense that no counterstrategy will be able to defeat it. To be more precise, the optimal strategy guarantees that the player will get at least the game-theoretic value of the game, which in headsup norake holdem is zero (assuming that each player has the button the same number of times).

Consider the game of "matching pennies" often used as an example (e.g. http://william-king.www.drexel.edu/top/eco/game/zerosum.html). The optimal strategy is the mixed strategy of choosing heads with probability 0.5 and tails with probability 0.5. This is guaranteed to get the value of the game.