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Poker, solved (Posted on 2008-03-29) Difficulty: 2 of 5

Tables are available giving the probabilities of various poker hands. Such tables can readily be adapted to give the chances that the hand you are holding will be the strongest among N random opponent hands.

Based on such a table, it is a simple matter to develop an algorithm to determine a handís chances even before the "drawing round". The algorithm could also advise on which cards to throw down and can refine the win probability calculation in consideration of the number of cards your opponent has drawn.

Suppose you are playing poker while using such an algorithm. You decide to completely ignore your opponents' bidding behaviour as you regard this information as unreliable. I.e., we can assume that your opponent is a skilled gambler who understand how to undermine the information content of his bids with carefully calibrated bluffs. Rather, you determine your bids based solely on the algorithms probability calculation: Whenever the probability indicates a positive payout for your participation, you bid to the limit, otherwise you fold.

As soon as your opponent comes to understand your approach - for instance, because he is an astute observer, or because the nature of your algorithm is publicly known - he can be expected to abandon bluffing as well. (Why would he bid on a weak hand when you canít be influenced?) Next, he will be pressed to adopt your algorithm himself, since deviations from the optimal participation decision only lead to a lower payout.

Iíve presented the case for 1-vs-1 play, but it works as well for 1-vs-N if a large enough number of players are using the same algorithm.

Consider: Have I shown that there is no point in bluffing? Is there any point in relying on inferences drawn from oneís opponentsí bidding behaviour? Is there ever any reason to bid less than the maximum? and, most fundamentally, why hasnít poker been set aside (like checkers) to the pile of solved games?

See The Solution Submitted by FrankM    
Rating: 2.5000 (2 votes)

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Hints/Tips You're not there yet - Reply to reviewers remarks | Comment 3 of 11 |
(In reply to As for you last question ... by Steve Herman)

Without wanting to contradict your statement that Any optimal strategy must take into account the opponent's betting behavior, I would like to draw your attention to your later statement about the practical difficulty in writing an algorithm which uses opponent bidding behaviour effectively in all circumstances. Remember: it is not enough for the algorithm to perform well, even for much of the time. Once an opponent recognises some systematic fault in your algorithm, he can work to drive the game in the direction of this weakness.

Perhaps it would be useful to clarify that i do not make any claim as to the optimality of the proposed program. However, it is robust; can be easily realised, and is especially effective against strong opponents, against whom (as you correctly point out) the task of reliably extracting useful information from bidding behaviour may be insurmountable.

In fact, my claim for this algorithm's strength lies precisely in the difficulty in producing a truly optimal algorithm. If we accept the notion that this goal is out of reach, then my proposed algorithm ought to be dominant. (Or so I claimed). 

In consideration of your remarks I'd like to take one half step back from my suggestion about relegating poker to the solved game pile. This claim is misleading, although (assuming my argument holds) it would still appear to have practical validity.

And yet, there is a fallacy in my claim, although up until now, no one has found it.


  Posted by FrankM on 2008-03-30 20:54:51
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