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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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Hints/Tips Response to Dej Mar | Comment 5 of 11 |
(In reply to No Subject by Dej Mar)

Dej Mar's suggestion that no bidding would take place is in error.

The algorithm advises on behaviour based on normal events. At times, two or more players will simultaneously have abnormally good hands. In this case both would bet.

No claim is raised to suggest that the algorithm is able to deliver excellent results for every hand. Rather, there is a claim of superior performance over a large number of hands against a collection of players who have not yet mastered the art of reliably extracting information from opponents bidding behaviour.

This claim however is invalid (hence placement of this problem in the category of paradoxes).

I still hope that someone will uncover the incorrect assumption that undermines the claim for the algorithms superiority.


  Posted by FrankM on 2008-03-31 10:35:34
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