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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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re: Response to Dej Mar | Comment 7 of 11 |
(In reply to Response to Dej Mar by FrankM)

I did not claim that no bets would ever take place, only they would be less common.  As to when one does make a bet based upon the strength of this hand meeting the algorithm, more often the other players following the same no-bluff algorithm would fold. Still, not always, as you said, there will be times in which two or more will have hands will meet the algorithm's advised betting requirement. 

As one can bluff even with a strong hand, to invalidate the use of bluffing the strength of one's hand would almost need be one of the most improbable strong hands. Of course, even if one has a Royal Straight Flush,  it does not guarantee a sole win.

I disagree with the claim (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) being invalid. One who has the superior capability at calculating the probabilities and applying this informaton properly into an algorithm that determines when to bet and when to fold, though it might be ever so slight of an advantage, should, over the long run, have "bettor" odds at winning.  Of course, for the long run to exist, we must assume whatever "bad luck" the player may have does not prevent him from participating in future games.

  Posted by Dej Mar on 2008-03-31 23:20:38
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