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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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Exchange with Dej Mar | Comment 9 of 11 |
(In reply to re: Response to Dej Mar by Dej Mar)

I did not claim that no bets would ever take place, only they would be less common. 

We are more or less in agreement then (and sorry for misinterpreting your statement!)

I think bidding would still be a reasonably frequent occurence, if only because of the broad unpredictability of final results before the drawing round. For instance: a player who drew 6,10,J,Q,K might reasonably be told to bid before drawing. And even if he only were to draw a second K, he might stay "in" during the second round, if there were to be a large sum in the pot, and if each second round bid increment were small.

I agree though that there would be some reduction in completed hands, as occasions where a player stays in because he suspects his opponent to be bluffing would be eliminated.

I didn't understand your comments in either of the last two paragraphs,

or, perhaps, you've misunderstood the reason for abandoning bluffing - which has nothing to do with having a strong hand. The reason for abandoning bluffing - so the claim - is that it carries a cost (namely, the cost of being "caught out") without bringing along any compensating benefit. Since the algorithm driven player is impervious from drawing any conclusion about the strength of an opponents hand based on his bidding behaviour, he will ignore the significance of your bid as a strength symbol, and concentrating instead on the question of whether his own hand is strong enough to merit meeting the opponent's bid.





  Posted by FrankM on 2008-04-01 00:09:15
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