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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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Some Thoughts Imperviousness to aggressive bluffing Comment 11 of 11 |
(In reply to re: Exchange with Dej Mar by Dej Mar)

The bluff .. not being used to trick the algorithmic player into thinking the opponent's hand is better, but to raise the stakes high enough that the bet would exceed the merit of the gamble, and thus lead the player to fold.

that's an interesting point and merits discussion..

of course, the bluffer would have to contend with the risk that the ADP (algorithmic driven player) may have a good enough hand to merit staying in. presumably that is what you meant by saying in an earlier comment that the ADP may need a very good hand to be impervious to such subtrefuge.

would a strategy of aggressive bluffing (i.e. frequent bluffing without consideration of the strength of one's own hand) targeted at persuading an ADP to fold be effective? I don't think so. This may be most readily grasped by considering the simple case of 1-on-1 play. here, half the time the ADP would anticipates a better than 50% chance of having the best hand, so that he would (rightfully) welcome seeing his opponent increase the stakes.

it seems in order for aggressive bidding to be generally effective against an ADP, the opponent needs strength in his own hand. in the problem text though, we already introduced the idea that a player might chose between only two betting alternatives: either fold/pass or bidding the limit. so this type of aggressive bidding, backed by strength, hardly seems to merit the name: bluffing.  

of course, to formalise the concept of an ADP'S imperviousness to aggressive bluffing, one would have to do the numbers. but, still, i don't doubt the outcome.

  Posted by FrankM on 2008-04-01 11:54:00
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