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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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No Subject | Comment 4 of 11 |
Assuming all players are using the same algorithm as proposed in the problem, most games would be composed of calls with no bets, and, thus, seldom would one need even fold. Following this thinking the game may as well be played with the hands completely open with the game reduced to a simple lottery, the pot being split between the players with the best hand. 
  Posted by Dej Mar on 2008-03-31 00:19:00
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