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 hand’s 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 can’t 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.

I’ve 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 one’s opponents’ bidding behaviour? Is there ever any reason to bid less than the maximum? and, most fundamentally, why hasn’t poker been set aside (like checkers) to the pile of solved games?

Well, first, I agree 100% with Leming.  Any optimal strategy must take into account the opponent's betting behavior.  And bluffing can be used somewhat to take advantage of any strategy which does take into account the opponent's betting behavior, so any optimal strategy must also incorporate bluffing.

Having said all that, it is still theoretically possible (although much harder) to come up with an optimal strategy.  Online, Computers already play better than most people.

So let me address myself to your last question: "why hasn’t poker been set aside (like checkers) to the pile of solved games".  Several reasons:

a) There is a world of difference between perfect information games like checkers and chess and tic-tac-toe, and games where players can hide information about the state of the game (like Stratego and Battleships).

b) And there is another world of difference between hidden-information games (like Stratego and Battleships) and games like poker, where there is an element of chance involved.

c) And even if in theory there is a perfect strategy for poker, the game isn't solved until somebody reduces that theory to practice.  According to Wikepdia (look up "Solved Game"), "Checkers is the largest game that has been solved to date, with a search space of 5x10²⁰. The number of calculations involved were 10¹⁴ and were done over a period of 18 years."  

Posted by Steve Herman on 2008-03-30 08:48:37