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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 6 of 11 |

Are your odds always for a new deck, after shuffle?  Do you have separate odds for each number of opponents?

Presumably you would not often get a 100% probability of winning, yet you do not say how your system suggests you act if the probability is lower. Do you bet the max if you hold cards with a 50% probability?

It is interesting that chequers has been "solved" but that has virtually no impact on players continuing to enjoy the game. Everyone knew from the start decades ago that there would be A solution if all combinations were explored (either a forced draw, or a forced win).  Unless someone created a searchable table of the correct (safest) move for every combination, there seems no impact on live play. Similarly, chess and Go  may one day be "solved" but hardly likely to have an impact. (I realize that studying these games may have academic benefits, in devising more efficient search and evaluate functions.)


  Posted by ed bottemiller on 2008-03-31 12:10:55
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