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?

Hi Ed,

I'll try to answer your questions.

The algorithm makes use of whatever information is publicly available and which can reasonably and reliably produce better play. So, the algorithm will taken into account that there has been a new shuffle/new deck, if that is the case. Otherwise it would taken into account any publicly available information about cards that are not available.

You are right in saying that you dont often see 100% probability of winning. presumably this only happens when you draw a royal straight flush in spades. But you should stay in whenever the cost of matching an opponents bid is justified based on the size of the pot and that calculated chances of having/getting a hand in the top 100/N percentile, where N is the number of opponents still "in". for example, if the pot holds $100 and you've been raised $1, it seems reasonable for you to stay in, even if you only have a 20% chance of having the best hand from among 4 typical hands (assuming there remain 3 active opponents).

i agree with you that chequers can still be fun. solved or not, its still intransparent enough so that (aside from performance issues) victory ought to depend on the best heurestics. in another way i even enjoy naughts and crosses, when i see that it amuses my 8 year old daughter!