Poker and Other Bluffing Games

In July of 2007, a computer program called Polaris played against two of the best human poker players, Phil Laak and Ali Aslami, in a four day tournament of limit Texas Hold ‘em. Oddly for poker, the match was carefully designed so that the result could not be attributable to good or bad cards.

To do this, two copies of the program played on separate computers in separate rooms. One computer played against Phil Laak in one room and the other computer played against Ali Aslami in the other room. The decks were arranged so that the cards dealt in one room were identical to the cards dealt in the other room. However, the cards given to Polaris in one room were given to the human in the other, and vice versa. This ensured that if the cards favored Polaris in one room, then they favored the human in the other room by exactly the same amount (since they were the same cards). As a result, whether Polaris won or not, it could not be because it received good or bad cards.

The first day of the match was a draw. What this actually means is that Polaris won, but only by $70, and a win by less than $250 was considered statistically insignificant. The second day Polaris won, and the third and fourth days the humans won. Although Polaris lost the tournament, it put on an unquestionably good showing.

What is particularly interesting about this is that in order to win at poker, one must be able to bluff well, and it seems to run rather counterintuitive to most people’s opinions of machines that they should actually be able to bluff. This was highlighted in the Star Trek Next Generation episode “The Measure of a Man” when Riker bluffed Data¹:

DATA: I will fold.
Riker rakes in the chips, then turns over his cards revealing a busted hand. Data turns his cards face up. He held a winning hand.
DATA: You had nothing.
GEORDI: He bluffed you, Data.
DATA: It makes very little sense to bet when you cannot win.
RIKER: But I did win. I was gambling that you wouldn’t call.
DATA: But how can you tell?
O’BRIEN: Instinct, Data, instinct.

I like Star Trek, but they got it wrong here. In fact, there is a mathematically perfect way to bluff, and no instinct is required at all. Unfortunately, perfect bluffing in a complex game like Texas Hold ‘Em requires more computational power than will ever be available. Fortunately, near perfect bluffing can be done on a desktop. It appears that the best humans bluff in a fashion that is still nearer perfect than a computer can yet achieve, but this is almost certain to reverse in the next few years (at least for two player limit Texas Hold ‘Em).

Unlike humans, who often bluff on instinct, a computer bluffs based on random chance. This ensures that its opponent cannot guess when it is bluffing. Of course, the probability of the computer bluffing is dependent on its cards, the opponent’s play, and so forth, but there are optimal values for these probabilities, and the “art” of bluffing really becomes a science.

1. Excerpted from Star Trek Minutiae