Much has been written about game theoretic optimal frequencies for bluffing, and calling a possible bluff. This serves as a nice example of how the underlying principles of game theory can be used as a starting point for a poker algorithm, but then must eventually be transcended to achieve the highest playing levels.
It can be shown that the theoretical maximum guaranteed profit from a given poker situation can be attained by bluffing, or calling a possible bluff, with a predetermined probability. The relative frequency of these actions is based only on the size of the bet in relation to the size of the pot. To ensure the best result against perfect play, the action must be unpredicatable, and one way to accomplish this is by selecting a particular range of hands to act upon, which will occur uniformly at random.
In the following example, we imagine two players involved in a hand of pot-limit Draw poker (where either may bet an amount up to the current size of the pot). Player B has called with a one-card draw to a flush, against Player A who currently has the best hand. To simplify the math, we will assume that Player B will win the showdown if she makes the flush, but will lose otherwise. We will further assume that the probability of completing the flush is exactly 0.20, or one in five. The question is how the hand should be played after the draw.
The first principle is that Player A should not bet, because Player B will simply fold if she missed on the draw, but will call (or raise) if she made the flush. Since there is no profit in Player A betting, we can assume without loss of generality that Player B is first to act after the draw. The correct strategy for Player B is to bet (the size of the pot) whenever she makes the flush, and also to bet occasionally when the draw failed. The optimal frequency of bluffs by Player B and calls by Player A are computed with a game theoretic analysis. For each pair of frequencies the overall expectation (expressed as a fraction of the total pot before the draw) can be calculated. Table 2 gives a sampling of these values over the full range of bluffing and calling frequencies.
Table 2: Expected Values for a Four-Flush Draw: Bluffing vs Calling
Frequencies
Legend:
BR = ratio of bluffs to legitimate bets
ABF = absolute bluff frequency (fraction)
CFr = absolute calling frequency (fraction)
(expected values are expressed as a fraction of the total pot,
given a 0.2 legitimate betting frequency, and pot-sized bet)
If Player B never bluffs and Player A never calls, it has the same effect as having no betting round after the draw, and the expected value is 0.20 of the pot for B and 0.80 for A. We can see from the table that to obtain the guaranteed maximum, Player B should bet 30% of the time - 20% with the flush and an additional 10% as bluffs, selected at random. Player A can always ensure his optimal expectation of 0.70 by calling exactly 50% of the time Player B bets. The bluffing ratio of one bluff for every two legitimate bets and the calling frequency of 50% are a general results for all situations in which Player B will bet or bluff the size of the pot. The optimal ratios will change depending on the size of the bet in relation to the pot, but are independent of other factors.
While these are optimal strategies, they are not maximal strategies. A maximal strategy is directed toward exploiting weaknesses in the opponent, whereas an optimal strategy implicitly assumes perfect play on the part of the opponent.
The game theoretic approach is valid if the opponent is a very strong player, or perhaps an unknown player, but is certainly not the way to maximize net profit in the long run. In a typical game of poker, game theory is not an appropriate strategy, because it also guarantees that a player makes no more than the expected value from the particular game situation. This effectively ensures that the opponent also plays optimally, regardless of her approach to the game.
As an example of maximizing strategies, we observe how a strong poker player handles this type of situation. If faced with a bet from a player who never bluffs, a strong player will usually fold a marginal hand, knowing she cannot win. Conversely, she will often call a chronic bluffer, even with only a mediocre holding. In the role of Player B, a strong player will frequently bluff against an overly conservative player, but will seldom try to bluff a player who almost always calls. The net result is an expectation higher than the optimal 0.3, and the table demonstrates just how profitable these strategy adjustments can be in practice.
An algorithm based on game theoretic principles will provide a solid basis for betting strategy. Nevertheless, to advance to the highest levels, a program must be able to understand each opponent's playing style, and be able to adapt to the specific game conditions.