Betting Curves

As described above, the calling threshold $\theta$ identifies the probability of winning at which the expected values of calling a bet versus folding are equal. Probabilities substantially higher than $\theta$ should in general lead to bets and raises; probabilities much lower to passes and folds. However, if a player invariably bets strongly given a strong hand and weakly given a weak hand, other players will quickly learn of this association; this will allow them to better assess their chances of winning and so to maximise their profit at the expense of the more predictable player. Therefore, BPP used betting curves, such as that of Figures 2, to randomise the actions of BPP in a way dependent upon the probability of winning. The horizontal axis shows the difference between that probability and $\theta$; the vertical axis is the (unnormalised) probability of a given action: fold, call or bet (raise). The normalised probabilities were then used to stochastically select an action in any situation. The playing curves were generated by exponential functions with a parameter adjustable for each round of play. Ideal parameters selected the optimal horizontal displacement of each curve relative to the difference of $\theta$ and the probability of winning, and thereby the optimal balance between conservative and aggressive play. For example, if the folding curve was shifted to the right relative to the calling curve, more conservative play would result, with even moderately strong hands perhaps being dropped. Or if the betting/raising curve were to be shifted to the left, more aggressive play would result. In order to find good parameters, a stochastic search of the parameter space when running BPP against a rule-based opponent was employed. Since the space being searched is 12 dimensional (three types of curves, four each for the rounds of play) and the score function is highly noisy (wins/losses in actual poker play), it is not clear that the search for optimality was successful. Nevertheless, the curves produced by our stochastic search appeared to provide a reasonable answer to such questions as how much greater the probability of winning must be over the threshold for active bets and raises to be rewarded. Their use also provided good camouflage for playing behaviour by their introduction of random play.

  

Figure 2: Final round betting curves (fold, call, bet). The horizontal axis is the difference between the probability of winning and $\theta$; the vertical axis is the (unnormalised) probability of the action.

One apparent anomaly is that the point at which the probability of folding equals the probability of calling should occur when the probability of winning is equal to $\theta$ (i.e., at 0 on the horizontal axis of Figure 2). It was believed the explanation was that the estimate of the pot odds (being dependent upon an estimate of the expected cost to a showdown) was inexact with the optimisation process of the playing curves compensating for the estimation error by displacement. Since the calling curve was displaced to the left, this suggested that $\theta$ was being overestimated.