7.2.3 Post-Flop Re-Weighting

For the three betting rounds after the flop, we infer a mean and variance ($\mu$ and $\sigma$) of the threshold for the opponent's observed action and rank all hands according to the effective hand strength (EHS') calculation from Equation 6.4. Although the relative ranking of all hands is not affected, it is an optimistic view, because hands with an EHS' above $\mu = .80$ represent more than 20% of all hands (due to the inclusion of PPOT in the equation). Additionally, we use HS₁ instead of HS_n so the number of opponents can instead be addressed as part of the context of the action frequencies (although presently it is ignored in the interest of a simplified definition of context).

To calculate EHS' we must compute both hand strength (HS) and positive potential (PPOT). Since PPOT is a computationally expensive operation and we must obtain PPOT values for about 1,000 hands every call to the re-weighting function, we use a crude but fast estimating function (PPOT_c). In terms of cost, PPOT_c << PPOT₁ << PPOT₂. It is crude because it ignores much of the available information (such as weight arrays) and looks for a few features about the hand, using heuristics to associate a value to each. For example, if the board has only 2 diamonds of our 4-card diamond flush draw, then we have 9 outs (1 out per remaining card), but if the board has 3 diamonds, each of the remaining cards is worth 0.5 to 1 out, depending on the rank of your suited hole card (a Two is worth 0.5 and an Ace is worth 1). This is because we are more likely to be up against other diamond flush draws. PPOT_c approximates very roughly our winning chances given each possible card to be dealt). This function produces values within 5% of PPOT₁, 95% of the time (this is from a random sampling of five card unweighted cases). Since these values are amortized over about 1,000 possible hands, the overall effect of this approximation is small.

However, the opponent is also likely to play some hands based on the pot odds required for the action ( i.e. the decision is not always based on EHS'). For this reason we have introduced an adjustment to the re-weighting algorithm for the post-flop. When PPOT_c is sufficient to warrant calling the pot odds, which is

$$\frac{amount\_put\_in}{pot\_size + amount\_put\_in},$$

the weight for that subcase is not reduced. This is a simplification of the betting decision the opponent could be making based on the pot size, however we feel it is sufficient to capture enough cases to prevent hands with low EHS' but high potential from being severely under-estimated. The new re-weighting algorithm can be found in Figure 7.3.

Figure 7.3: Post-Flop Re-weighting Function

What value do we use for $\sigma$? We chose to use a typical value of $\sigma = .2$ at $\mu = .5$ (interpolating over the range .3 to .7) and to increase $\sigma$ with smaller $\mu$ while decreasing it for larger $\mu$. This reflects the tendency for loose players (with low $\mu$) to exhibit more uncertainty and tight players (with high $\mu$) to adhere more consistently to the threshold. Hence, we use $\sigma = 0.4 * (1 - \mu)$.

However, as was the case with pre-flop re-weighting, there is a clear source of error when $\mu$ is very low. Observe that the area of the re-weighting function is a rectangle of height 1 and width $1 - (\mu + \sigma)$, with a triangle between $\mu-\sigma$ and $\mu+\sigma$ (Figure 7.2). That is,

$$A = (1 - (\mu + \sigma)) + \frac{2*\sigma}{2} = 1 - \mu$$ (7.5)

(if we were to ignore the area due to the minimum re-weighting factor of 0.01). But, since the function domain is bounded by 0, when $\sigma > \mu$ ($\mu < 0.2857$) we have $A < 1 - \mu$. For the weaker hands, the re-weighting factor is too low. When we are in this special case, we compensate by changing the re-weighting function so that at EHS' = 0 the re-weighting factor is r instead of 0.01 (and the function interpolates linearly between r and 1 when the hand value is between 0 and $\mu+\sigma$).

The value r is calculated to give $A = 1 - \mu$. The function now looks like a rectangle to the right of $\mu+\sigma$, a triangle of height 1 - r (width $\mu+\sigma$) and a second rectangle of height r below it (Figure 7.4). The total area of the function is now the area of the triangle subtracted from 1:

$$A = 1 - \frac{(1-r)(\mu + \sigma)}{2}.$$ (7.6)

Since we want $A = 1 - \mu$ we then see that

$$1 - \mu = 1 - \frac{(1-r)(\mu + \sigma)}{2}$$ (7.7)

which leads to the conclusion that

$$r = 1 - \frac{2*\mu}{\mu + \sigma}.$$ (7.8)

Figure 7.4: Re-weighting Function With $\mu < \sigma$