7.2.2 Pre-Flop Re-Weighting

In the pre-flop we do not have the convenience of a percentile hand valuing system. So $\mu$ needs to be converted from a percentile value to a value in the IR scale. To achieve this we use $\mu$ to index into a sorted array (sample the nearest point) of the 1326 IR₇ values from Appendix A (for simplicity IR₇ is always used). For example, suppose an opponent raises 30% of all hands, this translates to hand with IR = +118 (roughly corresponding to an average of 0.118 bets won per hand played) so we now use $\mu = 118$.

We do not observe the consistency of an opponent adhering to the estimated threshold, or of any other specific tendencies. While two separate opponents may call on average with a 118 hand, they both may have a very different standard deviation to the distribution of hands they call with (one may rarely call with a hand below 0 while another may sometimes play a hand as low as -200). For the present implementation, we have selected $\sigma = 330$ in an ad hoc manner: 68.26% of the hands (or two standard deviations in a normal distribution) lie in the income rate range -323 to +336. However, we only use a linear interpolation for re-weighting values within the range ($\mu-\sigma$,$\mu+\sigma$) rather than an S-curve based on a normal distribution.

However, there is one clear source of error when $\mu$ is very low. Consider a player who has been observed to take a certain action in a certain situation 100% of the time. We re-weight with $\mu = -495$ (the lowest value), but this means that even the hands with the lowest IR will only be re-weighted with a factor of 0.5 (which should be 1). For this reason, we do not re-weight when $\mu$ is below the 5th percentile (-433 in IR₇). A more accurate fix is possible, but is unlikely to be worth the added complexity.