7.3.1 The Field Array

In Chapter 5 we were presented with a problem: how to use the information available in multiple weight arrays to get a general value for strength and potential. The solution was the field array, an intuitive approach that fits into the existing framework. We average (or add, since in the relative sense they are the same operation) the weight arrays of all the opponents to give a new array representing the entire table. However, since the weights represent relative probabilities, we must normalize the individual weight arrays (to the same scale) before they can be combined. Specifically, when we need to calculate hand values for player p, we calculate a field array that is the average of the normalized weight arrays of all players except p.

This approach does introduce a small amount of error by abandoning some second-order considerations with respect to intersection cases, but again this error is minor when compared to the reduction in complexity. For example, if we want to compute the probability, given the appropriate normalized weight arrays, that either statement A (player 1 holds A$\spadesuit$-A$\heartsuit$) or statement B (player 2 holds A$\spadesuit$-A$\heartsuit$) is true we need to compute

P(A or B) = P(A) + P(B) - P(A and B).

In fact, it is significantly more complex than this. For each subcase we would need to also rule out all the other intersecting subcases ( e.g. the probability that player 1 holds A$\spadesuit$-A$\heartsuit$ precludes player 2 holding either the A$\spadesuit$ or the A$\heartsuit$).