Algorithmic Calculation of the Optimality Probability of Decision Rules

We study the dichotomous choice model under conditions of uncertainty. In this model, a committee of decision makers is required to select one of two alternatives, of which exactly one is correct. A decision rule translates the individual opinions into a group decision. We focus on the direction concerned with the identification of the optimal decision rule under partial information on the decision skills. Namely, we assume the correctness probabilities of the committee members to be independent random variables, selected from some given distribution. In addition, we assume that the ranking of the members in the committee is known. Thus, one can follow rules based on this ranking. One of the commonly used measures of the efficiency of a decision rule is its probability of being optimal. Here we provide a method for an explicit calculation of this probability for any given weighted majority rule for a wide family of distribution functions. Moreover, under the assumption of exponentially distributed decision skills, we provide an improved algorithm for this calculation. We illustrate our results with various examples.