Optimizing group judgmental accuracy in the presence of interdependencies

Consider a group of people confronted with a dichotomous choice (for example, a yes or no decision). Assume that we can characterize each person by a probability, p _i, of making the ‘better’ of the two choices open to the group, such that we define ‘better’ in terms of some linear ordering of the alternatives. If individual choices are independent, and if the a priori likelihood that either of the two choices is correct is one half, we show that the group decision procedure that maximizes the likelihood that the group will make the better of the two choices open to it is a weighted voting rule that assigns weights, w _i, such that $$w_i propto log frac{{p_i }} {{1 - p_i }}.$$ We then examine the implications for optimal group choice of interdependencies among individual choices.