The accuracy of group majority decisions in groups with added members

Analytic results indicate that many heads often can be better than one even when none o f the added heads is as individually compe ten t as the first . We define individual competence (accuracy) in a dichotomous choice situation as the probability Pi of making the 'correct' or 'better' choice.~ Condorcet (1785) shows that if each person has equal competence, and if that competence, pi, is greater than chance (.5), then the accuracy of majority choice increases monotonically towards infallibility with increasing numbers of voters. Owen, Grofman and Feld (1981) generalize the Condorcet Theorem to verify that as long as the average competence of a group is greater than chance, then the competence of the group majority increases towards one as the group becomes larger. However, the practical problem in collective decision making often is to ascertain whether adding a particular set of persons to the decision-making group will add or detract from the competence of the decisions made. Shapley and Grofman (1981) and Nitzan and Paroush (1980, 1982) show that we can obtain the maximum competence with a weighted voting rule, with the weights monotonically related to the individual competences (according to a simple function that they specify). However, the competence of particular persons often is unknown; and in any case, weighting according to competence often is impractical. Consequently, we follow Grofman (1975, 1978); Margolis (1976); Owen, Grofman and Feld (1981); and Grofman, Owen and Feld