The range of collective accuracy for binary classifications under majority rule

Many decisions and actions can be framed as binary classification problems in which an outcome function maps states of the world into one of two outcomes and in which individuals use models (interpreted signals) to classify the state. For this class of binary classification problems, we fully characterize the range of possible group accuracies as a function of group size, average individual accuracy and diversity (average pairwise disagreement) or groups using majority rule. Our characterization yields five implications (i) the range of possible collective accuracies can be large, especially for groups of low accuracy individuals, (ii) up to moderate levels of diversity, the maximal collective accuracy gain equals the maximal collective accuracy loss, (iii) possible group accuracy set-wise improves in the average accuracy of their members, (iv) larger groups increase the range of possible collective accuracies unless diversity is high, and (v) for groups to be guaranteed to be more accurate than their average member, diversity must be high.