This chapter deals with models of collective choice in which individual agents’ preferences are aggregated into collective behavior. The first class of models use some fixed method to aggregate preferences. We assume that collective choices can be observed, but that individual agents’ preferences are unobserved. The second class of models are more structured models of voting in political economy and political science. A common idea in political science is that voters' preferences are “Euclidean”; we present the testable implications of this notion. Finally, we consider models of individual voter behavior and work out the corresponding observable implications.

TESTABLE IMPLICATIONS OF PREFERENCE AGGREGATION FUNCTIONS

The main questions in this section take the following form. Suppose that a group preference (or choice) is observable. Is this group preference consistent with a collection of rational agents whose preferences are aggregated according to some rule? We may, for example, wonder when a group's collective behavior is consistent with majority rule.

There are three ways to interpret the material that we are about to present. First, if we know the aggregation rule that the agents use, we may want to test the hypothesis that a society of agents behave rationally as individuals, when the only observable data come in the form of aggregate preference. Second, when the aggregation rule is unknown, we may want to test the joint hypotheses that a group of agents use a certain aggregation rule, and that they each behave rationally as individuals. Finally, a different interpretation of these results is that we might want to characterize all possible “paradoxes” that we might expect from using a given aggregation rule. Condorcet's paradox (a cycle on three alternatives) illustrates the problems that can arise from using majority rule. The results in this section describe all possible paradoxes of this type.

The model is as follows. Let X be a set of possible alternatives. We shall assume that we observe all possible binary comparisons of elements in X; that is, we observe a complete binary relation on X.