On Variable Majority Rule and Kramer's Dynamic Competitive Process

Abstract

This paper defines and characterizes axiomatically two rational voting procedures which are variants of simple majority rule. The rules are closely related and coincide when individual indifference is excluded. Some of the conditions used are then discussed and related to similar properties that have been studied in the social choice literature. It has long been known that a majority winner in a field of three or more candidates need not exist in general unless the preferences of the electorate display some degree of similarity. Sen and Pattanaik (1969) derived a set of necessary and sufficient conditions for a majority preference relation to be well behaved. Kramer (1973) studied these conditions in the context of quasi-concave differentiable preferences over a multidimensional Euclidean choice space, the natural setting for most economic decision models, and showed that they were extremely restrictive. Even the most modest diversity in preferences causes a violation of all the Sen-Pattanaik conditions in Rn. At least two reactions are possible to Kramer's important theorem. One response is to deny its relevance to actual political systems; few collectivities in fact make decisions by pairwise comparisons of all feasible alternatives using direct majority rule. Building on the work of Downs (1956), much effort has been devoted to the analysis of spatial models in which the electorate votes only on alternatives advocated by a small number of parties or candidates. But the existence of equilibria in these models has until recently been equally problematic (see Riker and Ordeshook (1972) for an overview of spatial models). In another paper Kramer (1977) has recently developed a model in this tradition which is the most general structure yet examined in which majority voting is at all well behaved. The model is a dynamic one in which two parties are assumed to compete for votes in a series of elections over time. The parties choose as platforms points in a multidimensional policy space with the objective of maximizing the number of votes the party receives in the next election. Each voter chooses his most preferred party according to his spherical indifference map. Kramer shows that the sequence of successively-enacted policies tends to converge to a relatively small subset of the feasible policy alternatives. It is of considerable interest to know what normative properties are exhibited by this decision process. The social preference relation which the competing parties implicitly maximize is the second of the rules to be axiomatized in this paper. There is a second possible response to the difficulties posed by the restrictiveness of the Sen-Pattanaik conditions in economic models. One might look for ways to repair defective (i.e. cyclic) majority preference relations without straying too far from simple majority rule and thereby losing its desirable normative characteristics. Craven (1971), Schwartz (1972), Brown (forthcoming), Ferejohn and Grether (1974) and Young (1977), for example, have investigated such procedures. Virtually nothing is known about the