Majority and Positional Voting in a Probabilistic Framework

Abstract

A lot of attention has been devoted recently to the study of social decision-making procedures which combine voting with chance (for different approaches to the subject, see Zeckhauser (1969), Fishburn (1972a, b, 1978) Intriligator (1973) and Barbera and Sonnenschein (1977).) Decision schemes are procedures of this kind which assign a lottery on the set of alternatives to each N-tuple of rankings of alternatives. It is interpreted that they specify the probability with which each of the alternatives open to society is to be chosen, on the basis of the ordinal preferences on these alternatives expressed by the members of this society. In this paper I define two wide classes of decision schemes-supporting size and point voting decision schemes-which can be viewed as natural adaptations to the probabilistic framework of two basic principles used in making deterministic choices: majority and positional voting. It is noted that these two principles, which are in general incompatible within a deterministic framework, can be jointly satisfied within the setting of decision schemes by what I call simple decision schemes, a third class which is the intersection of the two above. The theorems in the paper characterize these three classes in terms of the properties they satisfy. Some of these properties are, in turn, adaptations to the new framework of standard conditions in social choice theory: anonymity, neutrality and strategyproofness. Two other new properties are introduced, which I call alternative independence and individual independence. The following results are obtained: