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Towards a Theory of Elections with Probabilistic Preferences
Peter C. Fishburn
William V. Gehrlein
Abstract
Social choice lottery rules are analyzed for two-candidate elections with voters who may be uncertain about whom they prefer. A voter's uncertainty is reflected by a nonobservable choice probability of voting for candidate A rather than candidate B, given that he votes. Lottery rules are based on the votes for A and B; they are to be monotonic and symmetric in voters and in candidates. Given n voters, all lottery rules are convex combinations of about n/2 basic rules ranging from the coin-flip rule to simple majority. Candidate A's win probability and two measures of expected voter satisfaction are examined as functions of the individuals' choice probabilities and the lottery rules. Comparisons are made between simple majority and the proportional lottery rule which assigns social choice probability of j/n to A when A gets j of n votes. Each of simple majority and the proportional lottery rule satisfies attractive properties that are not satisfied by the other rule.
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