Manipulation of Schemes that Mix Voting with Chance

Abstract

MANIPULATION OF SCHEMES THAT MIX VOTING WITH CHANCE' BY ALLAN GIBBARD A decision scheme makes the probabilities of alternatives depend on individual strong orderings of them. It is strategy-proof if it logically precludes anyone's advantageously misrepresenting his preferences. It is unilateral if only one individual can affect the outcome, and duple if it restricts the final outcome to a fixed pair of alternatives. Any strategy-proof decision scheme, it is shown, is a probability mixture of schemes each of which is unilateral or duple. If it guarantees Pareto optimal outcomes, it is a probability mixture of dictatorial schemes. If it guarantees ex ante Pareto optimal lotteries, it is dictatorial. ' I have been helped in revising this paper by conversations with Mark Satterthwait Schwartz, and Hugo Sonnenschein, and by letters from Peter Fishburn and Richard Zeckhau grateful to the referee for remarkably detailed suggestions for shortening the proof of th theorem.

Probabilistic strategy-proof rules over single-peaked domains

“…An equivalent interpretation of strategy-proofness is the following: the expected utility under truth-telling is (weakly) greater than the the expected utility under misrepresentation for any utility representation of the agent's true preferences and for any preferences of the other agents. Here we are following the approach of Gibbard (1977). Strategy-proofness of a deterministic rule is a special case of strategy-proofness of a probabilistic rule.…”

On the fairness of random procedures

“…The two procedures are versions of the "random dictator" which Gibbard (1977) showed to be the only social choice rule satisfying strategy-proofness, ex ante efficiency, neutrality and unanimity. Heyd (2000) argued that the merit of the random dictator procedure is that it provides the members of the minority, who are "suppressed" in the majority rule system, with a say, though he did not make a distinction between the two procedures described above.…”

Strategy-Proof Voting on Compact Ranges

“…Their results apply, respectively, to the choice of level for one public good, the distribution of a …xed amount of private good, and the stable solution of matching problems, under appropriate domain restrictions. Many other environments allowing for nondictatorial and strategy-proof but not necessarily e¢ cient social choice functions have been studied: see for example, Serizawa (1996) on economies with one public and one private good, Barberà and Jackson (1995) on exchange economies, Gibbard (1977), and Barberà, Bogomolnaia, and van der Stel (1997) on the choice of lotteries as social outcomes. 1 Another family of interesting environments arises when alternatives can be described as points in the Euclidean space.…”