STRATEGY-PROOF PROBABILISTIC RULES FOR EXPECTED UTILITY MAXIMIZERS

We consider social choice rules which select a lottery over outcomes f or each profile of individual preferences. Agents are assumed to have preferences over lotteries satisfying the axioms of expected utility. We exhibit a large class of rules satisfying strategy-proofness. All t hese rules are obtained by combining one of the following principles: (1) start from a fixed subset of lotteries, and for each profile let o ne fixed agent choose her preferred lottery from that subset (we call them unilateral rules); or, (2) start from two fixed lotteries and a r ule assigning weights to each of them depending on the coalition of ag ents which prefer one of the two lotteries to the other; let the outco me at each profile be the convex combination of these two given lotter ies according to the weights which correspond to them at that profile (these rules are called duples). All probabilistic mixtures (convex co mbinations or integrals) of unilateral and duple rules satisfying some additional and natural requirements are strategy-proof. Because we ar e facing a wide class of procedures, we investigate the possibility of designing some which are not only strategy-proof but also continuous or even smooth in their responses to changes in preferences. Smoothnes s requirements are not only attractive per se, but they can also be ex pected to help in telling apart different types of rules, Notice that unilateral rules can be very smooth, while no duple can even be contin uous. Yet, continuity can be regained by combining a continuum of dupl es, we provide an example of a continuous strategy-proof probabilistic rule which is an integral of duples. However, there is a limit as to how smooth a rule can be without resorting to unilateral schemes. We p rove that any strategy-proof probabilistic function of class C-2 must indeed be also a convex combination of unilateral schemes. (C) 1998 El sevier Science B.V.