We consider social choice rules which select a lottery over outcomes for 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 these rules are obtained by combining one of the following principles: (1) start from a fixed subset of lotteries, and for each profile let one fixed agent choose her preferred lottery from that subset (we call them unilateral rules); or, (2) start from two fixed lotteries and a rule assigning weights to each of them depending on the coalition of agents which prefer one of the two lotteries to the other; let the outcome at each profile be the convex combination of these two given lotteries according to the weights which correspond to them at that profile (these rules are called duples). All probabilistic mixtures (convex combinations or integrals) of unilateral and duple rules satisfying some additional and natural requirements are strategy-proof. Because we are 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. Smoothness requirements are not only attractive per se, but they can also be expected to help in telling apart different types of rules. Notice that unilateral rules can be very smooth, while no duple can even be continuous. Yet, continuity can be regained by combining a continuum of duples: 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 prove that any strategy-proof probabilistic function of class C2 must indeed be also a convex combination of unilateral schemes.
- Social choice rules