Ordinally Bayesian incentive compatible probabilistic voting rules

We study probabilistic voting rules in a two-voter model. The notion of incentive compatibility we consider is ordinal Bayesian incentive compatibility (OBIC) as introduced in d'Aspremont and Peleg (1988). We show that there exist anonymous and ex-post efficient probabilistic voting rules that are not random dictatorships and at the same time are OBIC with respect to an independently distributed generic prior. This contrasts with the results obtained for deterministic voting mechanisms obtained in Majumdar and Sen (2004) and in Mishra (2016). In case of neutral and efficient rules, there are two kinds of results. First we show that imposing OBIC with respect to some generic prior leads to random dictatorship when there are three alternatives. Second, we show that the result is no longer true when there are four or more alternatives and consequently we provide sufficient conditions on the priors for the result to be true. (C) 2021 Elsevier B.V. All rights reserved.