The structure of (local) ordinal Bayesian incentive compatible random rules

Abstract

We explore the structure of locally ordinal Bayesian incentive compatible (LOBIC) random Bayesian rules (RBRs). We show that under lower contour monotonicity, for almost all prior profiles (with full Lebesgue measure), a LOBIC RBR is locally dominant strategy incentive compatible (LDSIC). We further show that for almost all prior profiles, a unanimous and LOBIC RBR on the unrestricted domain is random dictatorial, and thereby extend the result in Gibbard (1977) for Bayesian rules. Next, we provide sufficient conditions on a domain so that for almost all prior profiles, unanimous RBRs on it (i) are Pareto optimal, and (ii) are tops-only. Finally, we provide a wide range of applications of our results on single-peaked (on arbitrary graphs), hybrid, multiple single-peaked, single-dipped, single-crossing, multi-dimensional separable domains, and domains under partitioning. We additionally establish the marginal decomposability property for both random social choice functions and RBRs (for almost all prior profiles) on multi-dimensional domains, and thereby generalize Breton and Sen (1999). Since OBIC implies LOBIC by definition, all our results hold for OBIC RBRs