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

We explore the structure of local ordinal Bayesian incentive compatible (LOBIC) random Bayesian rules (RBRs). We show that under lower contour monotonicity, almost all (with Lebesgue measure 1) LOBIC RBRs are local dominant strategy incentive compatible (LDSIC). We also provide conditions on domains so that unanimity implies lower contour monotonicity for almost all LOBIC RBRs. We provide sufficient conditions on a domain so that almost all unanimous RBRs on it (i) are Pareto optimal, (ii) are tops-only, and (iii) are only-topset. Finally, we provide a wide range of applications of our results on the unrestricted, single-peaked (on graphs), hybrid, multiple single-peaked, single-dipped, single-crossing, multidimensional separable, lexicographic, and domains under partitioning. We additionally establish the marginal decomposability property for both random social choice functions and almost all RBRs on multi-dimensional domains, and thereby generalize Breton and Sen (1999). Since OBIC implies LOBIC by definition, all our results hold for OBIC RBRs.