We explore the structure of local ordinal Bayesian incentive compatible (LO-BIC) 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, singlepeaked (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. KEYWORDS. random Bayesian rules; random social choice functions; (local) ordinal Bayesian incentive compatibility; (local) dominant strategy incentive compatibility JEL CLASSIFICATION CODES. D71; D82 * The authors would like to thank Arunava Sen and Debasis Mishra for their invaluable suggestions.
In this paper, we consider choice functions that are unanimous, anonymous, symmetric, and group strategy-proof and consider domains that are single-peaked on some tree. We prove the following three results in this setting. First, there exists a unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. Second, a choice function is unanimous, anonymous, symmetric, and group strategy-proof on a single-peaked domain on a tree if and only if it is the pairwise majority rule (also known as the tree-median rule) and the number of agents is odd. Third, there exists a unanimous, anonymous, symmetric, and strategy-proof choice function on a strongly path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. As a corollary of these results, we obtain that there exists no unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if the number of agents is even.
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