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

Karmokar, Madhuparna and Roy, Souvik (2020): The structure of (local) ordinal Bayesian incentive compatible random rules.

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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.

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