A Characterization of Single-Peaked Preferences via Random Social Choice Functions

Published in Theoretical Economics https://doi.org/10.3982/TE1972</p

The Structure of Strategy-Proof Random Social Choice Functions over Product Domains and Lexicographically Separable Preferences

Ministry of Education, Singapore under its Academic Research Funding Tier

Random Dictatorship Domains

Published in Games and Economic Behavior https://doi.org/10.1016/j.geb.2014.03.017</p

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

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

On the equivalence of strategy-proofness and upper contour strategy-proofness for randomized social choice functions

We consider a weaker notion of strategy-proofness called upper contour strategy-proofness (UCSP) and investigate its relation with strategy-proofness (SP) for random social choice functions (RSCFs). Apart from providing a simpler way to check whether a given RSCF is SP or not, UCSP is useful in modeling the incentive structures for certain behavioral agents. We show that SP is equivalent to UCSP and elementary monotonicity on any domain satisfying the upper contour no restoration (UCNR) property. To analyze UCSP on multi-dimensional domains, we consider some block structure over the preferences. We show that SP is equivalent to UCSP and block monotonicity on domains satisfying the block restricted upper contour preservation property. Next, we analyze the relation between SP and UCSP under unanimity and show that SP becomes equivalent to UCSP and multi-swap monotonicity on any domain satisfying the multi-swap UCNR property. Finally, we show that if there are two agents, then under unanimity, UCSP alone becomes equivalent to SP on any domain satisfying the swap UCNR property. We provide applications of our results on the unrestricted, single-peaked, single-crossing, single-dipped, hybrid, and multi-dimensional domains such as lexicographically separable domains with one component ordering and domains under committee formation

Unanimous and strategy-proof probabilistic rules for single-peaked preference profiles on graphs

Finitely many agents have preferences on a finite set of alternatives, single-peaked with respect to a connected graph with these alternatives as vertices. A probabilistic rule assigns to each preference profile a probability distribution over the alternatives. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles at which all peaks are on leaves of the tree and, thus, extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leaves. Finally, the two results are combined to obtain a general characterization for every connected graph by using its block tree representation

The single-peaked domain revisited: A simple global characterization

It is proved that, among all restricted preference domains that guarantee consistency (i.e. transitivity) of pairwise majority voting, the single-peaked domain is the only minimally rich and connected domain that contains two completely reversed strict preference orders. It is argued that this result explains the predominant role of single-peakedness as a domain restriction in models of political economy and elsewhere. The main result has a number of corollaries, among them a dual characterization of the single-dipped do-main; it also implies that a single-crossing (`order-restricted\u27) domain can be minimally rich only if it is a subdomain of a single-peaked domain. The conclusions are robust as the results apply both to domains of strict and of weak preference orders, respectively