Testing Top Monotonicity

Top monotonicity is a relaxation of various well-known domain restrictions such as single-peaked and single-crossing for which negative impossibility results are circumvented and for which the median-voter theorem still holds. We examine the problem of testing top monotonicity and present a characterization of top monotonicity with respect to non-betweenness constraints. We then extend the definition of top monotonicity to partial orders and show that testing top monotonicity of partial orders is NP-complete

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

On Top-connected Single-peaked and Partially Single-peaked Domains

We characterize all domains for which the set of unanimous and strategy-proof social choice functions coincides with the set of min-max rules. As an application of our result, we obtain a characterization of unanimous and strategy-proof social choice functions on maximal single-peaked domains (Moulin (1980), Weymark (2011)), minimally rich single-peaked domains (Peters et al. (2014)), maximal regular single-crossing domain (Saporiti (2009)), and distance based single-peaked domains. We further consider domains that exhibit single-peaked property only over a subset of alternatives. We call such domains top-connected partially single-peaked domains. We characterize the unanimous and strategy-proof social choice function on such domains. As an application of this result, we obtain a characterization of the unanimous and strategy-proof social choice functions on multiple single-peaked domains (Reffgen (2015)), single-peaked domains on graphs, and several other domains of practical significance

Strategy-proof Rules on Top-connected Single-peaked and Partially Single-peaked Domains

We characterize all domains on which (i) every unanimous and strategy-proof social choice function is a min-max rule, and (ii) every min-max rule is strategy-proof. As an application of our result, we obtain a characterization of unanimous and strategy-proof social choice functions on maximal single-peaked domains (Moulin (1980), Weymark (2011)), minimally rich single-peaked domains (Peters et al. (2014)), maximal regular single-crossing domain (Saporiti (2009),Saporiti(2014)), and distance based single-peaked domains. We further consider domains that exhibit single-peaked property only over a subset of alternatives. We call such domains top-connected partially single-peaked domains and provide a characterization of the unanimous and strategy-proof social choice function on these domains. As an application of this result, we obtain a characterization of the unanimous and strategy-proof social choice functions on multiple single-peaked domains (Reffgen (2015)) and single-peaked domains on graphs. As a by-product of our results, it follows that strategy-proofness implies tops-onlyness on these domains. Moreover, we show that strategy-proofness and group strategy-proofness are equivalent on these domains

Strategy-proof Rules on Top-connected Single-peaked and Partially Single-peaked Domains

We characterize all domains on which (i) every unanimous and strategy-proof social choice function is a min-max rule, and (ii) every min-max rule is strategy-proof. As an application of our result, we obtain a characterization of unanimous and strategy-proof social choice functions on maximal single-peaked domains (Moulin (1980), Weymark (2011)), minimally rich single-peaked domains (Peters et al. (2014)), maximal regular single-crossing domain (Saporiti (2009),Saporiti(2014)), and distance based single-peaked domains. We further consider domains that exhibit single-peaked property only over a subset of alternatives. We call such domains top-connected partially single-peaked domains and provide a characterization of the unanimous and strategy-proof social choice function on these domains. As an application of this result, we obtain a characterization of the unanimous and strategy-proof social choice functions on multiple single-peaked domains (Reffgen (2015)) and single-peaked domains on graphs. As a by-product of our results, it follows that strategy-proofness implies tops-onlyness on these domains. Moreover, we show that strategy-proofness and group strategy-proofness are equivalent on these domains

A Unified Characterization of Randomized Strategy-proof Rules

This paper presents a unified characterization of the unanimous and strategy-proof random rules on a class of domains that are based on some prior ordering over the alternatives. It identifies a condition called top-richness so that, if a domain satisfies top-richness, then an RSCF on it is unanimous and strategy-proof if and only if it is a convex combination of tops-restricted min-max rules. Well-known domains like single-crossing, single-peaked, single-dipped etc. satisfy top-richness. This paper also provides a characterization of the random min-max domains. Furthermore, it offers a characterization of the tops-only and strategy-proof random rules on top-rich domains satisfying top-connectedness. Finally, it presents a characterization of the unanimous (tops-only) and group strategy-proof random rules on those domains

Optimal Voting Rules

We study dominant strategy incentive compatible (DIC) and deterministic mechanisms in a social choice setting with several alternatives. The agents are privately informed about their preferences, and have single-crossing utility functions. Monetary transfers are not feasible. We use an equivalence between deterministic, DIC mechanisms and generalized median voter schemes to construct the constrained-efficient, optimal mechanism for an utilitarian planner. Optimal schemes for other welfare criteria such as, say, a Rawlsian maximin can be analogously obtained

Are there any nicely structured preference~profiles~nearby?

We investigate the problem of deciding whether a given preference profile is close to having a certain nice structure, as for instance single-peaked, single-caved, single-crossing, value-restricted, best-restricted, worst-restricted, medium-restricted, or group-separable profiles. We measure this distance by the number of voters or alternatives that have to be deleted to make the profile a nicely structured one. Our results classify the problem variants with respect to their computational complexity, and draw a clear line between computationally tractable (polynomial-time solvable) and computationally intractable (NP-hard) questions