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

Agenda manipulation-proofness, stalemates, and redundant elicitation in preference aggregation. Exposing the bright side of Arrow's theorem

This paper provides a general framework to explore the possibility of agenda manipulation-proof and proper consensus-based preference aggregation rules, so powerfully called in doubt by a disputable if widely shared understanding of Arrow's `general possibility theorem'. We consider two alternative versions of agenda manipulation-proofness for social welfare functions, that are distinguished by `parallel' vs. `sequential' execution of agenda formation and preference elicitation, respectively. Under the `parallel' version, it is shown that a large class of anonymous and idempotent social welfare functions that satisfy both agenda manipulation-proofness and strategy-proofness on a natural domain of single-peaked `meta-preferences' induced by arbitrary total preference preorders are indeed available. It is only under the second, `sequential' version that agenda manipulation-proofness on the same natural domain of single-peaked `meta-preferences' is in fact shown to be tightly related to the classic Arrowian `independence of irrelevant alternatives' (IIA) for social welfare functions. In particular, it is shown that using IIA to secure such `sequential' version of agenda manipulation-proofness and combining it with a very minimal requirement of distributed responsiveness results in a characterization of the `global stalemate' social welfare function, the constant function which invariably selects universal social indifference. It is also argued that, altogether, the foregoing results provide new significant insights concerning the actual content and the constructive implications of Arrow's `general possibility theorem' from a mechanism-design perspective

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

Maximal Domain for Strategy-Proof Rules in Allotment Economies

We consider the problem of allocating an amount of a perfectly divisible good among a group of n agents. We study how large a preference domain can be to allow for the existence of strategy-proof, symmetric, and efficient allocation rules when the amount of the good is a variable. This question is qualified by an additional requirement that a domain should include a minimally rich domain. We first characterize the uniform rule (Bennasy, 1982) as the unique strategy-proof, symmetric, and efficient rule on a minimally rich domain when the amount of the good is fixed. Then, exploiting this characterization, we establish the following: There is a unique maximal domain that includes a minimally rich domain and allows for the existence of strategy-proof, symmetric, and efficient rules when the amount of good is a variable. It is the single-plateaued domain.

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