The structure of non-manipulable social choice rules on a tree

doi:10.1016/0165-4896(93)00720-F

Mathematical Social Sciences

Volume 27, Issue 2, April 1994, Pages 123-131

https://doi.org/10.1016/0165-4896(93)00720-F Get rights and content

Abstract

In this paper we give a theorem describing a structure of any non-manipulable social choice rule on a tree. In particular, any such rule is a median of dictatorial and constant rules.

References (4)

G. Demange
Single peaked orders on a tree
Math. Soc. Sci.
(1982)
V.I. Danilov et al.
Mechanisms of Social Choice
(1991)

There are more references available in the full text version of this article.

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Strategy-proofness of the unanimity with status-quo rule over restricted domains
2022, Economics Letters
Single peaked domains with tree-shaped spectra
2020, Mathematical Social Sciences
Citation Excerpt :
(ii) thus, in particular, at any tree-based single peaked profile the set of Condorcet winners of a simple majority game is non-empty, and some appropriate5 version of the median voter theorem consequently holds; (iii) any sincere single peaked preference profile of a TSP domain yields a a strong Nash equilibrium outcome of the game resulting from the combination of such profile with the social choice function which is defined on the aforementioned domain and selects the median of the top alternatives at any preference profile (i.e. equivalently such a social choice function is coalitionally strategy-proof : see Danilov (1994) and Vannucci (2016)).6 Such remarkable properties shared by all TSP domains largely explain the extensive body of literature devoted to them, and to the related problems of their characterization and recognition.
On strategy-proofness and semilattice single-peakedness
2020, Games and Economic Behavior
We study social choice rules defined on the domain of semilattice single-peaked preferences. Semilattice single-peakedness has been identified as the condition that a set of preferences must satisfy so that the set can be the domain of a strategy-proof, tops-only, anonymous and unanimous rule. We characterize the class of all such rules on that domain and show that they are deeply related to the supremum of the underlying semilattice structure.
Strategy-proof location of public facilities
2018, Games and Economic Behavior
Consider the problem of locating a public facility taking into account the agents' preferences. To construct strategy-proof social choice rules, we propose a new preference domain that allows agents to have any single-peaked or any single-dipped preference on the location of the facility such that the peak/dip of the preference is in her own location. We characterize all strategy-proof rules in this general framework and study the conditions under which this family of strategy-proof rules includes non-dictatorial rules that have more than two alternatives in the range or that are Pareto efficient. Finally, we characterize for some focal cases all strategy-proof and Pareto efficient rules.
Weakly unimodal domains, anti-exchange properties, and coalitional strategy-proofness of aggregation rules
2016, Mathematical Social Sciences
Citation Excerpt :
Indeed, some facts about equivalence of simple and coalitional strategy-proofness (or its failure) on full unimodal domains in some specific median interval spaces are well-known. That is largely due to the circumstance that the structure of strategy-proof aggregation rules in those spaces is now well understood: in fact, it has been established that strategy-proof aggregation rules on unimodal domains in median interval spaces can be represented by iterated medians of projections (i.e. dictatorial rules) and constants (see e.g. Moulin, 1980, Danilov, 1994, Savaglio and Vannucci, 2014). Let us then start with a quick review of the best known classes of examples:
It is shown that simple and coalitional strategy-proofness of an aggregation rule on any rich weakly unimodal domain of an idempotent interval space are equivalent properties if that space satisfies interval anti-exchange, a basic property also shared by a large class of convex geometries including–but not reducing to–trees and Euclidean convex spaces. Therefore, strategy-proof location problems in a vast class of networks fall under the scope of that proposition.
It is also established that a much weaker minimalanti-exchangeproperty is necessary to ensure equivalence of simple and coalitional strategy-proofness in that setting. An immediate corollary to that result is that such equivalence fails to hold both in certain median interval spaces including those induced by bounded distributive lattices that are not chains, and in certain non-median interval spaces including those induced by partial cubes that are not trees.
Thus, it turns out that anti-exchange properties of the relevant interval space provide a powerful general common principle that explains the varying relationship between simple and coalitional strategy-proofness of aggregation rules for rich weakly unimodal domains across different interval spaces, both median and non-median.
On domains that admit well-behaved strategy-proof social choice functions
2013, Journal of Economic Theory
Citation Excerpt :
An example of this is the domain of single-peaked domains on trees Demange [13] that we discuss further in Section 3.3. There are several papers (Danilov [12], Schummer and Vohra [28]) that investigate the structure of strategy-proof social choice functions that choose locations on trees where preferences are single-peaked-like (such as quadratic). It is clear that our focus is different from that in these papers.
In this paper, we investigate domains that admit “well-behaved” strategy-proof social choice functions. We show that if the number of voters is even, then every domain that satisfies a richness condition and admits an anonymous, tops-only, unanimous and strategy-proof social choice function, must be semi-single-peaked. Conversely every semi-single-peaked domain admits an anonymous, tops-only, unanimous and strategy-proof social choice function. Semi-single-peaked domains are generalizations of single-peaked domains on a tree introduced by Demange (1982) [13].

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The structure of non-manipulable social choice rules on a tree

Abstract

Math. Soc. Sci.

Mechanisms of Social Choice