Necessary and sufficient conditions for pairwise majority decisions on path-connected domains

In this paper, we consider choice functions that are unanimous, anonymous, symmetric, and group strategy-proof and consider domains that are single-peaked on some tree. We prove the following three results in this setting. First, there exists a unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. Second, a choice function is unanimous, anonymous, symmetric, and group strategy-proof on a single-peaked domain on a tree if and only if it is the pairwise majority rule (also known as the tree-median rule) and the number of agents is odd. Third, there exists a unanimous, anonymous, symmetric, and strategy-proof choice function on a strongly path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. As a corollary of these results, we obtain that there exists no unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if the number of agents is even.

[1]  H. Moulin.  On strategy-proofness and single peakedness , 1980 .

[2]  V. Danilov.  The structure of non-manipulable social choice rules on a tree , 1994 .

[3]  P. Pattanaik,et al.  Necessary and Sufficient Conditions for Rational Choice under Majority Decision , 1969 .

[4]  Kenneth O. May,et al.  A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision , 1952 .

[5]  Arunava Sen,et al.  A Characterization of Single-Peaked Preferences via Random Social Choice Functions , 2016 .

[6]  H.C.M. de Swart.  Navin Aswal e.a., Dictatorial Domains. [Review of the book Dictatorial Domains (Journal of Economic Theory 22), , 2003] , 2005 .

[7]  Souvik Roy,et al.  On single-peaked domains and min–max rules , 2018, Soc. Choice Welf..

[8]  Arunava Sen,et al.  Dictatorial domains , 2003 .

[9]  Jacques-François Thisse,et al.  Outcomes of voting and planning: Condorcet, Weber and Rawls locations , 1981 .

[10]  Arunava Sen,et al.  Tops-only domains , 2011 .

[11]  D. Black.  On the Rationale of Group Decision-making , 1948, Journal of Political Economy.

[12]  Salvador Barberà,et al.  Individual versus group strategy-proofness: When do they coincide? , 2010, J. Econ. Theory.

[13]  Rakesh V. Vohra,et al.  Strategy-proof Location on a Network , 2002, J. Econ. Theory.

[14]  Ken-ichi Inada.  THE SIMPLE MAJORITY DECISION RULE , 1969 .

[15]  Gabrielle Demange,et al.  Single-peaked orders on a tree , 1982, Math. Soc. Sci..

[16]  John A. Weymark,et al.  A unified approach to strategy-proofness for single-peaked preferences , 2011 .

[17]  Salvador Barberà,et al.  Voting by Committees , 1991 .

[18]  P. Laplace.  Théorie analytique des probabilités , 1995 .

[19]  Clemens Puppe,et al.  The single-peaked domain revisited: A simple global characterization , 2018, J. Econ. Theory.

[20]  Klaus Nehring,et al.  The structure of strategy-proof social choice - Part I: General characterization and possibility results on median spaces , 2007, J. Econ. Theory.