Abstract
In this paper we consider the exogenous indifference classes model of Barberá and Ehlers (2011) and Sato (2009) and analyze further the relationship between the structure of indifference classes across agents and dictatorship results. The key to our approach is the pairwise partition graph. We provide necessary conditions on these graphs for strategy-proofness and unanimity (or efficiency) to imply dictatorship. These conditions are not sufficient; we also provide separate stronger conditions that are sufficient. A full characterization is obtained in the case of two agents for domains where strategy-proofness and efficiency imply dictatorship.
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Notes
In addition, Sato (2012) considers social choice problem where every individual has an exogenous partition of the set of alternatives and analyses the structure of strategy-proof SCFs. However, in Sato (2012), exogenous partitions across agents are same and agents have strict preferences over alternatives.
Since \(G(S_i,S_j)\) is not connected, there exist two vertices u and v which are not connected. Let \(V'=\{u': u \textit{ and } u' \textit{are connected}\}\cup \{u\}\) and \(V''=V\setminus V'\). Note that \(u\in V'\) and \(v\in V''\). Since u and v are not isolated vertices, \(|V'|\), \(|V''|\ge 2\). Also note that \(\{u,v\}\notin E\) for all \(u\in V'\) and \(v\in V''\). Therefore the existence of \(V'\) and \(V''\) satisfying such conditions is guaranteed.
Suppose that \(l\ne i,j\) is the dictator. Consider any indifference class \(s_i\in S_i\). Note that there exists a profile \(R\in \mathbb {R}(S)\) such that \(f(R)\in s_i\). Since l is the dictator, \(f(R)\in r_1(R_l)\). Also note that at any \(R'_l\), \(f(R'_l,R_{-l})\in s_i\) and \(f(R'_l,R_{-l})\in r_1(R'_l)\). Therefore \(s_i\) has a non-empty intersection with any indifference class of agent l. Since \(s_i\) is an arbitrary indifference class in \(S_i\), the graph \(G(S_i,S_l)\) is discrete.
By our assumption, \(G(S_i, S_j)\) is not a discrete graph.
Note that \(G(S_i,S_j)\) has two non-isolated vertices \(u'\) and \(v'\) which are not connected. Let \(V'=\{u: u \textit{ and } u' \textit{are connected}\}\cup \{u'\}\) and \(V''=V\setminus V'\). It is clear that \(u'\in V'\) and \(v'\in V''\). Also, \(\{u,v\}\notin E\) for all \(u\in V'\) and \(v\in V''\). Since \(u'\) and \(v'\) are non-isolated vertices, \(|V'|\), \(|V''|\ge 2\).
A cycle graph is a connected graph with the degree of every vertex being 2.
These arguments closely follow counterparts in Sen (2001).
Unanimity and ontoness are equivalent for strategy-proof SCFs defined over a suitably rich domain of strict preferences (see Dogan and Sanver (2007)). In our model, strategy-proof and onto SCFs are not necessarily unanimous. Similarly, there are strategy-proof and unanimous SCFs that are not onto. We omit the details.
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We would like to thank Debasis Mishra, Hans Peters, Soumendu Sarkar, Rakesh Vohra, participants of the 2012 Society for Social Choice and Welfare meeting and participants of the 2013 Society for Economic Design meeting for their useful comments and suggestions. In addition, we are also grateful to two referees of the journal for several helpful comments and suggestions. Pramanik gratefully acknowledges financial assistance from Indian Statistical Institute and JSPS KAKENHI - 15K17021. This paper is based on a chapter of Pramanik’s Ph.D dissertation submitted to the Indian Statistical Institute in 2014.
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Pramanik, A., Sen, A. Pairwise partition graphs and strategy-proof social choice in the exogenous indifference class model. Soc Choice Welf 47, 1–24 (2016). https://doi.org/10.1007/s00355-015-0944-x
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DOI: https://doi.org/10.1007/s00355-015-0944-x