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Social Choice and Welfare

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01-01-2015

Bounded response and the equivalence of nonmanipulability and independence of irrelevant alternatives

Author: Shin Sato

Published in: Social Choice and Welfare | Issue 1/2015

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Abstract

A social welfare function satisfies Bounded Response if the smallest change in the variable (i.e., preference profile) leads to the smallest change, if any, in the value (i.e., social preference). We show that each social welfare function on each connected domain satisfies Bounded Response and a nonmanipulability condition if and only if it satisfies a monotonicity condition and independence of irrelevant alternatives. Moreover, under Bounded Response, we show the equivalence of various notions of nonmanipulability of social welfare functions.

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See, for example, Young (1995) for arguments supporting IIA.

See Barberà, (2010) for a survey. The Gibbard–Satterthwaite theorem (Gibbard 1973; Satterthwaite 1975) is the seminal result.

Let \(f\) be a social welfare function and \(X\) a set of all alternatives. For each preference profile \(\varvec{R}\) and each set \(A\subset X\), let \(F(\varvec{R}, A)\) be the first ranked alternative in \(A\) according to \(f(\varvec{R})\). Then, \(F(\cdot , A)\) is a social choice function on \(A\), and we say that \(F(\cdot , A)\) is derived from \(f\).

Blair and Muller (1983)’s theorem involves other properties as well. We borrow this simpler version from Moulin (1988).

Bossert and Storcken (1992) assume that the number of alternatives is at least \(4\). Additional properties are nonimposition and weak extrema independence for even number of agents, and nonimposition and extrema independence for any number of agents.

A binary relation \(T\) on \(X\) is a subset of \(X\times X\). For each pair \(x, y\in X\), we write \(x~T~y\) for \((x, y)\in T\). A binary relation \(T\) is complete if for each pair \(x, y\in X\), either \(x~T~y\) or \(y~T~x\), transitive if for each triple \(x, y, z\in X\), [\(x~T~y\) and \(y~T~z\)] imply \(x~T~z\), antisymmetric if for each pair \(x, y\in X\), [\(x~T~y\) and \(y~T~x\)] imply \(x = y\). A binary relation is called a linear order if it is complete, transitive, and antisymmetric.

See footnote 3 for the definition of derived social choice functions.

Sato (2013) shows that on a connected domain, each social choice function is strategy-proof if and only if it is nonmanipulable by preferences which are adjacent to the true preference.

\(f\) is Agreement Manipulable because \(f(R_i^3, \varvec{R}_{-i})\) agrees more with \(R_i^1\) than \(f(R_i^1, \varvec{R}_{-i})\). To see Adjacency-restricted Agreement Nonmanipulability, consider the case \(R_i =R_i^1\). In \(\mathcal {D}\), only \(R_i^2\) is adjacent to \(R_i\). Because \((y, z) \in ( f(R_i^1, \varvec{R}_{-i})\cap R_i^1)\) and \((y, z)\not \in (f(R_i^2, \varvec{R}_{-i})\cap R_i^1)\), \(( f(R_i^1, \varvec{R}_{-i})\cap R_i^1) \subsetneq (f(R_i^2, \varvec{R}_{-i})\cap R_i^1)\) does not hold. Similar arguments hold in the other cases \(R_i =R_i^2\) or \(R_i = R_i^3\). Thus, \(f\) is Adjacency-restricted Agreement Nonmanipulable.

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Title: Bounded response and the equivalence of nonmanipulability and independence of irrelevant alternatives
Author: Shin Sato
Publication date: 01-01-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 1/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0825-8

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