Abstract
We show that the class of social welfare functions that satisfy a weak independence condition identified by Campbell (J Econ 12:259–272, 1976) and Baigent (J Econ 47(4):407–411, 1987) is fairly rich and freed of a power concentration on a single individual. This positive result prevails when a weak Pareto condition is imposed. Moreover, under weak independence, an impossibility of the Wilson (J Econ 5:478–486, 1972) type vanishes.
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Notes
See Campbell (1976) for a discussion of the computational advantages of quasi IIA. Note that when social indifference is not allowed, IIA and quasi IIA are equivalent.
Baigent (1987) claims this impossibility in an environment with at least three alternatives. Nevertheless, Campbell and Kelly (2000b) show the existence of Pareto optimal and quasi IIA SWF when there are precisely three alternatives. They also show that the impossibility announced by Baigent (1987) prevails when there are at least four alternatives and even under restricted domains.
As usual, for any distinct \(x,y\in A\), we intepret \(xP_{i}y\) as \(x\) being preferred to \(y\) in view of \(i\).
So for any distinct \(x,y\in A\), we have \(x \upsilon ^{*}(P) y\) whenever \(x \upsilon (P) y\) and not \(y \upsilon (P) x\).
We interpret \({\displaystyle {\begin{array}{l}x\\ y\\ \end{array}}}\) as \(x\) being preferred to \(y\); \({\displaystyle {\begin{array}{l}y\\ x\\ \end{array}}}\) as \(y\) being preferred to \(x\); and \(xy\) as indifference between \(x\) and \(y\). Note that all preferences over two alternatives are always transitive.
So for any \(i\in N\), we have \(P_{i}^{\{x,y\}}={\displaystyle {\begin{array}{l}x\\ y\\ \end{array}}} \Longleftrightarrow xP_{i}y\).
This is first stated by Gibbard (1968).
For example the SWF \(\alpha \) where \(x\) \(\alpha (P)\) \(y\) for all \(x,y\in A\) and for all \(P\in L(A)^{N}\) is a weak dictatorship but not Pareto optimal.
See Footnote 3.
We use the definition of “cycle” as stated by Peris and Subiza (1999).
The top-cycle, introduced by Good (1971) and Schwartz (1972), has been explored in details. Moreover, Peris and Subiza (1999) extend this concept to weak tournaments. In their setting, as \(K(X,S)\) is a cycle, there does not exist \(Y\subset K(X,S)\) with \(yS^{*}x\) for all \(y\in Y\) and for all \( x\in K(X,S)\backslash Y\).
We omitted the proof since the properties of top-cycles has been well discussed in the literature (see Deb 1977).
As a remark, these results can also be obtained by using graph theory. Any complete binary relation \(S\) over \(A\) is a directed graph (digraph) \(G(S)\) defined as follows; the elements of \(A\) becomes the vertices of digraph \( G(S) \) and the edges are determined by \(S.\) (\(xSy\) means that there is an edge from \(x\) to \(y\) and another one from \(y\) to \(x\).) Then, the ordered partitions of the transitive closure of the complete relation \(S\) are the strong components of the digraph \(G(S).\) Hence, it is already well known as an elementary result that the set of these strong components are linearly ordered and the maximal strong component is equivalent to the notion of top-cycle (see Bang-Jensen and Gutin 2007).
We say that \(\alpha :L(A)^{N}\rightarrow \textit{CT}(A)\) is a singleton-valued selection of \(\rho \circ \upsilon \) iff \(\alpha (P)\in \rho \circ \upsilon (P)\) \(\forall P\in L(A)^{N}\).
By “taking the transitive closure”, we mean to replace cycles with indifference classes. Formally speaking, writing \((A_{1},A_{2},\ldots ,A_{k})\) for the ordered partition induced by \(\upsilon (P)\) \(\in \Theta \) at \(P\in L(A)^{N}\), take \(\alpha (P)\in \rho (\upsilon (P))\) where \(x\) \(\alpha ^{*}(P)\) \(y\) for all \(x\in A_{i}\) and for all \( y\in A_{j}\) with \(i<j\). One can see Sen (1986) for a general discussion of the “closure methods”.
\(\alpha :L(A)^{N}\rightarrow \textit{CT}(A)\) is non-imposed iff \(\forall x,y\in A\) \( \exists P\in L(A)^{N}\) with \(x\) \(\alpha (P)\) \(y\).
\(\alpha \) is anti-dictatorial iff \(\exists i\in N\) such that \( xP_{i}y\Longrightarrow y\) \(\alpha ^{*}(P)\) \(x\) \(\forall P\in \) \( L(A)^{N},\forall x,y\in A.\)
\(\alpha :L(A)^{N}\rightarrow \textit{CT}(A)\) is null iff \(x\) \(\alpha (P)\) \(y\) \( \forall x,y\in A\) and \(\forall P\in L(A)^{N}\).
As a matter of fact, the SWF in Example 2 of Campbell and Kelly (2000b), which shows the failure of Theorem 3.1 for \(\#A=3\), belongs to this class.
We conjecture, by relying on Dasgupta and Maskin (2008), that this will be the pairwise majority rule.
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Acknowledgments
We thank Goksel Asan, Nicholas Baigent, Donald Campbell, Semih Koray, Gilbert Laffond, Jean Laine, Ipek Ozkal-Sanver and Jack Stecher for their constructive suggestions. Our research is part of a project entitled “Social Perception - A Social Choice Perspective”, supported by Istanbul Bilgi University Research Fund. Remzi Sanver acknowledges the support of the Turkish Academy of Sciences Distinguished Young Scientist Award Program (TUBA-GEBIP). Of course, the authors are responsible from all possible errors.
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Coban, C., Sanver, M.R. Social choice without the Pareto principle under weak independence. Soc Choice Welf 43, 953–961 (2014). https://doi.org/10.1007/s00355-014-0812-0
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DOI: https://doi.org/10.1007/s00355-014-0812-0