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Independence of irrelevant alternatives revisited

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Abstract

This paper aims to reexamine the axiom of the independence of irrelevant alternatives in the theory of social choice. A generalized notion of independence is introduced to clarify an informational requirement of binary independence which is usually imposed in the Arrovian framework. We characterize the implication of binary independence.

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Notes

  1. The “choice-function” formulation misleads some authors about an interpretation of the axiom: the independence of irrelevant alternatives is confused with Arrow’s choice axiom. This generates controversy over the meaning of the independence of irrelevant alternatives. See Hansson (1973), Ray (1973), Plott (1973), McLean (1995), Bordes and Tideman (1991), and Denicolò (2000).

  2. See also Saari (2001).

  3. Muller and Satterthwaite (1985) show that binary independence is equivalent to strategy proofness under a certain setting. Therefore, the independence of irrelevant alternatives must be satisfied for any truthful voting method. In this sense, binary independence can be regarded as a condition of information “revelation.” See also Blin (1976) and Muller and Satterthwaite (1977).

  4. Several authors focus on aspects of the axiom other than the informational one. See, for example, Sen (1993), Denicolö (1998), Baigent (1987), and Campbell and Kelly (2000b).

  5. Cato (2011) examines conditions under which the weak Pareto principle implies the strong Pareto principle.

  6. See Sen (1970). It is also known that the finiteness of \(N\) is necessary for the theorem. Fishburn (1970) shows that if \(N\) is infinite, then there exists a non-dictatorial social welfare function satisfying the two axioms.

  7. For a set \(A, 2^A\) denotes the power set of \(A\).

  8. See also Blair et al. (1975) and Cato (2010, 2012a, b).

  9. A collective choice rule \(f\) is oligarchical if and only if there exists \(A \subseteq N\) such that for all \(i \in A\), for all \(x, y \in X\), and for all \(\mathbf{R} \in \mathcal R ^{N}\),

    $$\begin{aligned} (x,y) \in P(R_i) \Rightarrow (x, y) \in f(\mathbf{R}), \end{aligned}$$

    and

    $$\begin{aligned}{}[(x,y) \in P(R_i) \text{ for } \text{ all } i \in A ] \Rightarrow (x, y) \in P(f(\mathbf{R})). \end{aligned}$$

  10. Brown (1975) provides another fundamental result for a social decision function. His theorem states that if a social decision function satisfies binary independence and weak Pareto, then it is collegial. Although Brown (1975) imposes binary independence, Banks (1995) points out that binary independence is redundant. Cato (2013) shows related results.

  11. An individual has veto over \((x,y)\) if and only if for all \(\mathbf{R} \in \mathcal R ^{N}, (x,y) \in P(R_i) \Rightarrow (x, y) \in f(\mathbf{R})\).

  12. Vickrey (1960) claims that “[w]here the number of alternatives is small, omission or inclusion of one or two alternatives may be crucial; where the number of alternatives is large, however, omission or inclusion even of a number of alternatives constituting the same proportion of the total number may be much less likely to affect the result” (Vickrey 1960, p. 517).

  13. I am grateful to an anonymous referee for informing me about the works by Campbell and Kelly.

  14. They prove the intersection property of the information function. The property is closely related to connectedness.

  15. For a permutation \(\mu \) on \(X, \mu (R)\) is defined by: \((\mu (x),\mu (y)) \in \mu (R) \Leftrightarrow (x,y) \in R\). Then, \(f\) is neutral if and only if \(f(\mu (\mathbf{R}))=\mu (f(\mathbf{R}))\), where \(\mu (\mathbf{R})=(\mu (R_i))_{i \in N}\). This definition of neutrality differs from Sen’s (1970) definition of neutrality, which directly implies binary independence.

  16. The proof is as follows: take any \(S \in \mathcal S \) such that \(S \subsetneq X\) and \(|S|>1\). For all \(x,y \in S\),

    $$\begin{aligned} \mathbf{R}|_S=\mathbf{R}^{\prime }|_S \Rightarrow \left[ f(\mathbf{R})|_{\{ x,y \}}=f(\mathbf{R}^{\prime })|_{\{ x,y \}} \text{ for } \text{ all } x,y \in S \right] . \end{aligned}$$

    Then, weakest independence is satisfied.

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Acknowledgments

I am grateful to an anonymous referee for his/her constructive comments and suggestions. This paper was financially supported by Grant-in-Aids for Young Scientists (B) from the Japan Society for the Promotion of Science and the Ministry of Education, Culture, Sports, Science and Technology.

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  1. Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, Tokyo, 192-0397, Japan

    Susumu Cato

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Cato, S. Independence of irrelevant alternatives revisited. Theory Decis 76, 511–527 (2014). https://doi.org/10.1007/s11238-013-9384-1

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