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On some oligarchy results when social preference is fuzzy

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Abstract

We consider a model in which individual preferences are orderings of social states, but the social preference relation is fuzzy. We motivate interest in the model by presenting a version of the strong Pareto rule that is suited to the setting of a fuzzy social preference. We prove a general oligarchy theorem under the assumption that this fuzzy relation is quasi-transitive. The framework allows us to make a distinction between a “strong” and a “weak” oligarchy, and our theorem identifies when the oligarchy must be strong and when it can be weak. Weak oligarchy need not be undesirable.

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Notes

  1. Intuitively, a quasi-transitive preference relation is one where the strict preference relation is transitive, whereas the indifference relation need not be. An acyclic relation is one with no cycles in the strict preference relation. A Suzumura consistent relation rules out not only strict preference cycles, but all cycles that involve at least one strict preference. Acyclicity is the weakest concept of the three. Suzumura consistency and quasi-transitivity are logically independent.

  2. Gibbard’s paper was written in the 1969–1970 academic year but remained unpublished until Gibbard (2014). An introduction to the paper is given by Weymark (2014).

  3. A collective choice rule is a function that takes a profile of preference orderings as its input (one ordering for each individual) and produces a social preference relation as its output.

  4. The idea that preferences can be numerically valued (or “fuzzy”) has, of course, been considered before. A sample of the literature is Orlovsky (1978), Ovchinnikov (1981), Basu (1984), Billot (1995), Dutta et al. (1986), Dutta (1987), Ponsard (1990), Dasgupta and Deb (1991, 1996, 2001), Ovchinnikov and Roubens (1992) and Banerjee (1993, 1994). Salles (1998) surveys the field and Piggins and Salles (2007) discuss some underlying philosophical issues.

  5. Formally, let \(n(xP_{i}y)\) denote the number of people who strictly prefer x to y. Then, under this rule, \(p(x,y)=\frac{n(xP_{i}y)}{n(xP_{i}y)+n(yP_{i}x)}.\) If everyone is indifferent between x and y then we set \(p(x,y)=1\).

  6. A simple proof of Gibbard’s theorem is given by Piggins (2017).

  7. Barrett and Salles (2011) survey the field. A sample of the literature is Barrett et al. (1986), Dutta (1987), Ovchinnikov (1991), Banerjee (1994), Billot (1995), Richardson (1998), Dasgupta and Deb (1999), García-Lapresta and Llamazares (2000, 2001), Fono and Andjiga (2005), Perote-Peña and Piggins (2007), Duddy et al. (2010, 2011) and Gibilisco et al. (2014). See also Leclerc (1984, 1991) and Leclerc and Monjardet (1995). Note that our model bears some formal similarities with the interesting framework of sophisticated social welfare functions introduced by Sanver and Selçuk (2009). They interpret the values assigned to ordered pairs of distinct alternatives as a measure of the “weight” by which one alternative is strictly preferred to another. They assume that, for all distinct x and y, the weight of preference for x vs. y plus the weight of preference for y vs. x must sum to one. We do not assume that any adding-up constraint holds for our central theorem. Further, our formulation of transitivity differs. In our fuzzy model, we assume that social preferences are “max-star” transitive. This assumption, in fact, admits a range of different possible transitivity conditions as special cases. The fact that this family of cases can be partitioned into two parts is central to our characterization theorem.

  8. The proportional rule satisfies unrestricted domain, independence of irrelevant alternatives, and the Pareto principle.

  9. Note that Barrett et al. (1986) use a fuzzy strict social preference relation as their primitive concept, not a weak one. The transitivity condition they use is Condition 2.3 in their paper.

  10. One possible objection to the proportional rule is that it does not agree with the majority rule at profiles where the latter is transitive.

  11. If X is an arbitrary set then a fuzzy relation is a function \(t:X\times X\rightarrow [0,1]\). We say that t is connected if \(t(x,y)+t(y,x)\ge 1\) for all \(x,y\in X\). Note that the only time we assume connectedness in our results section (Sect. 3) is in Corollary 6 and Corollary 8. In the literature, a related condition called “reciprocity” is often assumed (see De Baets et al. 2006; Llamazares et al. 2013). This condition says that \(t(x,y)+t(y,x)=1\) for all \(x,y\in X\). This condition is stronger than connectedness and is natural in models of preference intensity. However, we are not using fuzzy set theory here to model the intensity of preference, but rather the degree of truth. In the intensity literature, if x and y are socially indifferent then we would have \(t(x,y)=t(y,x)=\frac{1}{2}\). However, for us, indifference means that it is true that x is at least as good as y, and true that y is at least as good as x. Hence, \(t(x,y)=t(y,x)=1.\)

  12. Interesting characterizations of some approaches to factorization are contained in Llamazares (2005).

  13. Note that Banerjee’s method of factorization satisfies (C) only if connectedness of r is assumed.

  14. Klement et al. (2000) is a comprehensive survey of triangular norms.

  15. The role of these concepts in judgment aggregation is explored in Duddy and Piggins (2013). In Duddy et al. (2011) we considered these concepts with respect to a fuzzy version of Arrow’s theorem. In the current paper, we are concerned with Weymark’s theorem, not Arrow’s. Additionally, in our earlier paper we used a stronger version of the independence condition. The version we use here is weaker than that one, and is closer to the standard version employed in social choice theory.

  16. The independence condition in Barrett, Pattanaik and Salles (1986, p. 4), Dutta (1987, p. 222) and Banerjee (1994, p. 125) is equivalent to ours under the assumption of crisp individual preferences. Translating the independence condition of Duddy et al. (2011) into the setting of this paper yields: For all \(x,y\in X\) and all \((R_{1},...,R_{n}),(R_{1}',...,R_{n}')\in D\), \([xR_{i}y\leftrightarrow xR'_{i}y]\) for all \(i\in N\) implies \(r(x,y)=r'(x,y)\). To see the difference between the two formulations, consider the following collective choice rule. Suppose there are two people and one strictly prefers x to y, the other y to x. A natural outcome is a tie, i.e. \(r(x,y)=r(y,x)=1\). However, if the first person is indifferent then the natural outcome is to let the second break the tie, i.e. \(r(x,y)=0\) and \(r(y,x)=1\). Our original formulation prohibits this since each person’s \(xR_{i}y\) preference is the same in both profiles. This problem does not arise under our new formulation.

  17. The \(wP_{i}x\rightarrow wP'_{i}z\) direction comes from the preceding sentence. The \(wP_{i}'z\rightarrow wP_{i}x\) direction comes from taking the contrapositive of \(xP_{i}w\rightarrow zP'_{i}w\) (which again appears in the preceding sentence).

  18. The idea of a pivotal voter originates with Barberà (1980). This technique has been used in various proofs of Arrow’s theorem, e.g. Geanakoplos (2005) and Perote-Peña and Piggins (2002). We use the same technique here to prove the existence of an oligarchy.

  19. The proof of this is straightforward.

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Authors and Affiliations

  1. J.E. Cairnes School of Business and Economics and the Whitaker Institute, National University of Ireland Galway, Galway, Ireland

    Conal Duddy & Ashley Piggins

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  1. Conal Duddy
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  2. Ashley Piggins
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Correspondence to Ashley Piggins.

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We are grateful to participants at the 13th Meeting of the Society for Social Choice and Welfare, in particular Christian List and Bill Zwicker, for their helpful comments on this paper. We also thank the Managing Editor, Clemens Puppe, the Associate Editor, and two anonymous referees for their comments on the original version of this paper. Financial support from the Irish Research Council for the Humanities and Social Sciences co-funding from the European Commission, and from the Spanish Ministry of Economy and Competitiveness through MEC/FEDER Grant ECO2013-44483-P is gratefully acknowledged.

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Duddy, C., Piggins, A. On some oligarchy results when social preference is fuzzy. Soc Choice Welf 51, 717–735 (2018). https://doi.org/10.1007/s00355-018-1134-4

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