Fuzzy Arrow’s Theorem

Abstract In a recent paper Samir Okasha has suggested an application of Arrow’s impossibility theorem to theory choice. When epistemic virtues are interpreted as ‘voters’ in charge of ranking competing theories, and there are more than two theories at stake, the final ordering is bound to coincide with the one proposed by one of the voters (the dictator), provided a number of seemingly reasonable conditions are in place. In a similar spirit, Jacob Stegenga has shown that Arrow’s theorem applies to the amalgamation of evidence; the ‘voters’ here are the different sources of evidence. As with Okasha’s proposal, it is not clear how to avoid Arrow’s pessimistic conclusion. In this paper we develop a novel argument that purports to show that, in typical examples, Arrow’s result does not obtain when dealing with evidence amalgamation. The reason is that we cannot escape the well-known Duhem problem: the evidence actually confirms (or disconfirms) complex conjunctions that include various auxiliary hypotheses. We argue that confirmational holism induces us to restrict the domain of a reasonable amalgamation function, thus violating one of Arrow’s conditions. The upshot is that we are now able to see the Duhem problem under a different, positive light – namely, as a phenomenon that makes the aggregation of the evidence possible in the first place, when there are at least three options on the table.

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Arrow's theorem for economic environments and effective social preferences

Social Choice and Welfare ◽

10.1007/bf00446989 ◽

1989 ◽

Vol 6 (4) ◽

pp. 325-329 ◽

Cited By ~ 8

Author(s):

D. E. Campbell

Keyword(s):

Social Preferences ◽

Arrow’S Theorem ◽

Arrow's Theorem

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Arrow’s Theorem

The New Palgrave Dictionary of Economics ◽

10.1057/978-1-349-95121-5_137-1 ◽

1987 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Kenneth J. Arrow

Keyword(s):

Arrow’S Theorem ◽

Arrow's Theorem

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Arrow’s Theorem

Oxford Research Encyclopedia of Politics ◽

10.1093/acrefore/9780190228637.013.936 ◽

2020 ◽

Author(s):

Conal Duddy ◽

Ashley Piggins

Keyword(s):

Social Welfare ◽

Social Choice ◽

Social Welfare Function ◽

Social Preference ◽

Social Choice Theory ◽

Impossibility Theorem ◽

Preference Ordering ◽

Individual Values ◽

Arrow’S Theorem ◽

Arrow's Theorem

Kenneth Arrow’s “impossibility” theorem is rightly considered to be a landmark result in economic theory. It is a far-reaching result with implications not just for economics but for political science, philosophy, and many other fields. It has inspired an enormous literature, “social choice theory,” which lies on the interface of economics, politics, and philosophy. Arrow first proved the impossibility theorem in his doctoral dissertation—Social Choice and Individual Values—published in 1951. It is a remarkable result, and had Arrow not proved it, it is unlikely that the theorem would be known today. A social choice is simply a choice made by, or on behalf of, a group of people. Arrow’s theorem is concerned more specifically with the following problem. Suppose that we have a given set of options to choose from and that each member of a group of individuals has his or her own preference over these options. By what method should we construct a single ranking of the options for the group as a whole? Any such method may be represented mathematically by a “social welfare function.” This is a function that receives as its input the preference ordering of each individual and then generates as its output a social preference ordering. Arrow defined some properties that would seem to be essential to any reasonable social welfare function. These properties are called “unrestricted domain,” “weak Pareto,” “independence of irrelevant alternatives,” and “non-dictatorship.” Each of these properties, when taken alone, does appear to be very necessary indeed. Yet, Arrow proved that these properties are in fact mutually incompatible. This troubling fact has been central to the study of social choice ever since.

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