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Arrow’s theorem and theory choice

Abstract

In a recent paper (Okasha, Mind 120:83–115, 2011), Samir Okasha uses Arrow’s theorem to raise a challenge for the rationality of theory choice. He argues that, as soon as one accepts the plausibility of the assumptions leading to Arrow’s theorem, one is compelled to conclude that there are no adequate theory choice algorithms. Okasha offers a partial way out of this predicament by diagnosing the source of Arrow’s theorem and using his diagnosis to deploy an approach that circumvents it. In this paper I explain why, although Okasha is right to emphasise that Arrow’s result is the effect of an informational problem, he is not right to locate this problem at the level of the informational input of a theory choice rule. Once the informational problem is correctly located, Arrow’s theorem may be dismissed as a problem.

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Notes

  1. 1.

    Kuhn does not assume this list to be complete. The following discussion is not affected by incompleteness since it applies equally well to any finite number of criteria, provided that there are at least three. Kuhn lists five.

  2. 2.

    Kuhn points out that scientists in agreement on the criteria to adopt may make different choices because they attach different weights to the same criteria. His argument would, at least theoretically, go through without references to weighting, on account of the existence of distinct (non-weighted) aggregation procedures.

  3. 3.

    This is not Kuhn’s view, as will be explained in Sect. 4.

  4. 4.

    The collective in question being that of the several evaluation criteria.

  5. 5.

    By an ordering I here mean a relation that is transitive and complete, i.e., defined on each pair of theories.

  6. 6.

    This is equivalent to saying that there does not exist any aggregation procedure that satisfies the given properties and sends profiles into (transitive) orderings. I have chosen the alternative formulation in terms of cycles because the distinction between cycles and transitive orderings is central to the discussion to be carried out in the following sections. Also note that, throughout this discussion, when I talk about aggregations I implicitly refer to aggregations that satisfy the assumption of non-dictatorhip. Dictatorial aggregations clearly do not give rise to cyclic outcomes but, if they were plausible, i.e., if the use of a single, fixed criterion afforded an adequate method of theory choice, then Okasha’s problem would immediately vanish. The problem has a non-trivial solution only if one cannot immediately help oneself to dictatorial aggregations.

  7. 7.

    Restrictions on the set of profiles that avoid cyclic outcomes have been described, in decreasing order of strength, in Black (1958), Sen (1966) and Saari (1994).

  8. 8.

    In principle, however, one might try to find domain restrictions under which non-dictatorial aggregations exist [Kalai and Muller (1977) contains a partial classification] and that it is plausible to impose upon rankings of theories. This strategy will not be pursued in this paper, which evaluates the significance of Arrow’s theorem from an informational point of view.

  9. 9.

    In general, the availability of a profile of rankings that are more strongly structured than orderings does not affect Arrow’s theorem, unless one is able to rely on a structured space of alternatives (as in Weymark and Tsui (1997), a work cited by Okasha and where the set of alternatives is a \(n\)-dimensional Euclidean space), which is to say that alternatives, and thus profile orderings, are interrelated.

  10. 10.

    Such a sequence may include ties, e.g., it may be of the form \(\mathrm {a }\sim _{i}\,\mathrm {b} >_{i}\mathrm { c},\mathrm { if} '\sim '\) is the i-th criterion’s relation of being tied and \(>_\mathrm{i}\) strict ordering in the same criterion.

  11. 11.

    And has been made apparent in the proof of this theorem developed by Donald Saari [see Saari (1991, 1994, 1995)].

  12. 12.

    Two of these procedures are the Condorcet Improvement [see Saari (1995, p. 79–80)] and the Borda Count (see fn. 15).

  13. 13.

    With \(n\) competing theories, a positional method is specified by a vector \((\mathrm{u}_{1}, \ldots ,\mathrm{u}_n)\), with \(\mathrm{u}_\mathrm{i} = \mathrm{u}_\mathrm{j}\mathrm{iff }\;\mathrm{i} \le j,\mathrm{u}_n = 0\) and u \(\mathrm{u}_{1} \ne \mathrm{u}_n\). Given a profile p, the positional vector assigns, for each ordering in p, the score \(\mathrm {u}_\mathrm{i}\) to the alternative that occupies the i-th position in the ordering. The aggregate ordering is obtained by summing the score of each alternative across the orderings in p.

  14. 14.

    With \(n\) alternatives, the Borda count is a positional method with vector \((n - 1,\,n - 2, \ldots , 2, 1, 0)\).

  15. 15.

    The same conclusion may be extended to another recent re-interpretation of Arrow’s theorem, developed in Stegenga (2013), that applied it to the aggregation of modes of evidence. If modes of evidence generate transitive profiles and the information encoded in these profiles matter to evidence aggregation, then binary independent aggregations are not adequate and Arrow’s theorem may be, again, dismissed.

References

  1. Arrow, K. (1951). Social choice and individual values. New York: Wiley.

  2. Black, D. (1958). The theory of committees and election. London: Cambridge University Press.

  3. Kalai, E., & Muller, E. (1977). Characterization of domains admitting non-dictatorial social welfare functions and non-manipulable voting procedures. Journal of Economic Theory, 16, 457–467.

  4. Kuhn, T. (1977). Objectivity, value judgment, and theory choice. The essential tension (pp. 320–339). Chicago: University of Chicago Press.

  5. Okasha, S. (2011). Theory choice and social choice: Arrow versus Kuhn. Mind, 120, 83–115.

  6. Saari, D. (1991). Calculus and extensions of Arrow’s theorem. Journal of Mathematical Economics, 20, 271–306.

  7. Saari, D. (1994). Geometry of voting. New York: Springer.

  8. Saari, D. (1995). Basic geometry of voting. New York: Springer.

  9. Saari, D. (1998). Connecting and resolving Sen’s and arrow’s theorem. Social choice and welfare, 15, 239–261.

  10. Saari, D. (2001). Decisions and elections: Explaining the unexpected. Cambridge: Cambridge University Press.

  11. Sen, A. (1966). A possibility theorem on majority decisions. Econometrica, 34, 491–499.

  12. Sen, A. (1977). On weights and measures: Informational constraints in social welfare analysis. Econometrica, 45, 1539–1572.

  13. Stegenga, J. (2013). An impossibility theorem for amalgamating evidence. Synthese, 190, 2391–2411.

  14. Weymark, J. A. And, & Tsui, K. (1997). Social welfare orderings for ratio-scale measurable utilities. Economic Theory, 10, 241–256.

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Correspondence to Davide Rizza.

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Rizza, D. Arrow’s theorem and theory choice. Synthese 191, 1847–1856 (2014). https://doi.org/10.1007/s11229-013-0372-3

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Keywords

  • Arrow
  • Kuhn
  • Okasha
  • Saari
  • Theory choice