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Mr. Fit, Mr. Simplicity and Mr. Scope: From Social Choice to Theory Choice

Abstract

An analogue of Arrow’s theorem has been thought to limit the possibilities for multi-criterial theory choice. Here, an example drawn from Toy Science, a model of theories and choice criteria, suggests that it does not. Arrow’s assumption that domains are unrestricted is inappropriate in connection with theory choice in Toy Science. There are, however, variants of Arrow’s theorem that do not require an unrestricted domain. They require instead that domains are, in a technical sense, ‘rich’. Since there are rich domains in Toy Science, such theorems do constrain theory choice to some extent—certainly in the model and perhaps also in real science.

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Notes

  1. 1.

    See Forster and Sober (1994).

  2. 2.

    John Bigelow (1977, p. 462) found this same notion useful in accounting for probability within a possible-worlds framework.

  3. 3.

    As usual, ≈ means that both ≥ and ≤ ; and > means that ≥ but not ≤ .

  4. 4.

    I take this point from Forster and Sober (1994).

  5. 5.

    For a precise statement of the assumptions of Arrow's theorem and a proof, see any standard text such as Gaertner (2009).

  6. 6.

    See Bailey (1998).

  7. 7.

    Forster (2004) discusses examples of nested models in science. For more examples of domain restrictions in real science, see Morreau forthcoming.

  8. 8.

    The chain property is an example. See Campbell and Kelly (2002, p. 41).

  9. 9.

    I leave intuitive, for now, the idea of ordering some alternatives in some particular ‘way’. It will become precise in the next section, with the notions of patterns and their realization by profiles.

  10. 10.

    I do not know of any explicit discussion in the literature of the representational interpretation, but it does seem to be a part of the folklore. An anonymous reviewer of Morreau forthcoming, rejecting the argument of the previous section that inherent domain restrictions make Arrow’s theorem inapplicable to theory choice, wrote that we ‘must’ understand Arrow’s alternatives as names or labels, even though Arrow does not seem to have done so. Perhaps it is easy to fall into a representational interpretation of Arrow’s alternatives if one thinks of choice rules computationally, as procedures to be applied to different sets of alternatives on different occasions, in much the same way that a voting procedure is used year after year with different slates of candidates.

  11. 11.

    For a precise statement, see a standard text such as Gaertner (2009), or see Morreau forthcoming.

  12. 12.

    Fact 14 is a variant of the ‘single profile’ impossibility theorems pioneered by Parks (1976) and (independently) by Kemp and Ng (1976). These respond to an objection leveled against Arrow’s ‘multi-profile’ framework, according to which there is only a single profile of individual preferences that needs to be taken into account in social choice, namely the preferences that the members of society happen actually to have. In theory choice, as we have seen, there might be a certain amount of variety among the profiles that it makes sense to submit to a theory-choice rule, though it falls well short of what is required by Arrow’s domain assumption.

  13. 13.

    This is an adaptation and simplification of one of John Geanakoplos’s (2005) proofs of Arrow’s theorem.

References

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

  2. Arrow, K., & Raynaud, H. (1986). Social choice and multicriterion decision-making. Cambridge Ma. and London: The MIT Press.

  3. Bailey, R. W. (1998). The number of weak orderings of a finite set. Social Choice and Welfare, 15, 559–562.

  4. Bigelow, J. C. (1977). Semantics of probability. Synthese, 36, 459–472.

  5. Campbell, D. E., & Kelly, J. S. (2002). Impossibility theorems in the Arrovian framework. In K.J. Arrow, A.K. Sen & K. Suzumura (Eds.), (pp. 35–94).

  6. Forster, M. (2004). Chapter 3: Simplicity and unification in model selection. http://philosophy.wisc.edu/forster/520/Chapter%203.pdf.

  7. Forster, M., & Sober, E. (1994). How to tell when simpler, more unified or less ad hoc theories will provide more accurate predictions. British Journal for the Philosophy of Science, 45, 1–35.

  8. Gaertner, W. (2009). A primer in social choice theory (revised ed.). Oxford: Oxford University Press.

  9. Geanakoplos, J. (2005). Three brief proofs of Arrow’s theorem. Economic Theory, 26, 211–215.

  10. Hurley, S. (1985). Supervenience and the possibility of coherence. Mind, 94, 501–525.

  11. Kemp, M. C., & Ng, Y. K. (1976). On the existence of social welfare functions, social orderings and social decision functions. Economica, 43, 59–66.

  12. Kuhn, T. (1970). The structure of scientific revolutions (2nd ed.). Chicago: University of Chicago Press.

  13. Kuhn, T. (1977a). Objectivity, value judgment, and theory choice. In his 1977b, (pp. 320–339).

  14. Kuhn, T. (1977b). The essential tension. Chicago: University of Chicago Press.

  15. May, K. O. (1954). Intransitivity, utility, and the aggregation of preference patterns. Econometrica, 22, 1–13.

  16. Morreau, M. Theory choice and social choice: Kuhn vindicated. Mind (forthcoming).

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

  18. Parks, R. P. (1976). An impossibility theorem for fixed preferences: A dictatorial Bergson–Samuelson welfare function. Review of Economic Studies, 43, 447–450.

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Acknowledgments

I thank for helpful comments Malcolm Forster, Aidan Lyon, Samir Okasha, John Weymark, and an anonymous reviewer.

Author information

Correspondence to Michael Morreau.

Appendix

Appendix

Proof of Fact 14

Suppose the theoretical criteria are finite in number and that some theory-choice rule f has a rich domain and satisfies neutrality and weak Pareto. To be shown is that f has a dictator.

First, letting the criteria be 1,…,n, because the domain is rich we can find a series of profiles R 0,…, R n and pairs s j, t jA such that for each 0 ≤ j ≤ n:

$$ t_{\text{j}} <^{\text{j}}_{\text{i}} s_{\text{j}} \quad {\text{if}}\quad 1\le {\text{i}} \le {\text{j,}}\quad {\text{and}}\quad s_{\text{j}} <^{\text{j}}_{\text{i}} t_{\text{j}} \quad {\text{if}}\quad {\text{j}} < {\text{i}} \le {\text{n}}. $$

Here, \( \le^{\text{j}}_{\text{i}} \) is the ith criterial ordering of profile R j. Writing ≤j for f(Rj), by weak Pareto we have s0 <0t0 and tn <nsn. Counting up from 0 to n, let d be the first j such that not: sj <jtj. By choice of d and completeness of ≤d:

$$ {\text{s}}_{{{\text{d}} - 1}} <^{{{\text{d}} - 1}} {\text{t}}_{{{\text{d}} - 1}} ,\quad {\text{and}} $$
(1a)
$$ {\text{t}}_{\text{d}} \le^{\text{d}} {\text{s}}_{\text{d}} $$
(1b)

This criterion d is a dictator. Consider any profile P in the domain and any alternatives a, cA such that \( a \, < \, ^{P}_{d} c \) (which is to say that, according to P, c is strictly better than a by criterion d). To be shown is that:

$$ a <^{P} c $$
(2)

(in the sense of the overall ordering that f assigns to P, c is strictly better than a). Since the domain is rich, there is a profile Q and there are alternatives α, β, γ ∈ A, such that for all criteria i:

$$ a \le^{P}_{\text{i}} c\quad {\text{iff}}\quad \alpha \le^{Q}_{\text{i}} \gamma ;\quad {\text{and}}\quad c \le^{P}_{\text{i}} a\quad {\text{iff}}\quad \gamma \le^{Q}_{\text{i}} \alpha ; $$
(3)
$$ \alpha <^{\text{Q}}_{\text{i}} \beta \quad {\text{and}}\quad \gamma <^{\text{Q}}_{\text{i}} \beta ,\quad {\text{if}}\quad 1\le {\text{i}} < {\text{d;}} $$
(4)
$$ \alpha <^{\text{Q}}_{\text{d}} \beta <^{\text{Q}}_{\text{d}} \gamma ;\quad {\text{and}} $$
$$ \beta <^{\text{Q}}_{\text{i}} \alpha \quad {\text{and}}\quad \beta <^{\text{Q}}_{\text{i}} \gamma ,\quad {\text{if}}\quad {\text{d}} < {\text{i}} \le {\text{n}}. $$

By (3) and neutrality, it is sufficient for (2) that α < Q γ, and for this it is by transitivity of ≤Q sufficient that:

$$ \alpha \le^{Q} \beta ,\quad {\text{and:}} $$
(5)
$$ \beta <^{Q} \gamma . $$
(6)

By inspection of R d and (4), for all criteria i:

$$ \alpha \le^{Q}_{\text{i}} \beta \quad {\text{iff}}\quad {\text{t}}_{\text{d}} \le^{\text{d}}_{\text{i}} {\text{s}}_{\text{d}} ,\quad {\text{and}}\quad \beta \le^{Q}_{\text{i}} \alpha \quad {\text{iff}}\quad {\text{ s}}_{\text{d}} \le^{\text{d}}_{\text{i}} {\text{t}}_{\text{d}} . $$

Now (5) follows by neutrality from (1b). Also, by inspection of R d−1 and (4), for all i:

$$ \beta \le^{Q}_{\text{i}} \gamma \quad {\text{iff}}\quad {\text{s}}_{{{\text{d}} - 1}} \le^{{{\text{d}} - 1}}_{\text{i}} {\text{t}}_{{{\text{d}} - 1}} \quad {\text{and}}\quad \gamma \le^{Q}_{\text{i}} \beta \quad {\text{iff}}\quad {\text{t}}_{{{\text{d}} - 1}} \le^{{{\text{d}} - 1}}_{\text{i}} {\text{s}}_{{{\text{d}} - 1}} . $$

Now (6) follows by neutrality from (1a).

This completes the demonstration that (2). We have seen that d is a dictator.Footnote 13

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Morreau, M. Mr. Fit, Mr. Simplicity and Mr. Scope: From Social Choice to Theory Choice. Erkenn 79, 1253–1268 (2014). https://doi.org/10.1007/s10670-013-9549-x

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Keywords

  • Scientific Theory
  • Theory Choice
  • Real Science
  • Domain Restriction
  • Weak Ordering