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.

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 _j ∈ A 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(R_j), by weak Pareto we have s₀ <⁰t₀ and t_n <ⁿs_n. Counting up from 0 to n, let d be the first j such that not: s_j <^jt_j. 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, c ∈ A 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.¹³