# Do any social welfare functionals, other than maximin, meet all of Arrow's conditions plus invariance regarding ordinal level comparability?

In the literature on social welfare functionals, the only example I've seen of a functional which meets all of Arrow's conditions–––or at least utility analogues of Arrow's conditions–––plus invariance regarding ordinal level comparability is Rawls' maximin. E.g. Sen in On Weights and Measures (1977, p. 1544) cites maximin as his case of a functional meeting all of these conditions. Maximin orders the alternatives by the welfare of individual who is worst off. I assume that the inverse of maximin–––i.e. the alternatives are ordered by the welfare of individual who is best off–––would also meet these conditions.

Is there any work on other social welfare functionals which meet all these conditions? (I'm aware that if we tweak these conditions slightly we can derive other functionals, but I'm interested in the case in which we keep them unaltered.)

If not, is this evidence that maximin, and its inverse, are the only normatively sensible social welfare functionals that meets all these conditions? Or is it just evidence that people aren't so interested in this set of conditions? (If there is a clear reason why this set of conditions is uninteresting, I'd love to hear it).

Thanks for any help!

Utility analogues of Arrow’s conditions:

Utility analogues of Arrow’s conditions are Arrow’s conditions redefined for Sen’s welfare functional framework. Instead of taking a profile of orderings as input, Sen's functional takes a profile of utility functions as input: $$U \ = \ $$. $$U$$ is defined on $$X \times N$$; each individual, $$i \in N$$, is paired with each alternative, $$x \in X$$, and the result of each pairing is the utility derived by $$i$$ from $$x$$. $$\mathcal{U} \ = \ \{U^1, \ U^2, \ \dots \ , \ U^n \}$$ is the set of all possible utility profiles. $$\mathcal{U^*}$$ is the set of all utility profiles which meet a particular domain restriction. $$\mathcal{R}$$ is the set of all possible orderings of $$X$$. A social welfare functional can then be defined as: $$f: \ \mathcal{U^*} \longrightarrow \mathcal{R}$$. The final ordering given by profile $$U^1$$, $$f(U^1)$$, is denoted: $$R_{U^1}$$. We can then define utility analogues of Arrow's conditions:

Unrestricted Domain$$’$$: The domain of $$f$$ is the set of all possible utility profiles: $$\mathcal{U}^* \ = \ \mathcal{U}$$.

Weak Pareto$$’$$: $$\forall x, y \in X$$, $$\forall i \in N$$: $$( \ u_i(x) \ > \ u_i(y) \ ) \ \Longrightarrow \ (xPy)$$.

Non-Dictatorship$$’$$: $$f$$ does not single out one individual $$i \in N$$ such that, $$\forall U \in \mathcal{U^*}, \ \forall x, y \in X$$: $$( \ u_i(x) \ > \ u_i(y) \ ) \ \Longrightarrow \ (xPy)$$.

Independence of Irrelevant Utilities: $$\forall U^1$$ and $$U^2$$ $$\in \mathcal{U^*}, \ \forall x, y \in X$$: $$(\forall i \in N \ (( \ u^1_i(x) = u^2_i(x) \ ) \land ( \ u^1_i(y) = u^2_i(y) \ )) \ \Longrightarrow \ (( \ x R_{U^1} y \ ) \ \Longleftrightarrow \ ( \ x R_{U^2} y \ ))$$.

There are at least two other examples of SWFs that satisfy these conditions.

The first is a positional dictatorship. Let N be the number of individuals (assume it is fixed). For any k between 1 and N, the kth positional dictatorship SWF orders social alternatives in terms of the preferences of the "kth best off" agent. Formally, given any social alternative x, let v(x), be the vector of utilities of all individuals for x, but ordered from lowest to highest. The kth positional dictatorship SWF is then defined by the kth component of the function v. If k=1, we get the maximin. If k=N, then we get the "maximax" ---what you call the "inverse" of the maximin. If k=[N/2], we get effectively "dictatorship of the median individual". The point is not that these rules are normatively attractive (they aren't) ---but they satisfy your axioms.

Another possibility is the so-called leximin or lexicographical maximin rule. This is the lexicographical extension of the maximin, obtained by ranking social alternatives according to the vector-valued function v from the previous paragraph, but with coordinates treated lexicographically. Thus, alternative x is better than alternative y if it has a higher minimum utility value. If x and y yield the same minimum utility, then we compare them by looking at the utilities of the second-worst off individual in x and y. If these individuals also have the same utility, then we look at the third-worst off individuals, and so on.

This SWF is very similar to maximin, but it satisfies a stronger version of the Pareto axiom.

For more information, I suggest you look at the 2002 article by Claude d'Aspremont and Louis Gevers entitled "Social welfare functionals and interpersonal comparability", which is Chapter 10 of the Handbook of Social Choice and Welfare volume I (Arrow, Sen and Suzumura, eds.). You could also look at Chapter 2 of the book Axioms of Cooperative Decision-Making, by Hervé Moulin (1988). In particular, Theorem 2.4 on page 40 of Moulin's book might be useful to you: it says (roughly) that the positional dictatorships and their extensions (such as leximin) are the only SWFs satisfying ordinal level comparability and a few other mild conditions.

• Brilliant; thank you! I should clarify that I don’t take maximax to be `normatively sensible’. I shoehorned it in after it occurred that I had left it out, and did not edit carefully enough. However, I don’t want to edit the question now, as it would make your wonderful answer less precise. – Nikelmouse Dylar Jun 10 '19 at 19:04