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