Social evaluation functionals: a gateway to continuity in social choice

My former motivation for introducing this concept was to confront continuity to IIA in the topological aggregation of preferences problem.

For the concept of a SWFL see Sen (1970, 1977).

That is, a map \(H:\mathcal {R}^{n}\longrightarrow \mathcal {R}\) as in the Arrovian framework.

The case in which \(X\) is an infinite set has a different treatment and will not be considered here. In particular, if \(X\) is infinite, then the topological concepts that follow would need to be accommodated in order to provide a meaningful framework. For instance, if \(X\) is compact, then \(\mathcal {U}\) can be equipped with the supnorm topology.

For the basics on total preorders see the “Appendix”.

The cases \(n=1\), or #\(X=1\), have a slightly distinct mathematical consideration and, because their lack of interest in the social choice context, will not be studied here.

An alternative way to introduce OMIN axiom is the following: For every \(U\in \mathcal {U}^{n}, \Phi \in \Delta ^{n}\), there is \(\varphi \in \Delta \), which depends on \(U\) and \(\Phi \), such that \(F(\Phi U)=\varphi (F(U))\). Note that, since its definition only involves a single profile, OMIN turns out to be an intra-profile property on \(F\). In accordance with the terminology used in measurament theory this axiom is closely related to the concept comparison meaningful (see, e.g., Krantz et al. (1971) for a thorough treatment of this theory).

For a SEF \(F\) both notions OSP and OMIN can also be given for sub-domains of \(\Delta \), or \(\Delta ^{n}\). It should be noted that all results in this article remain true if both OSP and OMIN are restricted to the sub-domains \(\Delta _{spa}\) and \(\Delta _{spa}^{n}\), respectively.

Obviously, Pro implies SD and SD implies D.

The explanation that follows was suggested to me by a referee. The referee also pointed out that OMIN axiom could be given a stronger form by allowing neutrality of the aggregator to the class of all increasing funtions. However, and as a consequence of footnote #11 below, in the case considered here, where \(X\) is finite, it drives the same conclusion.

It is a standard result in utility theory (see e.g. Kreps (1988), p. 26) that if \(h\) and \(g\) represent a total preorder \(\precsim \) on a set \(X\), then there is an increasing function \(\varphi :\mathbb {R}\longrightarrow \mathbb {R}\) such that (a) \(\varphi \) is strictly increasing on \(g(X):=\{r:r=g(x), \text{ for } \text{ some }\ x\in X\}\), (b) \(h=\varphi \circ g\). Notice that if \(X\) is finite, then the function \(\varphi \) can be chosen to be strictly increasing. However, this is not longer true if \(X\) is infinite. In this case, not pursued in the present article, OMIN axiom should be changed accordingly.

The connection established between SEFs and SWFs remains true for the case in which \(X\) is a countably infinite set since every total preorder defined on such a set admits a utility function (for details see Bridges and Mehta (1995)). However, as suggested by a referee, in the uncountably infinite case this link breaks down and one should restrict the preferences domain in order to resume such a connection (for example continuous preferences over (compact subsets of) \(\mathbb {R}^{m}\)). Notice, in addition, that other results stated in this paper strongly depend upon the finiteness assumption of \(X\) (see footnote \(\#4\) above).

This example can be adapted to the case where \(X=\{x,y,z\}\) by defining, for each \(U=(u_{1}, u_{2})\in \mathcal {U}^{2}, F\) as follows:and

Since, clearly, \(F\) is not D, it follows that OMIN, OSP and U allow for something else than dictatorial aggregators.

In the economics literature a total preorder is also referred to as a preference and as a social welfare ordering in social choice theory.

Indeed, let \(a,b \in \mathbb {R}^{n}\) such that \(a\precsim _{W} b\) and let \(\Phi \in \Delta ^{n}\). Choose \(U\in \mathcal {U}^{n}\) so that \(U(x)=a, U(y)=b\), for some \(x,y\in X\). Note that, by universality of domain, this choice is possible. Then \(a\precsim _{W} b\Longleftrightarrow W(a)=F(U)(x)\le W(b)=F(U)(y)\). Now, since \(F\) fulfils OMIN, it holds that \(F(U)(x)\le F(U)(y)\Longleftrightarrow F(\Phi U)(x)\le F(\Phi U)(y)\). But, by definition, \(\Phi U(x)=\Phi (a)\), and \(\Phi U(y)=\Phi (b)\). So, \(W(\Phi (a))=F(\Phi U)(x)\), and \(W(\Phi (b))=F(\Phi U)(y)\). Therefore, \(a\precsim _{W} b\Longleftrightarrow W(\Phi (a))\le W(\Phi (b))\Longleftrightarrow \Phi (a)\precsim _{W} \Phi (b)\).

Theorem 3 of Candeal (2013) states that a nontrivial, zero-independent, and scale-independent total preorder on \(\mathbb {R}^{n}\) is two-serial. The fact that \(\precsim _{W}\) becomes strongly dictatorial follows, in addition to the conclusion of Theorem 3 in Candeal (2013), from the unanimity axiom and the key fact that it is representable (i.e., it admits a utility function).

This example shows that SEFs, that fulfil rational properties of aggregation such as OMIN, C or U, provide a framework for collective aggregation. In particular, I present an example of a nondictatorial SEF that satisfies OMIN, C, U and A. What is really amazing with this example is that anonymity (A) is still compatible with a large portion of dictatorship. More precisely, the set of profiles, \(\mathcal {U}^{2}\), is split into (pairwise disjoint) five subsets, four of which are open and the remaining has empty interior. In addition, over each of these four open subsets of \(\mathcal {U}^{2}\), the SEF exhibits a dictatorial behavior, the dictator changing from one subset to another. This situation is related to, actually can be considered as a vector-valued version of, the concept of a strong positional dictator for SWFLs that satisfy IIA, information invariance with respect to ordinally measurable and fully comparable utilities, weak Pareto, continuity and anonymity (for details see Theorem 9.4 in Bossert and Weymark (2004)).

The following function satisfies the requires properties: \(\phi (x)=(x+a_{1})/2\) for \(x\in (-\infty ,a_{1}), \phi (x)=(x+a_{j})/2\) for \(x\in (a_{j-1},a_{j}]\), and \(\phi (x)=(x+a_{p})/2\) for \(x\in (a_{p},\infty )\).

Viewed as functions from \(X\times N\) into \(\mathbb {R}\), if \(U,V\in \mathcal {U}^{n}\) are such that \(U\approx V\), then \(U,V\) are said to be comonotonic.