An Arrovian impossibility in combining ranking and evaluation

The term “alternative-wise” indicates a distinction from a weaker version of unanimity which requires that if all voters have exactly the same evaluations for all the alternatives, then the output evaluations are identical. We do not consider the weaker version of unanimity in this paper.

By g(R)|_{x,y}, we mean the restriction of g(R) over {x, y}; i.e., the ranking of {x, y} as in g(R). In a similar vein, R|_{x,y} is the restriction of R over {x, y}, i.e., the preference profile over {x, y} where each i ∈ N orders {x, y} as in R_i.

To ease presentation, we transpose the Cartesian product without comment, i.e., when we write (R, B), where R = (R_1, … R_n) and B = (B₁, …, B_n), we are technically referring to the profile ((R₁, B₁), … (R_n, B_n)).

The restriction to the domain Π^– is reasonable for many applications of the preference-approval framework. Suppose we need to select a date for a meeting; attendees may have preferences over those dates they can attend, but it seems reasonable to assume that they would be indifferent between those dates that they are unavailable.

For preference-approval aggregators, and other functions that return pairs, sometimes we will want to isolate the first or second coordinate of the returned value. We do this by subscripting the function with 1 or 2. Generally, for a function of type h: X → Y × Z, we write h₁ for the projection of the function onto the first coordinate and h₂ for the projection of the function onto the second coordinate, i.e., for h(x) = (y, z) we have h₁(x) = y and h₂(x) = z.

Our definitions here are given in functional style, i.e., functions are used as arguments. We assume left associativity of expressions, i.e., h(i)(x) is (h(i))(x), indeed we will typically write h_i(x); and that we assume right associativity of type definitions, i.e., h: X → Y → Z is implicitly h: X → (Y → Z).

As noted by Jean-François Laslier, methods that implicitly select the alternative with the highest median have been rediscovered many times. As well as Bucklin voting (used under this name in the early twentieth century) and majoritarian compromise (Sertel and Yılmaz 1999), such a method was proposed as early as 1793 by Condorcet (McLean et al. 1994).

Later work picks up this approach and compares different evaluative methods within an evaluative framework, for example work by Brams and Potthoff (2015).

Consider the back-and-forth between Saari and Van Newenhizen (1988), Brams et al. (1988b), Saari and Van Newenhizen (1988) and again Brams et al. (1988a).

For example: Balinski and Laraki (2007), Felsenthal and Machover (2008), Brams (2011), Edelman (2012), Balinski and Laraki (2016) and Laslier (2017).

The preference-approval framework that we treat in this paper can, as Sanver (2010) dis-cusses, be mathematically placed within the traditional literature of social welfare functionals (Sen 1977) where cardinal or interpersonal comparisons are allowed. Preference-approvals present a weak version of ordinal level comparisons (OLC) which are explored by Roberts (1980). The closest previous work of this type that we are aware of is due to List (2001), who considers a narrow informational addition that he calls OLC + 0, which only allows a single level of ordinal comparability. This is almost equivalent to the preference-approval framework, but it allows for alternatives to be on the zero-line, thus there is a third evaluative category within which indifference is forced. Also, List’s results concern functions that produce choice sets or ordinal rankings, not functions that output preference-approvals or their equivalent.

In particular we thank Phillipe Mongin for considerable efforts made in this direction.