Weighted tournament solutions

An obvious way to move beyond tournament solutions as studied in Chapter 3 is to take into account not only the direction of the majority preference between a pair of alternatives, but also its strength in terms of the margin by which one alternative is preferred. We focus in this chapter on social choice functions that are called C2 functions by Fishburn (1977) and could also be referred to as weighted tournament solutions: social choice functions that depend only on pairwise majority margins but are not tournament solutions. Consider a set A = {1, …, m} of alternatives and a set N = {1, …, n} of voters with preferences ≻_i∈ L (A) for all i ∈ N. Here, we denote by L (X) the set of all linear orders on a finite set X, that is, the set of all binary relations on X that are complete, transitive, and asymmetric. For a given preference profile R = (≻₁, …, ≻_n) ∈ L (A) ⁿ, the majority margin m_R(x, y) of x over y is defined as the difference between the number of voters who prefer x to y and the number of voters who prefer y to x, that is, We will routinely omit the subscript when R is clear from the context. The pairwise majority margins arising from a preference profile R can be conveniently represented by a weighted tournament (A, M_R), where M_R is the antisymmetric m×m matrix with (M_R) _xx = 0 for x ∈ A and (M_R) _xy = m_R(x, y) for x, y ∈ A with x≠y. An example of a weighted tournament and a corresponding preference profile is shown in Figure 4.1. Because C2 functions only depend on majority margins, they can be viewed as functions mapping weighted tournaments to sets of alternatives, or to linear orders of the alternatives. Clearly all majority margins will be even if the number of voters is even, and odd if the number of voters is odd. The following result, similarly to McGarvey’s Theorem for tournaments (McGarvey, 1953), establishes that this condition in fact characterizes the set of weighted tournaments induced by preference profiles.