A computational analysis of the tournament equilibrium set

Abstract

A recurring theme in the mathematical social sciences is how to select the "most desirable" elements given a binary dominance relation on a set of alternatives. Schwartz's tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions that have been proposed so far in this context. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem up to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive tournament solution concept refining both the Banks set and Dutta's minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NP-hard.

The minimal covering set in large tournaments

“…Given these results, an obvious open question is to establish the size of other tournament solutions. For the case of the tournament equilibrium set (Schwartz, 1990), Brandt et al (2010) reports on computational experiments that showed the tournament equilibrium set almost always was the entire set of alternatives in uniformly chosen large tournaments. This evidence suggests…”

A counterexample to a conjecture of Schwartz

“…By means of an exhaustive computer analysis, this counterexample was later shown to be of minimum cardinality [3]. The same analysis did not yield a counterexample to Schwartz' conjecture itself in all tournaments with less than 13 vertices and billions of random tournaments with up to 50 vertices.…”

“…Over the years, Schwartz' conjecture has been extensively studied [6,10,11,8,2,3]. For instance, it is known that Schwartz' conjecture is equivalent to τ having any one of several desirable properties of tournament solutions, including monotonicity, independence of unchosen alternatives, and the "strong superset property".…”

Bounds on the disparity and separation of tournament solutions

“…A general method for computingS, given an implementation for S, is to compute the corresponding relation S − → and return the maximal elements of that relation's transitive closure, as suggested by Brandt et al [9]. In order to computeTC , we consider This takes polynomial time.…”

“…Due to its recursive nature, computing TEQ is much harder than computingTC . The problem is known to be NP-hard while the best known upper bound is PSPACE [9]. For general tournaments with more than 100 alternatives, computing TEQ is currently out of reach.…”