Robust Bounds on Choosing from Large Tournaments

TL;DR

This paper extends the analysis of tournament solutions to a more general probabilistic model, showing that many solutions still rarely exclude alternatives in large tournaments, and provides bounds on when all alternatives are selected.

Contribution

It introduces a generalized probabilistic model for tournaments and proves that common solutions remain inclusive, offering tight bounds on their selection behavior.

Findings

Tournament solutions like the top cycle rarely exclude alternatives in large random tournaments.

The results hold under more flexible probabilistic models than previous rigid assumptions.

Tight asymptotic bounds are established for the probability ranges where all alternatives are selected.

Abstract

Tournament solutions provide methods for selecting the "best" alternatives from a tournament and have found applications in a wide range of areas. Previous work has shown that several well-known tournament solutions almost never rule out any alternative in large random tournaments. Nevertheless, all analytical results thus far have assumed a rigid probabilistic model, in which either a tournament is chosen uniformly at random, or there is a linear order of alternatives and the orientation of all edges in the tournament is chosen with the same probabilities according to the linear order. In this work, we consider a significantly more general model where the orientation of different edges can be chosen with different probabilities. We show that a number of common tournament solutions, including the top cycle and the uncovered set, are still unlikely to rule out any alternative under this model. This corresponds to natural graph-theoretic conditions such as irreducibility of the tournament. In addition, we provide tight asymptotic bounds on the boundary of the probability range for which the tournament solutions select all alternatives with high probability.