Fair Matchings with an Application to Chess

In this paper we consider the strategic problem of pairing opponents from two teams at a chess meet. We propose using a bias-minimizing stable matching and establish two main results, that a strict dominance relationship between the two teams can be determined in polynomial time, and if such a relationship exists, then a biasminimizing stable matching can be found by the simplex method. We also propose a bound on the computational complexity of finding a bias-minimizing stable matching if no such relationship exists. Introduction In a chess meet two teams A and B, each of five players, are paired off in 5 parallel matches. The scoring of each match is determined as follows: a player receives 1 point if he wins his match, 2 1 points if he ties, and 0 points if he loses. Each team’s score is them the sum of the scores of the individual players on that team, and the winning team is the team with the highest score. Typically the players are paired off as follows, the highest rated player from team A plays the highest rated player from team B, the 2 highest rated player from team A players the 2 highest rated player from team B, and so on. Such a matching is appropriate if the players are consistent, i.e. if player a1∈A is rated higher than player a2∈A, then there is no player b∈B such that player a2 is expected to perform strictly better against b than player a1. This paper will consider the case where players are erratic, i.e. where inversions of the expected performance relative to rating are permitted. A meet with erratic players introduces an element of strategy into the pairing of players from teams A and B, as now a team’s win or loss (or tie) depends not only on the “overall” ability of its players, but also upon the choice of opponent. Define f(i,j) = 1×Prob(ai beats bj) + 2 1 ×Prob(ai ties bj) + 0×Prob(bj beats ai) for ai∈A, bj∈B. That is, f(i,j) is the expected number of points ai will score if he plays against bj. Notice that since a chess match is a constant sum game with sum equal to one, the expected number of points bj will score if he plays ai, g(i,j), is equal to 1-f(i,j). Here we presume that the attendant probabilities are known or can be estimated from past performance for every (ai,bj) ∈A×B. Then for any matching μ between A and B, define: f(μ ) = ∑ ∈μ ) , ( ) , (