Algorithms for Leximin-Optimal Fair Policies in Repeated Games

Solutions to non-cooperative multiagent systems often require achieving a joint policy which is as fair to all parties as possible. There are a variety of methods for determining the fairest such joint policy. One approach, min fairness, finds the policy which maximizes the minimum average reward given to any agent. We focus on an extension, leximin fairness, which breaks ties among candidate policies by choosing the one which maximizes the second-to-minimum average reward, then the thirdto-minimum average reward, and so on. This method has a number of advantages over others in the literature, but has so far been little-used because of the computational cost in employing it to find the fairest policy. In this paper we propose a linear programming based algorithm for computing leximin fairness in repeated games which has a polynomial time complexity given certain reasonable assumptions.