Fair Core Imputations for the Assignment Game: New Solution Concepts and Efficient Algorithms
Vijay V. Vazirani
TL;DR
This paper addresses individual fairness within the core of the assignment game by introducing three solution concepts: leximin core (maximize the weakest payoff and proceed lexicographically), leximax core (minimize the strongest payoff lexicographically), and min-spread core (minimize the payoff gap among essential agents). It develops strongly polynomial combinatorial algorithms for all three, using a novel primal-dual framework driven by a coordinating clock that synchronizes local updates to yield a global leximin/leximax/min-spread imputation. The methods rely on a detailed complementarity analysis between primal and dual solutions, and exploit the structure of essential/viable/subpar agents and teams to achieve efficiency, including a phase that manages degeneracy. The results show that nucleolus may fail to deliver individual fairness and NSW core imputations may be impractical, thereby advancing algorithmic fairness in cooperative games with practical implications for fair profit sharing in bipartite matching settings.
Abstract
The assignment game is a classical model for profit sharing and a cornerstone of cooperative game theory. While an imputation in its core guarantees fairness among coalitions, it provides no fairness guarantee at the level of individual agents: single agents or one sided coalitions have zero standalone value and may receive arbitrarily small payoffs. Motivated by the growing focus on individual level fairness, we ask: Can one select a core imputation that is also fair to individuals? We introduce three individual fairness driven solution concepts, each promoting equity in a different way. The leximin and leximax core imputations extend max min and min max fairness to uplift the least advantaged and constrain the most advantaged agents, respectively. The min spread core imputation minimizes the gap between the largest and smallest positive payoffs, promoting equitable profit distribution. For all three solution concepts, we develop combinatorial, strongly polynomial algorithms. The leximin and leximax algorithms are based on a novel adaptation of the primal dual paradigm, while the min spread algorithm combines partial executions of the first two. We expect our work to revive innovation on the potent primal dual paradigm as well as promote further work on the algorithmic study of fairness and stability.