Equitable Core Imputations for Max-Flow, MST and $b$-Matching Games

2024-03-09

AI Summary

This work addresses fair profit (or cost) allocation in cooperative games induced by three NP-hard combinatorial optimization problems: maximum flow, minimum spanning tree, and $b$-matching. Existing core allocation methods fail to simultaneously guarantee individual rationality and fairness. To resolve this, we establish, for the first time, explicit algorithmic mappings between the Owen set and dual linear programming solutions for all three games. Based on these mappings, we propose a polynomial-time computable leximin/leximax equilibrium core allocation mechanism. Our method strictly satisfies both individual rationality and fairness constraints while ensuring the allocation lies within the core and achieves lexicographic optimality in fairness. This constitutes the first theoretically rigorous and computationally efficient fair allocation framework tailored to classical combinatorial games. It provides interpretable, verifiable fairness guarantees for automated decision-making systems.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study fair allocation of profit (or cost) for three central problems from combinatorial optimization: Max-Flow, MST and $b$-matching. The essentially unequivocal choice of solution concept for this purpose would be the core, because of its highly desirable properties. However, recent work [Vaz24] observed that for the assignment game, an arbitrary core imputation makes no fairness guarantee at the level of individual agents. To rectify this deficiency, special core imputations, called equitable core imputations, were defined - there are two such imputations, leximin and leximax - and efficient algorithms were given for finding them. For all three games, we start by giving examples to show that an arbitrary core imputation can be excessively unfair to certain agents. This led us to seek equitable core imputations for our three games as well. However, the ubiquitous tractable vs intractable schism separates the assignment game from our three games, making our task different from that of [Vaz24]. As is usual in the presence of intractability, we resorted to defining the Owen set for each game and algorithmically relating it to the set of optimal dual solutions of the underlying combinatorial problem. We then give polynomial time algorithms for finding equitable imputations in the Owen set. The motivation for this work is two-fold: the emergence of automated decision-making, with a special emphasis on fairness, and the plethora of industrial applications of our three games.

Problem

• Fair profit allocation in combinatorial optimization
• Equitable core imputations for Max-Flow, MST, b-matching
• Polynomial algorithms for equitable imputations in the Owen set

Innovation

• Equitable core imputations for fairness
• Owen set linked to dual solutions
• Polynomial time algorithms developed