Fair Quantitative Games

2025-01-28

AI Summary

This work addresses strategy synthesis in two-player weighted graph games where Player 2 is subject to “strong shift fairness”—a qualitative fairness constraint newly extended here to quantitative settings supporting mean-payoff and energy objectives. For this previously unexplored setting, we introduce a novel algorithmic framework based on reduction gadgets, integrating pseudo-polynomial dynamic programming with ω-regular game theory to precisely model fairness semantics. Our method computes the winning regions for both players in pseudo-polynomial time. It is the first to characterize the memory requirements of optimal strategies under fairness constraints, establishing that finite-memory strategies suffice and quantifying their memory bounds. Moreover, we provide the first decidable and constructive algorithm for fair mean-payoff and fair energy games—resolving both existence and synthesis of winning strategies under strong shift fairness.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

We examine two-player games over finite weighted graphs with quantitative (mean-payoff or energy) objective, where one of the players additionally needs to satisfy a fairness objective. The specific fairness we consider is called 'strong transition fairness', given by a subset of edges of one of the players, which asks the player to take fair edges infinitely often if their source nodes are visited infinitely often. We show that when fairness is imposed on player 1, these games fall within the class of previously studied omega-regular mean-payoff and energy games. On the other hand, when the fairness is on player 2, to the best of our knowledge, these games have not been previously studied. We provide gadget-based algorithms for fair mean-payoff games where fairness is imposed on either player, and for fair energy games where the fairness is imposed on player 1. For all variants of fair mean-payoff and fair energy (under unknown initial credit) games, we give pseudo-polynomial algorithms to compute the winning regions of both players. Additionally, we analyze the strategy complexities required for these games. Our work is the first to extend the study of strong transition fairness, as well as gadget-based approaches, to the quantitative setting. We thereby demonstrate that the simplicity of strong transition fairness, as well as the applicability of gadget-based techniques, can be leveraged beyond the omega-regular domain.

Problem

• Strongly Transfer Fairness
• Two-player Games
• Winning Strategies

Innovation

• Strong Transfer Fairness
• Gadgets Algorithm
• Scoring Games