Fair Quantitative Games

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.