arXiv subjects

Christophe Grandmont

Publications and source records attributed to Christophe Grandmont.

3 recordsLinked to original sources

arXiv2026

An Automata-Based Approach to Games with $\omega$-Automatic Preferences

This paper studies multiplayer turn-based games on graphs in which player preferences are modeled as $\omega$-automatic relations given by deterministic parity automata. This contrasts with most existing work, which focuses on specific reward functions. We conduct a computational analysis of these games, starting with the threshold problem in the antagonistic zero-sum case. As in classical games, we introduce the concept of value, defined here as the set of plays a player can guarantee to improve upon, relative to their preference relation. We show that this set is recognized by an alternating parity automaton APW of polynomial size. We also establish the computational complexity of several problems related to the concepts of value and optimal strategy, taking advantage of the $\omega$-automatic characterization of value. Next, we shift to multiplayer games and Nash equilibria, and revisit the threshold problem in this context. Based on an APW construction again, we close complexity gaps left open in the literature, and additionally show that cooperative rational synthesis is $\mathsf{PSPACE}$-complete, while it becomes undecidable in the non-cooperative case.

cs.GT↗

arXiv2026

Games with $ω$-Automatic Preference Relations

This paper investigates Nash equilibria (NEs) in multi-player turn-based games on graphs, where player preferences are modeled as $ω$-automatic relations via deterministic parity automata. Unlike much of the existing literature, which focuses on specific reward functions, our results apply to any preference relation definable by an $ω$-automatic relation. We analyze the computational complexity of determining the existence of an NE (possibly under some constraints), verifying whether a given strategy profile forms an NE, and checking whether a specific outcome can be realized by an NE. When a (constrained) NE exists, we show that there always exists one with finite-memory strategies. Finally, we explore fundamental properties of $ω$-automatic relations and their implications for the existence of equilibria.

cs.LO↗

arXiv2025

As Soon as Possible but Rationally

This paper addresses complexity problems in rational verification and synthesis for multi-player games played on weighted graphs, where the objective of each player is to minimize the cost of reaching a specific set of target vertices. In these games, one player, referred to as the system, declares his strategy upfront. The other players, composing the environment, then rationally make their moves according to their objectives. The rational behavior of these responding players is captured through two models: they opt for strategies that either represent a Nash equilibrium or lead to a play with a Pareto-optimal cost tuple.

cs.GT↗