Bounding Fixed Points of Set-based Bellman Operator and Nash Equilibria of Stochastic Games

Sarah H Q Li (1) , Assalé Adjé (2) , Pierre-Loïc Garoche (3) , Behçet Açikmeşe (1)

1 University of Washington [Seattle]
2 UPVD - Université de Perpignan Via Domitia
3 ENAC - Ecole Nationale de l'Aviation Civile

Motivated by uncertain parameters encountered in Markov decision processes (MDPs) and stochastic games, we study the effect of parameter uncertainty on Bellman operator-based algorithms under a set-based framework. Specifically, we first consider a family of MDPs where the cost parameters are in a given compact set; we then define a Bellman operator acting on a set of value functions to produce a new set of value functions as the output under all possible variations in the cost parameter. We prove the existence of a fixed point of this set-based Bellman operator by showing that it is contractive on a complete metric space, and explore its relationship with the corresponding family of MDPs and stochastic games. Additionally, we show that given interval set-bounded cost parameters, we can form exact bounds on the set of optimal value functions. Finally, we utilize our results to bound the value function trajectory of a player in a stochastic game.

Mots clés: Markov decision process; learning theory; stochastic control; multi-agent systems; learning in games; decision making and autonomy

Domaines: Systèmes et contrôle [cs.SY]; Informatique et théorie des jeux [cs.GT]; Théorie et langage formel [cs.FL]

Sarah H Q Li, Assalé Adjé, Pierre-Loïc Garoche, Behçet Açikmeşe. Bounding Fixed Points of Set-based Bellman Operator and Nash Equilibria of Stochastic Games. Automatica, 2021, 130, ⟨10.1016/j.automatica.2021.109685⟩. ⟨hal-03187698⟩