The operator approach to dynamic Strong Stackelberg Equilibria

The ISDG12-GTM2019 International Meeting on Game Theory: joint meeting of “12th International ISDG Workshop” and “13th International Conference on Game Theory and Management”International audienc

On the Value Iteration method for dynamic Strong Stackelberg Equilibria

International audienc

Leadership in Singleton Congestion Games: What is Hard and What is Easy

We study the problem of computing Stackelberg equilibria Stackelberg games whose underlying structure is in congestion games, focusing on the case where each player can choose a single resource (a.k.a. singleton congestion games) and one of them acts as leader. In particular, we address the cases where the players either have the same action spaces (i.e., the set of resources they can choose is the same for all of them) or different ones, and where their costs are either monotonic functions of the resource congestion or not. We show that, in the case where the players have different action spaces, the cost the leader incurs in a Stackelberg equilibrium cannot be approximated in polynomial time up to within any polynomial factor in the size of the game unless P = NP, independently of the cost functions being monotonic or not. We show that a similar result also holds when the players have nonmonotonic cost functions, even if their action spaces are the same. Differently, we prove that the case with identical action spaces and monotonic cost functions is easy, and propose polynomial-time algorithm for it. We also improve an algorithm for the computation of a socially optimal equilibrium in singleton congestion games with the same action spaces without leadership, and extend it to the computation of a Stackelberg equilibrium for the case where the leader is restricted to pure strategies. For the cases in which the problem of finding an equilibrium is hard, we show how, in the optimistic setting where the followers break ties in favor of the leader, the problem can be formulated via mixed-integer linear programming techniques, which computational experiments show to scale quite well

Stationary Strong Stackelberg Equilibrium in Discounted Stochastic Games

International audienceIn this work we study Stackelberg equilibria for discounted stochastic games. We consider two solution concepts for these games: Stationary Strong Stackelberg Equlibrium (SSSE) and Fixed Point Equilibrium (FPE) solutions. The SSSE solution is obtained by explicitly solving the Stackelberg equilibrium conditions, while the FPE can be computed efficiently using value or policy iteration algorithms. However, previous work has overlooked the relationship between these two different solution concepts. Here we investigate the conditions for existence and equivalence of these solution concepts. Our theoretical results prove that the FPE and SSSE exist and coincide for important classes of games, including Myopic Follower Strategy and Team games. This however does not hold in general and we provide numerical examples where one of SSSE or FPE does not exist, or when they both exist, they differ. Our computational results compare the solutions obtained by value iteration, policy iteration and a mathematical programming formulations for this problem. Finally, we present a discounted stochastic Stackelberg game for a security application to illustrate the solution concepts and the efficiency of the algorithms studied

Équilibre de Stackelberg Stationnaire Fort dans les jeux stochastiques actualisés

In this work we focus on Stackelberg equilibria for discounted stochasticgames. We begin by formalizing the concept of Stationary Strong Stackelberg Equlibrium(SSSE) policies for such games. We provide classes of games where the SSSE exists, andwe prove via counterexamples that SSSE does not exist in the general case. We definesuitable dynamic programming operators whose fixed points are referred to as FixedPoint Equilibrium (FPE). We show that the FPE and SSSE coincide for a class of gameswith Myopic Follower Strategy. We provide numerical examples that shed light on therelationship between SSSE and FPE and the behavior of Value Iteration, Policy Iterationand Mathematical programming formulations for this problem. Finally, we present asecurity application to illustrate the solution concepts and the efficiency of the algorithmsstudied in this article.Dans cet article, nous nous focalisons sur les équilibres de Stackelbergpour pour les jeux stochastiques actualisés. Nous commençons par formaliser le conceptd’équilibre fort de Stackelberg en politiques stationnaires (Strong Stationary StackelbergEquilibria, SSSE) pour ces jeux. Nous exhibons des classes de jeux pour lesquels le SSSEexiste et nous montrons par des contre-exemples que les SSSE n’existent pas dans le casgénéral. Nous définissons des opérateurs de programmation dynamique appropriés pour ceconcept dont les points fixes sont nommés FPE (Fixed Point Equilibria). Nous montronsque le FPE et le SSSE coïncident pour la classe de jeux stochastiques avec stratégiedu “follower” myope (Myopic Follower Strategy, MFS). Nous montrons des exemplesnumériques qui éclairent la relation entre SSSE et FPE ainsi que le comportement desalgorithmes Value Iteration, Policy Iteration, et la formulation par ProgrammationMathématique de ce problème. Finalement, nous décrivons une application dans ledomaine de la sécurité pour illustrer les concepts de solution et l’efficacité des algorithmesintroduits dans cet article

Computing Optimal Strategies to Commit to in Stochastic Games

Significant progress has been made recently in the following two lines of research in the intersection of AI and game theory: (1) the computation of optimal strategies to commit to (Stackelberg strategies), and (2) the computation of correlated equilibria of stochastic games. In this paper, we unite these two lines of research by studying the computation of Stackelberg strategies in stochastic games. We provide theoretical results on the value of being able to commit and the value of being able to correlate, as well as complexity results about computing Stackelberg strategies in stochastic games. We then modify the QPACE algorithm (MacDermed et al. 2011) to compute Stackelberg strategies, and provide experimental results

Computing optimal strategies to commit to in stochastic games

Significant progress has been made recently in the following two lines of research in the intersection of AI and game theory: (1) the computation of optimal strategies to commit to (Stackelberg strategies), and (2) the computation of correlated equilibria of stochastic games. In this paper, we unite these two lines of research by studying the computation of Stackelberg strategies in stochastic games. We provide theoretical results on the value of being able to commit and the value of being able to correlate, as well as complexity results about computing Stackelberg strategies in stochastic games. We then modify the QPACE algorithm (MacDermed et al. 2011) to compute Stackelberg strategies, and provide experimental results.