The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games

We study multiplayer quantitative reachability games played on a finite directed graph, where the objective of each player is to reach his target set of vertices as quickly as possible. Instead of the well-known notion of Nash equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE), a refinement of NE well-suited in the framework of games played on graphs. It is known that there always exists an SPE in quantitative reachability games and that the constrained existence problem is decidable. We here prove that this problem is PSPACE-complete. To obtain this result, we propose a new algorithm that iteratively builds a set of constraints characterizing the set of SPE outcomes in quantitative reachability games. This set of constraints is obtained by iterating an operator that reinforces the constraints up to obtaining a fixpoint. With this fixpoint, the set of SPE outcomes can be represented by a finite graph of size at most exponential. A careful inspection of the computation allows us to establish PSPACE membership

The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games

We study multiplayer quantitative reachability games played on a finite directed graph, where the objective of each player is to reach his target set of vertices as quickly as possible. Instead of the well-known notion of Nash equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE), a refinement of NE well-suited in the framework of games played on graphs. It is known that there always exists an SPE in quantitative reachability games and that the constrained existence problem is decidable. We here prove that this problem is PSPACE-complete. To obtain this result, we propose a new algorithm that iteratively builds a set of constraints characterizing the set of SPE outcomes in quantitative reachability games. This set of constraints is obtained by iterating an operator that reinforces the constraints up to obtaining a fixpoint. With this fixpoint, the set of SPE outcomes can be represented by a finite graph of size at most exponential. A careful inspection of the computation allows us to establish PSPACE membership

The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games

We study multiplayer quantitative reachability games played on a finite directed graph, where the objective of each player is to reach his target set of vertices as quickly as possible. Instead of the well-known notion of Nash equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE), a refinement of NE well-suited in the framework of games played on graphs. It is known that there always exists an SPE in quantitative reachability games and that the constrained existence problem is decidable. We here prove that this problem is PSPACE-complete. To obtain this result, we propose a new algorithm that iteratively builds a set of constraints characterizing the set of SPE outcomes in quantitative reachability games. This set of constraints is obtained by iterating an operator that reinforces the constraints up to obtaining a fixpoint. With this fixpoint, the set of SPE outcomes can be represented by a finite graph of size at most exponential. A careful inspection of the computation allows us to establish PSPACE membership

Equilibria in Multiplayer Games Played on Graphs

Today, as computer systems are ubiquitous in our everyday life, there is no need to argue that their correctness is of capital importance. In order to prove (in a mathematical sense) that a given system satisfies a given property, formal methods have been introduced. They include concepts such as model checking and synthesis. Roughly speaking, when considering synthesis, we aim at building a model of the system which is correct by construction. In order to do so, models are mainly borrowed from game theory. During the last decades, there has been a shift from two-player qualitative zero-sum games (used to model antagonistic interactions between a system and its environment) to multiplayer quantitative games (used to model complex systems composed of several agents whose objectives are not necessarily antagonistic). In the latter setting, the solution concepts of interest include numerous equilibria, such as Nash equilibrium (NE) and subgame perfect equilibrium (SPE). While the existence of equilibria is widely studied, it is also well known that several equilibria may coexist in the same game. Nevertheless, some equilibria are more relevant than others. For example, if we consider a game in which each player aims at satisfying a given qualitative objective, it is possible to have both an equilibrium in which no player satisfies his objective and another one in which each player satisfies it. In this case one prefers the latter equilibrium which is more relevant.In this thesis, we focus on multiplayer turn-based games played on graphs either with qualitative or quantitative objectives. Our contributions are twofold: (i) we provide equilibria characterizations and (ii) we use these characterizations to solve decision problems related to the existence of relevant equilibria; and characterize their complexities. Firstly, we provide a characterization of a weaker notion of SPE (weak SPE) in multiplayer games with omega-regular objectives based on the payoff profiles which are realizable by a weak SPE. We then adopt another point of view by characterizing the outcomes of equilibria instead of their payoff profiles. In particular we focus on weak SPE outcome characterization. As for some kinds of games (e.g. multiplayer quantitative Reachability games), weak SPEs and SPEs are equivalent, this characterization is useful in order to study SPEs in these games.Secondly, we use those different equilibrium characterizations to provide the exact complexity classes of different decision problems related to the existence of relevant equilibria. We mainly focus on the constrained existence problem: if each player aims at maximizing his gain, this problem asks whether there exists an equilibrium such that each resulting player’s gain is greater than a threshold (one per player). We also consider variants of relevant equilibria based on the social welfare and the Pareto optimality of the players’ payoff. In this way, we prove the exact complexity classes for (i) the weak SPE constrained existence problem in multiplayer games with classical qualitative objectives such as Büchi, co-Büchi and Safety and (ii) the NE and SPE constrained existence problems (and variants) for qualitative and quantitative reachability games. In the latter case, the upper bounds on the required memory for such relevant equilibria are studied and proved to be finite. Studying memory requirements of strategies is important since with the synthesis process those strategies have to be implemented.Finally, we consider multiplayer, non zero-sum, turn-based timed games with qualitative Reachability objectives together with the concept of SPE. We prove that the SPE constrained existence problem is EXPTIME-complete for qualitative Reachability timed games. In order to obtain an EXPTIME algorithm, we proceed in different steps. In the first step, we prove that the game variant of the classical region graph is a good abstraction for the SPE constrained existence problem. In fact, we identify conditions on bisimulations under which the study of SPE in a given game can be reduced to the study of its quotient.Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe

Constrained existence of weak subgame perfect equilibria in multiplayer Büchi games

In turn-based games played on graphs, the constrained existence problem of equilibria is a well studied problem. This problem is already known to be decidable in P for weak subgame perfect equilibria in Boolean games with Büchi objectives. In this paper, we prove that this problem is P-complete.SCOPUS: ar.jDecretOANoAutActifinfo:eu-repo/semantics/publishe

Non-blind Strategies in Timed Network Congestion Games

International audienceNetwork congestion games are a convenient model for reasoning about routing problems in a network: agents have to move from a source to a target vertex while avoiding congestion, measured as a cost depending on the number of players using the same link. Network congestion games have been extensively studied over the last 40 years, whiletheir extension with timing constraints were considered more recently. Most of the results on network congestion games consider blind strategies: they are static, and do not adapt to the strategies selected by the other players. We extend the recent results of [Bertrand et al., Dynamic network congestion games. FSTTCS’20] to timed network congestion games, in which the availability of the edges depend on (discrete) time. We prove that computing Nash equilibria satisfying some constraint on the total cost (and in particular, computing the best and worst Nash equilibria), and computing the social optimum, can be achieved in exponential space. The social optimum can be computed in polynomial space if all players have the same source and target

On relevant equilibria in reachability games

We study multiplayer reachability games played on a finite directed graph equipped with target sets, one for each player. In those reachability games, it is known that there always exists a Nash equilibrium. But sometimes several equilibria may coexist. For instance we can have two equilibria: a first one where no player reaches his target set and an other one where all the players reach their target set. It is thus very natural to identify “relevant” equilibria. In this paper, we consider different notions of relevant Nash equilibria including Pareto optimal equilibria and equilibria with high social welfare. We also study relevant subgame perfect equilibria in reachability games. We provide complexity results for various related decision problems for both Nash equilibria and subgame perfect equilibria.SCOPUS: ar.jDecretOANoAutActifinfo:eu-repo/semantics/publishe