Decision Problems for Nash Equilibria in Stochastic Games

We analyse the computational complexity of finding Nash equilibria in stochastic multiplayer games with $\omega$-regular objectives. While the existence of an equilibrium whose payoff falls into a certain interval may be undecidable, we single out several decidable restrictions of the problem. First, restricting the search space to stationary, or pure stationary, equilibria results in problems that are typically contained in PSPACE and NP, respectively. Second, we show that the existence of an equilibrium with a binary payoff (i.e. an equilibrium where each player either wins or loses with probability 1) is decidable. We also establish that the existence of a Nash equilibrium with a certain binary payoff entails the existence of an equilibrium with the same payoff in pure, finite-state strategies.Comment: 22 pages, revised versio

A survey of stochastic ω regular games

We summarize classical and recent results about two-player games played on graphs with ω-regular objectives. These games have applications in the verification and synthesis of reactive systems. Important distinctions are whether a graph game is turn-based or concurrent; deterministic or stochastic; zero-sum or not. We cluster known results and open problems according to these classifications

The Complexity of Nash Equilibria in Stochastic Multiplayer Games

We analyse the computational complexity of finding Nash equilibria in stochastic multiplayer games with $\omega$-regular objectives. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game~$\mathcal{G}$, does there exist a pure-strategy Nash equilibrium of~$\mathcal{G}$ where player 0 wins with probability~$1$. Moreover, this problem remains undecidable if it is restricted to strategies with (unbounded) finite memory. However, if randomised strategies are allowed, decidability remains an open problem; we can only prove NP-hardness in this case. One way to obtain a provably decidable variant of the problem is to restrict the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively. Finally, we single out a special case of the general problem that, in many cases, admits an efficient solution. In particular, we prove that deciding the existence of an equilibrium in which each player either wins or loses with probability~$1$ can be done in polynomial time for games where, for instance, the objective of each player is given by a parity condition with a bounded number of priorities

Computer aided synthesis: a game theoretic approach

In this invited contribution, we propose a comprehensive introduction to game theory applied in computer aided synthesis. In this context, we give some classical results on two-player zero-sum games and then on multi-player non zero-sum games. The simple case of one-player games is strongly related to automata theory on infinite words. All along the article, we focus on general approaches to solve the studied problems, and we provide several illustrative examples as well as intuitions on the proofs.Comment: Invitation contribution for conference "Developments in Language Theory" (DLT 2017

The Complexity of Nash Equilibria in Stochastic Multiplayer Games

We analyse the computational complexity of finding Nash equilibria in turn-based stochastic multiplayer games with omega-regular objectives. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game G, does there exist a Nash equilibrium of G where Player 0 wins with probability 1? Moreover, this problem remains undecidable when restricted to pure strategies or (pure) strategies with finite memory. One way to obtain a decidable variant of the problem is to restrict the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively. Finally, we single out a special case of the general problem that, in many cases, admits an efficient solution. In particular, we prove that deciding the existence of an equilibrium in which each player either wins or loses with probability 1 can be done in polynomial time for games where the objective of each player is given by a parity condition with a bounded number of priorities

Pure Nash Equilibria in Concurrent Deterministic Games

We study pure-strategy Nash equilibria in multi-player concurrent deterministic games, for a variety of preference relations. We provide a novel construction, called the suspect game, which transforms a multi-player concurrent game into a two-player turn-based game which turns Nash equilibria into winning strategies (for some objective that depends on the preference relations of the players in the original game). We use that transformation to design algorithms for computing Nash equilibria in finite games, which in most cases have optimal worst-case complexity, for large classes of preference relations. This includes the purely qualitative framework, where each player has a single omega-regular objective that she wants to satisfy, but also the larger class of semi-quantitative objectives, where each player has several omega-regular objectives equipped with a preorder (for instance, a player may want to satisfy all her objectives, or to maximise the number of objectives that she achieves.)Comment: 72 page