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

Concurrent Games and Semi-Random Determinacy

Consider concurrent, infinite duration, two-player win/lose games played on graphs. If the winning condition satisfies some simple requirement, the existence of Player 1 winning (finite-memory) strategies is equivalent to the existence of winning (finite-memory) strategies in finitely many derived one-player games. Several classical winning conditions satisfy this simple requirement. Under an additional requirement on the winning condition, the non-existence of Player 1 winning strategies from all vertices is equivalent to the existence of Player 2 stochastic strategies almost-sure winning from all vertices. Only few classical winning conditions satisfy this additional requirement, but a fairness variant of omega-regular languages does

Repairing Multi-Player Games

Synthesis is the automated construction of systems from their specifications. Modern systems often consist of interacting components, each having its own objective. The interaction among the components is modeled by a multi-player game. Strategies of the components induce a trace in the game, and the objective of each component is to force the game into a trace that satisfies its specification. This is modeled by augmenting the game with omega-regular winning conditions. Unlike traditional synthesis games, which are zero-sum, here the objectives of the components do not necessarily contradict each other. Accordingly, typical questions about these games concern their stability - whether the players reach an equilibrium, and their social welfare - maximizing the set of (possibly weighted) specifications that are satisfied. We introduce and study repair of multi-player games. Given a game, we study the possibility of modifying the objectives of the players in order to obtain stability or to improve the social welfare. Specifically, we solve the problem of modifying the winning conditions in a given concurrent multi-player game in a way that guarantees the existence of a Nash equilibrium. Each modification has a value, reflecting both the cost of strengthening or weakening the underlying specifications, as well as the benefit of satisfying specifications in the obtained equilibrium. We seek optimal modifications, and we study the problem for various omega-regular objectives and various cost and benefit functions. We analyze the complexity of the problem in the general setting as well as in one with a fixed number of players. We also study two additional types of repair, namely redirection of transitions and control of a subset of the players

Liveness of Randomised Parameterised Systems under Arbitrary Schedulers (Technical Report)

We consider the problem of verifying liveness for systems with a finite, but unbounded, number of processes, commonly known as parameterised systems. Typical examples of such systems include distributed protocols (e.g. for the dining philosopher problem). Unlike the case of verifying safety, proving liveness is still considered extremely challenging, especially in the presence of randomness in the system. In this paper we consider liveness under arbitrary (including unfair) schedulers, which is often considered a desirable property in the literature of self-stabilising systems. We introduce an automatic method of proving liveness for randomised parameterised systems under arbitrary schedulers. Viewing liveness as a two-player reachability game (between Scheduler and Process), our method is a CEGAR approach that synthesises a progress relation for Process that can be symbolically represented as a finite-state automaton. The method is incremental and exploits both Angluin-style L*-learning and SAT-solvers. Our experiments show that our algorithm is able to prove liveness automatically for well-known randomised distributed protocols, including Lehmann-Rabin Randomised Dining Philosopher Protocol and randomised self-stabilising protocols (such as the Israeli-Jalfon Protocol). To the best of our knowledge, this is the first fully-automatic method that can prove liveness for randomised protocols.Comment: Full version of CAV'16 pape

Randomness for Free

We consider two-player zero-sum games on graphs. These games can be classified on the basis of the information of the players and on the mode of interaction between them. On the basis of information the classification is as follows: (a) partial-observation (both players have partial view of the game); (b) one-sided complete-observation (one player has complete observation); and (c) complete-observation (both players have complete view of the game). On the basis of mode of interaction we have the following classification: (a) concurrent (both players interact simultaneously); and (b) turn-based (both players interact in turn). The two sources of randomness in these games are randomness in transition function and randomness in strategies. In general, randomized strategies are more powerful than deterministic strategies, and randomness in transitions gives more general classes of games. In this work we present a complete characterization for the classes of games where randomness is not helpful in: (a) the transition function probabilistic transition can be simulated by deterministic transition); and (b) strategies (pure strategies are as powerful as randomized strategies). As consequence of our characterization we obtain new undecidability results for these games

Non-Zero Sum Games for Reactive Synthesis

In this invited contribution, we summarize new solution concepts useful for the synthesis of reactive systems that we have introduced in several recent publications. These solution concepts are developed in the context of non-zero sum games played on graphs. They are part of the contributions obtained in the inVEST project funded by the European Research Council.Comment: LATA'16 invited pape

Qualitative Analysis of Concurrent Mean-payoff Games

We consider concurrent games played by two-players on a finite-state graph, where in every round the players simultaneously choose a move, and the current state along with the joint moves determine the successor state. We study a fundamental objective, namely, mean-payoff objective, where a reward is associated to each transition, and the goal of player 1 is to maximize the long-run average of the rewards, and the objective of player 2 is strictly the opposite. The path constraint for player 1 could be qualitative, i.e., the mean-payoff is the maximal reward, or arbitrarily close to it; or quantitative, i.e., a given threshold between the minimal and maximal reward. We consider the computation of the almost-sure (resp. positive) winning sets, where player 1 can ensure that the path constraint is satisfied with probability 1 (resp. positive probability). Our main results for qualitative path constraints are as follows: (1) we establish qualitative determinacy results that show that for every state either player 1 has a strategy to ensure almost-sure (resp. positive) winning against all player-2 strategies, or player 2 has a spoiling strategy to falsify almost-sure (resp. positive) winning against all player-1 strategies; (2) we present optimal strategy complexity results that precisely characterize the classes of strategies required for almost-sure and positive winning for both players; and (3) we present quadratic time algorithms to compute the almost-sure and the positive winning sets, matching the best known bound of algorithms for much simpler problems (such as reachability objectives). For quantitative constraints we show that a polynomial time solution for the almost-sure or the positive winning set would imply a solution to a long-standing open problem (the value problem for turn-based deterministic mean-payoff games) that is not known to be solvable in polynomial time

Termination Criteria for Solving Concurrent Safety and Reachability Games

We consider concurrent games played on graphs. At every round of a game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective to stay forever in a given set of states, and its dual, the reachability objective to reach a given set of states. We present in this paper a strategy improvement algorithm for computing the value of a concurrent safety game, that is, the maximal probability with which player~1 can enforce the safety objective. The algorithm yields a sequence of player-1 strategies which ensure probabilities of winning that converge monotonically to the value of the safety game. Our result is significant because the strategy improvement algorithm provides, for the first time, a way to approximate the value of a concurrent safety game from below. Since a value iteration algorithm, or a strategy improvement algorithm for reachability games, can be used to approximate the same value from above, the combination of both algorithms yields a method for computing a converging sequence of upper and lower bounds for the values of concurrent reachability and safety games. Previous methods could approximate the values of these games only from one direction, and as no rates of convergence are known, they did not provide a practical way to solve these games