This paper studies the rational synthesis problem for multi-player games played on graphs when rational players are following subgame perfect equilibria. In these games, one player, the system, declares his strategy upfront, and the other players, composing the environment, then rationally respond by playing strategies forming a subgame perfect equilibrium. We study the complexity of the rational synthesis problem when the players have ω-regular objectives encoded as parity objectives. Our algorithm is based on an encoding into a three-player game with imperfect information, showing that the problem is in 2ExpTime. When the number of environment players is fixed, the problem is in ExpTime and is NP- and coNP-hard. Moreover, for a fixed number of players and reachability objectives, we get a polynomial algorithm.

Fairness is a desirable and crucial property of many protocols that handle, for instance, exchanges of message. It states that if at least one agent engaging in the protocol is honest, then either the protocol will unfold correctly and fulfill its intended goal for all participants, or it will fail for everyone. In this work, we present a game-based framework for the study of fairness protocols, that does not define a priori an attacker model. It is based on the notion of strong secure equilibria, and leverages the conceptual and algorithmic toolbox of game theory. In the case of finite games, we provide decision procedures with tight complexity bounds for determining whether a protocol is immune to nefarious attacks from a coalition of participants, and whether such a protocol could exist based on the underlying graph structure and objectives.

We study a natural problem about rational behaviors in multiplayer non-zero-sum sequential infinite duration games played on graphs: rational verification, that consists in deciding whether all the rational answers to a given strategy satisfy some specification. We give the complexities of that problem for two major concepts of rationality: Nash equilibria and subgame-perfect equilibria, and for three major classes of payoff functions: energy, discounted-sum, and mean-payoff.

In this paper, we provide an effective characterization of all the subgame-perfect equilibria in infinite duration games played on finite graphs with mean-payoff objectives. To this end, we introduce the notion of requirement, and the notion of negotiation function. We establish that the plays that are supported by SPEs are exactly those that are consistent with a fixed point of the negotiation function. Finally, we use that characterization to prove that the SPE threshold problem, who status was left open in the literature, is decidable.

We study the complexity of problems related to subgame-perfect equilibria (SPEs) in infinite duration non zero-sum multiplayer games played on finite graphs with parity objectives. We present new complexity results that close gaps in the literature. Our techniques are based on a recent characterization of SPEs in prefix-independent games that is grounded on the notions of requirements and negotiation, and according to which the plays supported by SPEs are exactly the plays consistent with the requirement that is the least fixed point of the negotiation function. The new results are as follows. First, checking that a given requirement is a fixed point of the negotiation function is an NP-complete problem. Second, we show that the SPE constrained existence problem is NP-complete, this problem was previously known to be ExpTime-easy and NP-hard. Third, the SPE constrained existence problem is fixed-parameter tractable when the number of players and of colors are parameters. Fourth, deciding whether some requirement is the least fixed point of the negotiation function is complete for the second level of the Boolean hierarchy. Finally, the SPE-verification problem - that is, the problem of deciding whether there exists a play supported by a SPE that satisfies some LTL formula - is PSpace-complete, this problem was known to be ExpTime-easy and PSpace-hard.

We establish that the subgame perfect equilibrium (SPE) threshold problem for mean-payoff games is NP-complete. While the SPE threshold problem was recently shown to be decidable (in doubly exponential time) and NP-hard, its exact worst case complexity was left open.

In this paper, we provide an effective characterization of all the subgame-perfect equilibria in infinite duration games played on finite graphs with mean-payoff objectives. To this end, we introduce the notion of requirement, and the notion of negotiation function. We establish that the plays that are supported by SPEs are exactly those that are consistent with the least fixed point of the negotiation function. Finally, we show that the negotiation function is piecewise linear, and can be analyzed using the linear algebraic tool box. As a corollary, we prove the decidability of the SPE constrained existence problem, whose status was left open in the literature.

Abstract Discounted-sum games provide a formal model for the study of reinforcement learning, where the agent is enticed to get rewards early since later rewards are discounted. When the agent interacts with the environment, she may realize that, with hindsight, she could have increased her reward by playing differently: this difference in outcomes constitutes her regret value. The agent may thus elect to follow a regret- minimal strategy. In this paper, it is shown that (1) there always exist regret-minimal strategies that are admissible—a strategy being inadmissible if there is another strategy that always performs better; (2) computing the minimum possible regret or checking that a strategy is regret-minimal can be done in "Equation missing", disregarding the computational cost of numerical analysis (otherwise, this bound becomes "Equation missing").

Admissible strategies, i.e. those that are not dominated by any other strategy, are a typical rationality notion in game theory. In many classes of games this is justified by results showing that any strategy is admissible or dominated by an admissible strategy. However, in games played on finite graphs with quantitative objectives (as used for reactive synthesis), this is not the case. We consider increasing chains of strategies instead to recover a satisfactory rationality notion based on dominance in such games. We start with some order-theoretic considerations establishing sufficient criteria for this to work. We then turn our attention to generalised safety/reachability games as a particular application. We propose the notion of maximal uniform chain as the desired dominance-based rationality concept in these games. Decidability of some fundamental questions about uniform chains is established.

We investigate infinite games on finite graphs where the information flow is perturbed by nondeterministic signalling delays. It is known that such perturbations make synthesis problems virtually unsolvable, in the general case. On the classical model where signals are attached to states, tractable cases are rare and difficult to identify. Here, we propose a model where signals are detached from control states, and we identify a subclass on which equilibrium outcomes can be preserved, even if signals are delivered with a delay that is finitely bounded. To offset the perturbation, our solution procedure combines responses from a collection of virtual plays following an equilibrium strategy in the instant- signalling game to synthesise, in a Frankenstein manner, an equivalent equilibrium strategy for the delayed-signalling game.