Concurrent multi-player mean-payoff games are important models for systems of agents with individual, non-dichotomous preferences. Whilst these games have been extensively studied in terms of their equilibria in non-cooperative settings, this paper explores an alternative solution concept: the core from cooperative game theory. This concept is particularly relevant for cooperative AI systems, as it enables the modelling of cooperation among agents, even when their goals are not fully aligned. Our contribution is twofold. First, we provide a characterisation of the core using discrete geometry techniques and establish a necessary and sufficient condition for its non-emptiness. We then use the characterisation to prove the existence of polynomial witnesses in the core. Second, we use the existence of such witnesses to solve key decision problems in rational verification and provide tight complexity bounds for the problem of checking whether some/every equilibrium in a game satisfies a given LTL or GR(1) specification. Our approach is general and can be adapted to handle other specifications expressed in various fragments of LTL without incurring additional computational costs.

In rational verification , the aim is to verify which temporal logic properties will obtain in a multi-agent system, under the assumption that agents (“players”) in the system choose strategies for acting that form a game theoretic equilibrium. Preferences are typically defined by assuming that agents act in pursuit of individual goals, specified as temporal logic formulae. To date, rational verification has been studied using non-cooperative solution concepts—Nash equilibrium and refinements thereof. Such non-cooperative solution concepts assume that there is no possibility of agents forming binding agreements to cooperate, and as such they are restricted in their applicability. In this article, we extend rational verification to cooperative solution concepts, as studied in the field of cooperative game theory . We focus on the core , as this is the most fundamental (and most widely studied) cooperative solution concept. We begin by presenting a variant of the core that seems well-suited to the concurrent game setting, and we show that this version of the core can be characterised using ATL ⁎ . We then study the computational complexity of key decision problems associated with the core, which range from problems in PSpace to problems in 3ExpTime . We also investigate conditions that are sufficient to ensure that the core is non-empty, and explore when it is invariant under bisimilarity. We then introduce and study a number of variants of the main definition of the core, leading to the issue of credible deviations, and to stronger notions of collective stable behaviour. Finally, we study cooperative rational verification using an alternative model of preferences, in which players seek to maximise the mean-payoff they obtain over an infinite play in games where quantitative information is allowed.

Abstract We provide a survey of the state of the art ofrational verification: the problem of checking whether a given temporal logic formulaϕis satisfied in some or all game-theoretic equilibria of a multi-agent system – that is, whether the system will exhibit the behaviorϕrepresents under the assumption that agents within the system act rationally in pursuit of their preferences. After motivating and introducing the overall framework of rational verification, we discuss key results obtained in the past few years as well as relevant related work in logic, AI, and computer science.