Responsibility plays a key role in the development and deployment of trustworthy autonomous systems. In this paper, we focus on the problem of strategic reasoning in probabilistic multi-agent systems with responsibility-aware agents. We introduce the logic PATL+R, a variant of Probabilistic Alternating-time Temporal Logic. The novelty of PATL+R lies in its incorporation of modalities for causal responsibility, providing a framework for responsibility-aware multi-agent strategic reasoning. We present an approach to synthesise joint strategies that satisfy an outcome specified in PATL+R, while optimising the share of expected causal responsibility and reward. This provides a notion of balanced distribution of responsibility and reward gain among agents. To this end, we utilise the Nash equilibrium as the solution concept for our strategic reasoning problem and demonstrate how to compute responsibility-aware Nash equilibrium strategies via a reduction to parametric model checking of concurrent stochastic multi-player games.

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.

Concurrent stochastic games are an important formalism for the rational verification of probabilistic multi-agent systems, which involves verifying whether a temporal logic property is satisfied in some or all game-theoretic equilibria of such systems. In this work, we study the rational verification of probabilistic multi-agent systems where agents can cooperate by communicating over unbounded lossy channels. To model such systems, we present concurrent stochastic lossy channel games (CSLCG) and employ an equilibrium concept from cooperative game theory known as the core, which is the most fundamental and widely studied cooperative equilibrium concept. Our main contribution is twofold. First, we show that the rational verification problem is undecidable for systems whose agents have almost-sure LTL objectives. Second, we provide a decidable fragment of such a class of objectives that subsumes almost-sure reachability and safety. Our techniques involve reductions to solving infinite-state zero-sum games with conjunctions of qualitative objectives. To the best of our knowledge, our result represents the first decidability result on the rational verification of stochastic multi-agent systems on infinite arenas.

Mechanism design is a well-established game-theoretic paradigm for designing games to achieve desired outcomes. This paper addresses a closely related but distinct concept, equilibrium design. Unlike mechanism design, the designer’s authority in equilibrium design is more constrained; she can only modify the incentive structures in a given game to achieve certain outcomes without the ability to create the game from scratch. We study the problem of equilibrium design using dynamic incentive structures, known as reward machines. We use weighted concurrent game structures for the game model, with goals (for the players and the designer) defined as mean-payoff objectives. We show how reward machines can be used to represent dynamic incentives that allocate rewards in a manner that optimises the designer’s goal. We also introduce the main decision problem within our framework, the payoff improvement problem. This problem essentially asks whether there exists a dynamic incentive (represented by some reward machine) that can improve the designer’s payoff by more than a given threshold value. We present two variants of the problem: strong and weak. We demonstrate that both can be solved in polynomial time using a Turing machine equipped with an NP oracle. Furthermore, we also establish that these variants are either NP-hard or coNP-hard. Finally, we show how to synthesise the corresponding reward machine if it exists.

In game theory, mechanism design is concerned with the design of incentives so that a desired outcome of the game can be achieved. In this paper, we explore the concept of equilibrium design, where incentives are designed to obtain a desirable equilibrium that satisfies a specific temporal logic property. Our study is based on a framework where system specifications are represented as temporal logic formulae, games as quantitative concurrent game structures, and players' goals as mean-payoff objectives. We consider system specifications given by LTL and GR(1) formulae, and show that designing incentives to ensure that a given temporal logic property is satisfied on some/every Nash equilibrium of the game can be achieved in PSPACE for LTL properties and in NP/{\Sigma}P 2 for GR(1) specifications. We also examine the complexity of related decision and optimisation problems, such as optimality and uniqueness of solutions, as well as considering social welfare, and show that the complexities of these problems lie within the polynomial hierarchy. Equilibrium design can be used as an alternative solution to rational synthesis and verification problems for concurrent games with mean-payoff objectives when no solution exists or as a technique to repair concurrent games with undesirable Nash equilibria in an optimal way. Comment: arXiv admin note: substantial text overlap with arXiv:2106.10192

Rational verification is the problem of determining which temporal logic properties will hold in a multi-agent system, under the assumption that agents in the system act rationally, by choosing strategies that collectively form a game-theoretic equilibrium. Previous work in this area has largely focussed on deterministic systems. In this paper, we develop the theory and algorithms for rational verification in probabilistic systems. We focus on concurrent stochastic games (CSGs), which can be used to model uncertainty and randomness in complex multi-agent environments. We study the rational verification problem for both non-cooperative games and cooperative games in the qualitative probabilistic setting. In the former case, we consider LTL properties satisfied by the Nash equilibria of the game and in the latter case LTL properties satisfied by the core. In both cases, we show that the problem is 2EXPTIME-complete, thus not harder than the much simpler verification problem of model checking LTL properties of systems modelled as Markov decision processes (MDPs).

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.

In game theory, mechanism design is concerned with the design of incentives so that a desired outcome of the game can be achieved. In this paper, we study the design of incentives so that a desirable equilibrium is obtained, for instance, an equilibrium satisfying a given temporal logic property - a problem that we call equilibrium design. We base our study on a framework where system specifications are represented as temporal logic formulae, games as quantitative concurrent game structures, and players' goals as mean-payoff objectives. In particular, we consider system specifications given by LTL and GR(1) formulae, and show that implementing a mechanism to ensure that a given temporal logic property is satisfied on some/every Nash equilibrium of the game, whenever such a mechanism exists, can be done in PSPACE for LTL properties and in NP/Sigma^P_2 for GR(1) specifications. We also study the complexity of various related decision and optimisation problems, such as optimality and uniqueness of solutions, and show that the complexities of all such problems lie within the polynomial hierarchy. As an application, equilibrium design can be used as an alternative solution to the rational synthesis and verification problems for concurrent games with mean-payoff objectives whenever no solution exists, or as a technique to repair, whenever possible, concurrent games with undesirable rational outcomes (Nash equilibria) in an optimal way.

Rational verification involves checking which temporal logic properties hold of a concurrent and multiagent system, under the assumption that agents in the system choose strategies in game theoretic equilibrium. Rational verification can be understood as a counterpart of model checking for multiagent systems, but while model checking can be done in polynomial time for some temporal logic specification languages such as CTL, and polynomial space with LTL specifications, rational verification is much more intractable: it is 2EXPTIME-complete with LTL specifications, even when using explicit-state system representations. In this paper we show that the complexity of rational verification can be greatly reduced by restricting specifications to GR(1), a fragment of LTL that can represent most response properties of reactive systems. We also provide improved complexity results for rational verification when considering players' goals given by mean-payoff utility functions -- arguably the most widely used quantitative objective for agents in concurrent and multiagent systems. In particular, we show that for a number of relevant settings, rational verification can be done in polynomial space or even in polynomial time.