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.

We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model of multi-agent systems [Gutierrez et al. 2013, 2015a]. In such a game, each agent <?TeX $i$?> has a goal <?TeX $\gamma _i$?> , represented using (a fragment of) Linear Temporal Logic ( <?TeX $\mathrm{LTL}$?> ) . The goal <?TeX $\gamma _i$?> captures agent <?TeX $i$?> ’s preferences, in the sense that the models of <?TeX $\gamma _i$?> represent system behaviours that would satisfy <?TeX $i$?> . Each player controls a subset of Boolean variables <?TeX $\Phi _i$?> , and at each round in the game, player <?TeX $i$?> is at liberty to choose values for variables <?TeX $\Phi _i$?> in any way that she sees fit. Play continues for an infinite sequence of rounds, and so as players act they collectively trace out a model for <?TeX $\mathrm{LTL}$?> , which for every player will either satisfy or fail to satisfy their goal. Players are assumed to act strategically, taking into account the goals of other players, in an attempt to bring about computations satisfying their goal. In this setting, we apply the standard game-theoretic concept of (pure) Nash equilibria. The (possibly empty) set of Nash equilibria of an iterated Boolean game can be understood as inducing a set of computations, each computation representing one way the system could evolve if players chose strategies that together constitute a Nash equilibrium. Such a set of equilibrium computations expresses a temporal property—which may or may not be expressible within a particular <?TeX $\mathrm{LTL}$?> fragment. The new notion of expressiveness that we formally define and investigate is then as follows: What temporal properties are characterised by the Nash equilibria of games in which agent goals are expressed in specific fragments of <?TeX $\mathrm{LTL}$?> ? We formally define and investigate this notion of expressiveness for a range of <?TeX $\mathrm{LTL}$?> fragments. For example, a very natural question is the following: Suppose we have an iterated Boolean game in which every goal is represented using a particular fragment <?TeX $L$?> of <?TeX $\mathrm{LTL}$?> : is it then always the case that the equilibria of the game can be characterised within <?TeX $L$?> ? We show that this is not true in general.

Some game-theoretic solution concepts such as the Shapley value and the Banzhaf index have recently gained popularity as measures of node centrality in networks. While this direction of research is promising, the computational problems that surround it are challenging and have largely been left open. To date there are only a few positive results in the literature, which show that some game-theoretic extensions of degree-, closeness- and betweenness-centrality measures are computable in polynomial time, i.e., without the need to enumerate the exponential number of all possible coalitions. In this article, we show that these results can be extended to a much larger class of centrality measures that are based on a family of solution concepts known as semivalues. The family of semivalues includes, among others, the Shapley value and the Banzhaf index. To this end, we present a generic framework for defining game-theoretic network centralities and prove that all centrality measures that can be expressed in this framework are computable in polynomial time. Using our framework, we present a number of new and polynomial-time computable game-theoretic centrality measures.

Game theory provides a well-established framework for the analysis of concurrent and multi-agent systems. The basic idea is that concurrent processes (agents) can be understood as corresponding to players in a game; plays represent the possible computation runs of the system; and strategies define the behaviour of agents. Typically, strategies are modelled as functions from sequences of system states to player actions. Analysing a system in such a way involves computing the set of (Nash) equilibria in the game. However, we show that, with respect to the above model of strategies---the standard model in the literature---bisimilarity does not preserve the existence of Nash equilibria. Thus, two concurrent games which are behaviourally equivalent from a semantic perspective, and which from a logical perspective satisfy the same temporal formulae, nevertheless have fundamentally different properties from a game theoretic perspective. In this paper we explore the issues raised by this discovery, and investigate three models of strategies with respect to which the existence of Nash equilibria is preserved under bisimilarity. We also use some of these models of strategies to provide new semantic foundations for logics for strategic reasoning, and investigate restricted scenarios where bisimilarity can be shown to preserve the existence of Nash equilibria with respect to the conventional model of strategies in the literature.

The existence of (Nash) equilibria with undesirable properties is a well-known problem in game theory, which has motivated much research directed at the possibility of mechanisms for modifying games in order to eliminate undesirable equilibria, or induce desirable ones. Taxation schemes are a well-known mechanism for modifying games in this way. In the multi-agent systems community, taxation mechanisms for incentive engineering have been studied in the context of Boolean games with costs. These are games in which each player assigns truth-values to a set of propositional variables she uniquely controls in pursuit of satisfying an individual propositional goal formula; different choices for the player are also associated with different costs. In such a game, each player prefers primarily to see the satisfaction of their goal, and secondarily, to minimise the cost of their choice, thereby giving rise to lexicographic preferences over goal-satisfaction and costs. Within this setting, where taxes operate on costs only, however, it may well happen that the elimination or introduction of equilibria can only be achieved at the cost of simultaneously introducing less desirable equilibria or eliminating more attractive ones. Although this framework has been studied extensively, the problem of precisely characterising the equilibria that may be induced or eliminated has remained open. In this paper we close this problem, giving a complete characterisation of those mechanisms that can induce a set of outcomes of the game to be exactly the set of Nash Equilibrium outcomes.

Rational verification is concerned with establishing whether a given temporal logic formula φ is satisfied in some or all equilibrium computations of a multi-agent system – that is, whether the system will exhibit the behaviour φ under the assumption that agents within the system act rationally in pursuit of their preferences. After motivating and introducing the framework of rational verification, we present formal models through which rational verification can be studied, and survey the complexity of key decision problems. We give an overview of a prototype software tool for rational verification, and conclude with a discussion and related work.

Solution concepts from cooperative game theory, such as the Shapley value or the Banzhaf index, have recently been advocated as interesting extensions of standard measures of node centrality in networks. While this direction of research is promising, the computation of game-theoretic centrality can be challenging. In an attempt to address the computational issues of game-theoretic network centrality, we present a generic framework for constructing game-theoretic network centralities. We prove that all extensions that can be expressed in this framework are computable in polynomial time. Using our framework, we present the first game-theoretic extensions of weighted and normalized degree centralities, impact factor centrality,distance-scaled and normalized betweenness centrality,and closeness and normalized closeness centralities.

This paper presents a range of expressiveness and complexity results for the specification, computation, and verification of Nash equilibria in multi-player non-zero-sum concurrent games in which players have goals expressed as temporal logic formulae. Our results are based on a novel approach to the characterisation of equilibria in such games: a semantic characterisation based on winning strategies and memoryful reasoning. This characterisation allows us to obtain a number of other results relating to the analysis of equilibrium properties in temporal logic. We show that, up to bisimilarity, reasoning about Nash equilibria in multi-player non-zero-sum concurrent games can be done in ATL^* and that constructing equilibrium strategy profiles in such games can be done in 2EXPTIME using finite-memory strategies. We also study two simpler cases, two-player games and sequential games, and show that the specification of equilibria in the latter setting can be obtained in a temporal logic that is weaker than ATL^*. Based on these results, we settle a few open problems, put forward new logical characterisations of equilibria, and provide improved answers and alternative solutions to a number of questions.

We study the problem of computing possible and necessary winners for partially specified weighted and unweighted tournaments. This problem arises naturally in elections with incompletely specified votes, partially completed sports competitions, and more generally in any scenario where the outcome of some pairwise comparisons is not yet fully known. We specifically consider a number of well-known solution concepts---including the uncovered set, Borda, ranked pairs, and maximin---and show that for most of them, possible and necessary winners can be identified in polynomial time. These positive algorithmic results stand in sharp contrast to earlier results concerning possible and necessary winners given partially specified preference profiles.

An important subclass of social choice functions, so-called majoritarian (or C1) functions, only take into account the pairwise majority relation between alternatives. In the absence of majority ties--e.g., when there is an odd number of agents with linear preferences--the majority relation is antisymmetric and complete and can thus conveniently be represented by a tournament. Tournaments have a rich mathematical theory and many formal results for majoritarian functions assume that the majority relation constitutes a tournament. Moreover, most majoritarian functions have only been defined for tournaments and allow for a variety of generalizations to unrestricted preference profiles, none of which can be seen as the unequivocal extension of the original function. In this paper, we argue that restricting attention to tournaments is justified by the existence of a conservative extension, which inherits most of the commonly considered properties from its underlying tournament solution.

Restricting the preferences of the agents by assuming that their utility functions linearly depend on a payment allows for the positive results of the Vickrey auction and the Vickrey-Clarke-Groves mechanism. These results, however, are limited to settings where there is some commonly desired commodity or numeraire--money, shells, beads, etcetera--which is commensurable with utility. We propose a generalization of the Vickrey auction that does not assume that the agents' preferences are quasilinear, but nevertheless retains some of the Vickrey auction's desirable properties. In this auction, a bid can be any alternative, rather than just a monetary offer. As a consequence, the auction is also applicable to situations where there is a fixed budget, or no numeraire is available at all (or it is undesirable to use payments for other reasons)--such as, for example, in the allocation of the task of contributing a module to an open-source project. We show that in two general settings, this qualitative Vickrey auction has a dominant-strategy equilibrium, invariably yields a weakly Pareto efficient outcome in this equilibrium, and is individually rational. In the first setting, the center has a linear preference order over a finite set of alternatives, and in the second setting, the bidders' preferences can be represented by continuous utility functions over a closed metric space of alternatives and the center's utility is equipeaked. The traditional Vickrey auction turns out to be a special case of the qualitative Vickrey auction in this second setting.

Social choice rules are often evaluated and compared by inquiring whether they satisfy certain desirable criteria such as the Condorcet criterion, which states that an alternative should always be chosen when more than half of the voters prefer it over any other alternative. Many of these criteria can be formulated in terms of choice sets that single out reasonable alternatives based on the preferences of the voters. In this paper, we consider choice sets whose definition merely relies on the pairwise majority relation. These sets include the Copeland set, the Smith set, the Schwartz set, von Neumann-Morgenstern stable sets, the Banks set, and the Slater set. We investigate the relationships between these sets and completely characterize their computational complexity, which allows us to obtain hardness results for entire classes of social choice rules.