We introduce and study a computational version of the principal-agent problem -- a classic problem in Economics that arises when a principal desires to contract an agent to carry out some task, but has incomplete information about the agent or their subsequent actions. The key challenge in this setting is for the principal to design a contract for the agent such that the agent's preferences are then aligned with those of the principal. We study this problem using a variation of Boolean games, where multiple players each choose valuations for Boolean variables under their control, seeking the satisfaction of a personal goal, given as a Boolean logic formula. In our setting, the principal can only observe some subset of these variables, and the principal chooses a contract which rewards players on the basis of the assignments they make for the variables that are observable to the principal. The principal's challenge is to design a contract so that, firstly, the principal's goal is achieved in some or all Nash equilibrium choices, and secondly, that the principal is able to verify that their goal is satisfied. In this paper, we formally define this problem and completely characterise the computational complexity of the most relevant decision problems associated with it.

Boolean games are an expressive and natural formalism through which to investigate problems of strategic interaction in multiagent systems. Although they have been widely studied, almost all previous work on Nash equilibria in Boolean games has focused on the restricted setting of pure strategies. This is a shortcoming as finite games are guaranteed to have at least one equilibrium in mixed strategies, but many simple games fail to have pure strategy equilibria at all. We address this by showing that a natural decision problem about mixed equilibria: determining whether a Boolean game has a mixed strategy equilibrium that guarantees every player a given payoff, is NEXP-hard. Accordingly, the epsilon variety of the problem is NEXP-complete. The proof can be adapted to show coNEXP-hardness of a similar question: whether all Nash equilibria of a Boolean game guarantee every player at least the given payoff.

Boolean games are a game-theoretic framework in which propositional logic is used to describe agents' goals. In this paper we investigate how agents in Boolean games can reach an efficient and fair outcome through a simple negotiation protocol. We are particularly interested in settings where agents only have incomplete knowledge about the preferences of others. After explaining how generalized possibilistic logic can be used to compactly encode such knowledge, we analyze how a lack of knowledge affects the agreement outcome. In particular, we show how knowledgeable agents can obtain a more desirable outcome than others.

We introduce Boolean Observation Games, a subclass of multi-player finite strategic games with incomplete information and qualitative objectives. In Boolean observation games, each player is associated with a finite set of propositional variables of which only it can observe the value, and it controls whether and to whom it can reveal that value. It does not control the given, fixed, value of variables. Boolean observation games are a generalization of Boolean games, a well-studied subclass of strategic games but with complete information, and wherein each player controls the value of its variables. In Boolean observation games player goals describe multi-agent knowledge of variables. As in classical strategic games, players choose their strategies simultaneously and therefore observation games capture aspects of both imperfect and incomplete information. They require reasoning about sets of outcomes given sets of indistinguishable valuations of variables. What a Nash equilibrium is, depends on an outcome relation between such sets. We present various outcome relations, including a qualitative variant of ex-post equilibrium. We identify conditions under which, given an outcome relation, Nash equilibria are guaranteed to exist. We also study the complexity of checking for the existence of Nash equilibria and of verifying if a strategy profile is a Nash equilibrium. We further study the subclass of Boolean observation games with `knowing whether' goal formulas, for which the satisfaction does not depend on the value of variables. We show that each such Boolean observation game corresponds to a Boolean game and vice versa, by a different correspondence, and that both correspondences are precise in terms of existence of Nash equilibria.

We study hedonic games with dichotomous preferences. Hedonic games are cooperative games in which players desire to form coalitions, but only care about the makeup of the coalitions of which they are members; they are indifferent about the makeup of other coalitions. The assumption of dichotomous preferences means that, additionally, each player's preference relation partitions the set of coalitions of which that player is a member into just two equivalence classes: satisfactory and unsatisfactory. A player is indifferent between satisfactory coalitions, and is indifferent between unsatisfactory coalitions, but strictly prefers any satisfactory coalition over any unsatisfactory coalition. We develop a succinct representation for such games, in which each player's preference relation is represented by a propositional formula. We show how solution concepts for hedonic games with dichotomous preferences are characterised by propositional formulas.

Boolean games are a game-theoretic framework in which propositional logic is used to describe agents’ goals. In this paper we investigate how agents in Boolean games can reach an efficient and fair outcome through a simple negotiation protocol. We are particularly interested in settings where agents only have incomplete knowledge about the preferences of others. After explaining how generalized possibilistic logic can be used to compactly encode such knowledge, we analyze how a lack of knowledge affects the agreement outcome. In particular, we show how knowledgeable agents can obtain a more desirable outcome than others.

Two incentive mechanisms for Boolean games were proposed recently - taxation schemes and side payments. Both mechanisms have been shown to be able to secure a pure Nash equilibrium (PNE) for Boolean games. A complete characterization of outcomes that can be transformed to PNEs is given for each of the two incentive mechanisms. Side payments are proved to be a weaker mechanism in the sense that the outcomes that they can transform to PNEs are a subset of those transformable by taxation. A family of social-network-based Boolean games, which demonstrates the differences between the two mechanisms for securing a PNE, is presented. A distributed search algorithm for finding the side payments needed for securing a PNE is proposed. An empirical evaluation demonstrates the properties of the two mechanisms on the family of social-network-based Boolean games.

Boolean games are a framework for reasoning about the rational behaviour of agents, whose goals are formalized using propositional formulas. They offer an attractive alternative to normal-form games, because they allow for a more intuitive and more compact encoding. Unfortunately, however, there is currently no general, tailor-made method available to compute the equilibria of Boolean games. In this paper, we introduce a method for finding the pure Nash equilibria based on disjunctive answer set programming. Our method is furthermore capable of finding the core elements and the Pareto optimal equilibria, and can easily be modified to support other forms of optimality, thanks to the declarative nature of disjunctive answer set programming. Experimental results clearly demonstrate the effectiveness of the proposed method.

Boolean games are an expressive and natural formalism through which to investigate problems of strategic interaction in multiagent systems. Although they have been widely studied, almost all previous work on Nash equilibria in Boolean games has focused on the restricted setting of pure strategies. This is a shortcoming as finite games are guaranteed to have at least one equilibrium in mixed strategies, but many simple games fail to have pure strategy equilibria at all. We address this by showing that a natural decision problem about mixed equilibria: determining whether a Boolean game has a mixed strategy equilibrium that guarantees every player a given payoff, is NEXP-hard. Accordingly, the epsilon variety of the problem is NEXP-complete. The proof can be adapted to show coNEXP-hardness of a similar question: whether all Nash equilibria of a Boolean game guarantee every player at least the given payoff.

We investigate the problem of influencing the preferences of players within a Boolean game so that, if all players act rationally, certain desirable outcomes will result. The way in which we influence preferences is by overlaying games with taxation schemes. In a Boolean game, each player has unique control of a set of Boolean variables, and the choices available to the player correspond to the possible assignments that may be made to these variables. Each player also has a goal, represented by a Boolean formula, that they desire to see satisfied. Whether or not a player’s goal is satisfied will depend both on their own choices and on the choices of others, which gives Boolean games their strategic charac- ter. We extend this basic framework by introducing an external principal who is able to levy a taxation scheme on the game, which imposes a cost on every possible action that a player can choose. By designing a taxation scheme appropriately, it is possible to perturb the preferences of the players, so that they are incentivised to choose some equilibrium that would not otherwise be chosen. After motivating and formally presenting our model, we explore some issues surrounding it, including the complexity of finding a taxation scheme that implements some socially desirable outcome, and then discuss desirable properties of taxation schemes.