Rational Verification in Iterated Electric Boolean Games

Electric boolean games are compact representations of games where the players have qualitative objectives described by LTL formulae and have limited resources. We study the complexity of several decision problems related to the analysis of rationality in electric boolean games with LTL objectives. In particular, we report that the problem of deciding whether a profile is a Nash equilibrium in an iterated electric boolean game is no harder than in iterated boolean games without resource bounds. We show that it is a PSPACE-complete problem. As a corollary, we obtain that both rational elimination and rational construction of Nash equilibria by a supervising authority are PSPACE-complete problems.Comment: In Proceedings SR 2016, arXiv:1607.0269

Characterising the manipulability of Boolean games

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

Electric Boolean games : redistribution schemes for resource-bounded agents

In Boolean games, agents uniquely control a set of propositional variables, and aim at achieving a goal formula whose realisation might depend on the choices the other agents make with respect to the variables they control. We consider the case in which assigning a value to propositional variables incurs a cost, and moreover, we assume agents to be restricted in their choice of assignments by an initial endowment: they can only make choices with a lower cost than this endowment. We then consider the possibility that endowments can be redistributed among agents. Different redistributions may lead to Nash equilibrium outcomes with very different properties, and so certain redistributions may be considered more attractive than others. In this context we study centralised redistribution schemes, where a system designer is allowed to redistribute the initial energy endowment among the agents in order to achieve desirable systemic properties. We also show how to extend this basic model to a dynamic variant in which an electric Boolean game takes place over a series of rounds

Efficiency and complexity of price competition among single-product vendors

Motivated by recent progress on pricing in the AI literature, we study marketplaces that contain multiple vendors offering identical or similar products and unit-demand buyers with different valuations on these vendors. The objective of each vendor is to set the price of its product to a fixed value so that its profit is maximized. The profit depends on the vendor's price itself and the total volume of buyers that find the particular price more attractive than the price of the vendor's competitors. We model the behavior of buyers and vendors as a two-stage full-information game and study a series of questions related to the existence, efficiency (price of anarchy) and computational complexity of equilibria in this game. To overcome situations where equilibria do not exist or exist but are highly inefficient, we consider the scenario where some of the vendors are subsidized in order to keep prices low and buyers highly satisfied

Hard and soft preparation sets in Boolean games

A fundamental problem in game theory is the possibility of reaching equi- librium outcomes with undesirable properties, e.g., inefficiency. The economics literature abounds with models that attempt to modify games in order to avoid such undesirable properties, for example through the use of subsidies and taxation, or by allowing players to undergo a bargaining phase before their decision. In this paper, we consider the effect of such transformations in Boolean games with costs, where players control propositional variables that they can set to true or false, and are primarily motivated to seek the sat- isfaction of some goal formula, while secondarily motivated to minimise the costs of their actions. We adopt (pure) preparation sets (prep sets) as our basic solution concept. A preparation set is a set of outcomes that contains for every player at least one best re- sponse to every outcome in the set. Prep sets are well-suited to the analysis of Boolean games, because we can naturally represent prep sets as propositional formulas, which in turn allows us to refer to prep formulas . The preference structure of Boolean games with costs makes it possible to distinguish between hard and soft prep sets. The hard prep sets of a game are sets of valuations that would be prep sets in that game no matter what the cost function of the game was. The properties defined by hard prep sets typically relate to goal-seeking behaviour, and as such these properties cannot be eliminated from games by, for example, taxation or subsidies. In contrast, soft prep sets can be eliminated by an appropriate system of incentives. Besides considering what can happen in a game by unrestricted manipulation of players’ cost function, we also investigate several mechanisms that allow groups of players to form coalitions and eliminate undesirable outcomes from the game, even when taxes or subsidies are not a possibility

Endogenous games with goals : side-payments among goal-directed agents

Boolean games have been developed as a paradigm for modelling societies of goal-directed agents. In boolean games agents exercise control over propositional variables and strive to achieve a goal formula whose realization might require the opponents’ cooperation. The presence of agents that are goal-directed makes it difficult for an external authority to be able to remove undesirable properties that are inconsistent with agents’ goals, as shown by recent contributions in the multi-agent literature. What this paper does is to analyse the problem of regulation of goal-direct agents from within the system, i.e., what happens when agents themselves are given the chance to negotiate the strategies to be played with one another. Concretely, we introduce endogenous games with goals, obtained coupling a general model of goal-directed agents (strategic games with goals) with a general model of pre-play negotiations (endogenous games) coming from game theory. Strategic games with goals are shown to have a direct correspondence with strategic games (Proposition 1) but, when side-payments are allowed in the pre-play phase, display a striking imbalance (Proposition 4). The effect of side-payments can be fully simulated by taxation mechanisms studied in the literature (Proposition 7), yet we show sufficient conditions under which outcomes can be rationally sustained without external intervention (Proposition 5). Also, integrating taxation mechanisms and side-payments, we are able to transform our starting models in such a way that outcomes that are theoretically sustainable thanks to a pre-play phase can be actually sustained even with limited resources (Proposition 8). Finally, we show how an external authority incentivising a group of agents can be studied as a special agent of an appropriately extended endogenous game with goals (Proposition 11)