Hard and soft equilibria in Boolean games

A fundamental problem in game theory is the possibility of reaching equilibrium outcomes with undesirable properties, e.g., inefficiency. The economics literature abounds with models that attempt to modify games in order to eliminate such undesirable equilibria, for example through the use of subsidies and taxation, or by allowing players to undergo a preplay negotiation phase. In this paper, we consider the effect of such transformations in Boolean games with costs, where players are primarily motivated to seek the satisfaction of some goal, and are secondarily motivated to minimise the costs of their actions. The preference structure of these games allows us to distinguish between hard and soft equilibria, where hard equilibria arise from goal-seeking behaviour, and cannot be eliminated from games by, e.g., taxes or subsidies, while soft equilibria are those that arise from the desire of agents to minimise costs. We investigate several mechanisms which allow groups of players to form coalitions and eliminate undesirable equilibria from the game, even when taxes or subsidies are not a possibility

Fostering Cooperation in Structured Populations Through Local and Global Interference Strategies

We study the situation of an exogenous decision-maker aiming to encourage a population of autonomous, self-regarding agents to follow a desired behaviour at a minimal cost. The primary goal is therefore to reach an efficient trade-off between pushing the agents to achieve the desired configuration while minimising the total investment. To this end, we test several interference paradigms resorting to simulations of agents facing a cooperative dilemma in a spatial arrangement. We systematically analyse and compare interference strategies rewarding local or global behavioural patterns. Our results show that taking into account the neighbourhood's local properties, such as its level of cooperativeness, can lead to a significant improvement regarding cost efficiency while guaranteeing high levels of cooperation. As such, we argue that local interference strategies are more efficient than global ones in fostering cooperation in a population of autonomous agents.</p

Cost-effective external interference for promoting the evolution of cooperation.

The problem of promoting the evolution of cooperative behaviour within populations of self-regarding individuals has been intensively investigated across diverse fields of behavioural, social and computational sciences. In most studies, cooperation is assumed to emerge from the combined actions of participating individuals within the populations, without taking into account the possibility of external interference and how it can be performed in a cost-efficient way. Here, we bridge this gap by studying a cost-efficient interference model based on evolutionary game theory, where an exogenous decision-maker aims to ensure high levels of cooperation from a population of individuals playing the one-shot Prisoner's Dilemma, at a minimal cost. We derive analytical conditions for which an interference scheme or strategy can guarantee a given level of cooperation while at the same time minimising the total cost of investment (for rewarding cooperative behaviours), and show that the results are highly sensitive to the intensity of selection by interference. Interestingly, we show that a simple class of interference that makes investment decisions based on the population composition can lead to significantly more cost-efficient outcomes than standard institutional incentive strategies, especially in the case of weak selection.</p

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

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