Complexity of determining nonemptiness of the core

In many multiagent scenarios, agents distribute resources, such as time or energy, among several tasks. Having completed their tasks and generated profits, task payoffs must be divided among the agents in some reasonable manner. Cooperative games with overlapping coalitions (OCF games) are a recent framework proposed by , generalizing classic cooperative games to the case where agents may belong to more than one coalition. Having formed overlapping coalitions and divided profits, some agents may feel dissatisfied with their share of the profits, and would like to deviate from the given outcome. However, deviation in OCF games is a complicated matter: agents may decide to withdraw only some of their weight from some of the coalitions they belong to; that is, even after deviation, it is possible that agents will still be involved in tasks with non-deviators. This means that the desirability of a deviation, and the stability of formed coalitions, is to a great extent determined by the reaction of non-deviators. In this work, we explore algorithmic aspects of OCF games, focusing on the core in OCF games. We study the problem of deciding if the core of an OCF game is not empty, and whether a core payoff division can be found in polynomial time; moreover, we identify conditions that ensure that the problem admits polynomial time algorithms. Finally, we introduce and study a natural class of OCF games, the Linear Bottleneck Games. Interestingly, we show that such games always have a non-empty core, even assuming a highly lenient reaction to deviations.

Eumas, 2005

It is self-evident that in numerous Multiagent settings, selfish agents stand to benefit from cooperating by forming coalitions. Nevertheless, negotiating a stable distribution of the payoff among agents may prove challenging. The issue of coalition formation has been investigated extensively in the field of cooperative n-person game theory, but until recently little attention has been given to the complications that arise when the players are software agents. The bounded rationality of such agents has motivated researchers to study the computational complexity of the aforementioned problems.

2012

Much of the literature on multi-agent coalition formation has focused on Characteristic Function Games, where the effectiveness of a coalition is not affected by how the other agents are arranged in the system. In contrast, very little attention has been given to the more general class of Partition Function Games, where the emphasis is on how the formation of one coalition could influence the performance of other co-existing coalitions in the system. However, these intercoalitional dependencies, called externalities from coalition formation, play a crucial role in many real-world multi-agent applications where agents have either conflicting or overlapping goals.

Autonomous Agents and Multi-Agent Systems, 2018

The optimal coalition structure determination problem is in general computationally hard. In this article, we identify some problem instances for which the space of possible coalition structures has a certain form and constructively prove that the problem is polynomial time solvable. Specifically, we consider games with an ordering over the players and introduce a distance metric for measuring the distance between any two structures. In terms of this metric, we define the property of monotonicity, meaning that coalition structures closer to the optimal, as measured by the metric, have higher value than those further away. Similarly, quasi-monotonicity means that part of the space of coalition structures is monotonic, while part of it is non-monotonic. (Quasi)-monotonicity is a property that can be satisfied by coalition games in characteristic function form and also those in partition function form. For a setting with a monotonic value function and a known player ordering, we prove that the optimal coalition structure determination problem is polynomial time solvable and devise such an algorithm using a greedy approach. We extend this algorithm to quasi-monotonic value functions and demonstrate how its time complexity improves from exponential to polynomial as the degree of monotonicity of the value function increases. We go further and consider a setting in which the value function is monotonic and an ordering over the players is known to exist but ordering itself is unknown. For this setting too, we prove that the coalition structure determination problem is polynomial time solvable and devise such an algorithm.

Logic Journal of IGPL, 2009

We introduce a reciprocity criterion for coalition formation among goal-directed agents, which we call the indecomposable do-ut-des property. It refines an older reciprocity property, called the do-ut-des or give-to-get property by considering the fact that agents prefer to form coalitions whose components cannot be formed independently. A formal description of this property is provided as well as an analysis of algorithms and their complexity. We provide an algorithm to decide whether a coalition has the desired property, and we show that the problem to verify whether a single coalition satisfies the property is tractable. Moreover, we provide an algorithm to search all the sub-coalitions of a given coalition satisfying the new property. Even if this problem is not computationally tractable, we show that in several cases, also the complexity of this problem may decrease considerably.

Synthesis Lectures on Artificial Intelligence and Machine Learning, 2011

Cooperative game theory is a branch of (micro-)economics that studies the behavior of self-interested agents in strategic settings where binding agreements among agents are possible. Our aim in this book is to present a survey of work on the computational aspects of cooperative game theory. We begin by formally defining transferable utility games in characteristic function form, and introducing key solution concepts such as the core and the Shapley value. We then discuss two major issues that arise when considering such games from a computational perspective: identifying compact representations for games, and the closely related problem of efficiently computing solution concepts for games. We survey several formalisms for cooperative games that have been proposed in the literature, including, for example, cooperative games defined on networks, as well as general compact representation schemes such as MC-nets and skill games. As a detailed case study, we consider weighted voting games: a widely-used and practically important class of cooperative games that inherently have a natural compact representation. We investigate the complexity of solution concepts for such games, and generalizations of them. We briefly discuss games with non-transferable utility and partition function games. We then overview algorithms for identifying welfare-maximizing coalition structures and methods used by rational agents to form coalitions (even under uncertainty), including bargaining algorithms. We conclude by considering some developing topics, applications, and future research directions.

Intelligent Agents …, 2002

In the last few years the use of coalition formation algorithms in multi-agent systems has been proposed as a possible way of modelling autonomous agent cooperation. Game theory provides different concepts for the stability of solutions in cooperative games, regarding the fairness of the resultant payment configuration. One of these is the core. In this paper we present an analysis -based upon game theory-for a class of task-oriented problems arising from some Internet transactions. A procedure that guarantees an optimum task allocation is defined, and a new payoff division model is proposed for the corresponding value. We prove that the proposed payoff division lies inside the core. The whole computation (of the optimal value and the payoffs) has a polynomial complexity in the number of agents.

Artificial Intelligence, 1999

Coalition formation is a key topic in multiagent systems. One may prefer a coalition structure that maximizes the sum of the values of the coalitions, but often the number of coalition structures is too large to allow exhaustive search for the optimal one. Furthermore, finding the optimal coalition structure is -complete. But then, can the coalition structure found via a partial search be guaranteed to be within a bound from optimum?We show that none of the previous coalition structure generation algorithms can establish any bound because they search fewer nodes than a threshold that we show necessary for establishing a bound. We present an algorithm that establishes a tight bound within this minimal amount of search, and show that any other algorithm would have to search strictly more. The fraction of nodes needed to be searched approaches zero as the number of agents grows.If additional time remains, our anytime algorithm searches further, and establishes a progressively lower tight bound. Surprisingly, just searching one more node drops the bound in half. As desired, our algorithm lowers the bound rapidly early on, and exhibits diminishing returns to computation. It also significantly outperforms its obvious contenders. Finally, we show how to distribute the desired search across self-interested manipulative agents.

In the usual models of cooperative game theory, the outcome of a coalition formation process is either the grand coalition or a coalition structure that consists of disjoint coalitions. However, in many domains where coalitions are associated with tasks, an agent may be involved in executing more than one task, and thus may distribute his resources among several coalitions. To tackle such scenarios, we introduce a model for cooperative games with overlapping coalitions-or overlapping coalition formation (OCF) games. We then explore the issue of stability in this setting. In particular, we introduce a notion of the core, which generalizes the corresponding notion in the traditional (non-overlapping) scenario. Then, under some quite general conditions, we characterize the elements of the core, and show that any element of the core maximizes the social welfare. We also introduce a concept of balancedness for overlapping coalitional games, and use it to characterize coalition structures that can be extended to elements of the core. Finally, we generalize the notion of convexity to our setting, and show that under some natural assumptions convex games have a non-empty core. Moreover, we introduce two alternative notions of stability in OCF that allow a wider range of deviations, and explore the relationships among the corresponding definitions of the core, as well as the classic (non-overlapping) core and the Aubin core. We illustrate the general properties of the three cores, and also study them from a computational perspective, thus obtaining additional insights into their fundamental structure.

Artificial Intelligence, 2011

Coalitional games are mathematical models suited to analyze scenarios where players can collaborate by forming coalitions in order to obtain higher worths than by acting in isolation. A fundamental problem for coalitional games is to single out the most desirable outcomes in terms of appropriate notions of worth distributions, which are usually called solution concepts. Motivated by the fact that decisions taken by realistic players cannot involve unbounded resources, recent computer science literature reconsidered the definition of such concepts by advocating the relevance of assessing the amount of resources needed for their computation in terms of their computational complexity. By following this avenue of research, the paper provides a complete picture of the complexity issues arising with three prominent solution concepts for coalitional games with transferable utility, namely, the core, the kernel, and the bargaining set, whenever the game worth-function is represented in some reasonable compact form (otherwise, if the worths of all coalitions are explicitly listed, the input sizes are so large that complexity problems are-artificially-trivial). The starting investigation point is the setting of graph games, about which various open questions were stated in the literature. The paper gives an answer to these questions, and in addition provides new insights on the setting, by characterizing the computational complexity of the three concepts in some relevant generalizations and specializations.