The Cost of Stability in Coalitional Games

A key question in cooperative game theory is that of coalitional stability, usually captured by the notion of the \emph{core}--the set of outcomes such that no subgroup of players has an incentive to deviate. However, some coalitional games have empty cores, and any outcome in such a game is unstable. In this paper, we investigate the possibility of stabilizing a coalitional game by using external payments. We consider a scenario where an external party, which is interested in having the players work together, offers a supplemental payment to the grand coalition (or, more generally, a particular coalition structure). This payment is conditional on players not deviating from their coalition(s). The sum of this payment plus the actual gains of the coalition(s) may then be divided among the agents so as to promote stability. We define the \emph{cost of stability (CoS)} as the minimal external payment that stabilizes the game. We provide general bounds on the cost of stability in several classes of games, and explore its algorithmic properties. To develop a better intuition for the concepts we introduce, we provide a detailed algorithmic study of the cost of stability in weighted voting games, a simple but expressive class of games which can model decision-making in political bodies, and cooperation in multiagent settings. Finally, we extend our model and results to games with coalition structures.Comment: 20 pages; will be presented at SAGT'0

Taxation and stability in cooperative games

Cooperative games are a useful framework for modeling multi-agent behavior in environments where agents must collaborate in order to complete tasks. Having jointly completed a task and generated revenue, agents need to agree on some reasonable method of sharing their profits. One particularly appealing family of payoff divisions is the core, which consists of all coalitionally rational (or, stable) payoff divisions. Unfortunately, it is often the case that the core of a game is empty, i.e. there is no payoff scheme guaranteeing each group of agents a total payoff higher than what they can get on their own. As stability is a highly attractive property, there have been various methods of achieving it proposed in the literature. One natural way of stabilizing a game is via taxation, i.e. reducing the value of some coalitions in order to decrease their bargaining power. Existing taxation methods include the ε-core, the least-core and several others. However, taxing coalitions is in general undesirable: one would not wish to overly tamper with a given coalitional game, or overly tax the agents. Thus, in this work we study minimal taxation policies, i.e. those minimizing the amount of tax required in order to stabilize a given game. We show that games that minimize the total tax are to some extent a linear approximation of the original games, and explore their properties. We demonstrate connections between the minimal tax and the cost of stability, and characterize the types of games for which it is possible to obtain a tax-minimizing policy using variants of notion of the ε-core, as well as those for which it is possible to do so using reliability extensions. Copyright © 2013, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al. [ABEP11] analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10] for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, leaving the question about their exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time", and provide matching upper bounds for beneficial merging and, whenever the number of false identities is fixed, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely

Learning Cooperative Games

This paper explores a PAC (probably approximately correct) learning model in cooperative games. Specifically, we are given $m$ random samples of coalitions and their values, taken from some unknown cooperative game; can we predict the values of unseen coalitions? We study the PAC learnability of several well-known classes of cooperative games, such as network flow games, threshold task games, and induced subgraph games. We also establish a novel connection between PAC learnability and core stability: for games that are efficiently learnable, it is possible to find payoff divisions that are likely to be stable using a polynomial number of samples.Comment: accepted to IJCAI 201

Coalitional Games in MISO Interference Channels: Epsilon-Core and Coalition Structure Stable Set

The multiple-input single-output interference channel is considered. Each transmitter is assumed to know the channels between itself and all receivers perfectly and the receivers are assumed to treat interference as additive noise. In this setting, noncooperative transmission does not take into account the interference generated at other receivers which generally leads to inefficient performance of the links. To improve this situation, we study cooperation between the links using coalitional games. The players (links) in a coalition either perform zero forcing transmission or Wiener filter precoding to each other. The $\epsilon$-core is a solution concept for coalitional games which takes into account the overhead required in coalition deviation. We provide necessary and sufficient conditions for the strong and weak $\epsilon$-core of our coalitional game not to be empty with zero forcing transmission. Since, the $\epsilon$-core only considers the possibility of joint cooperation of all links, we study coalitional games in partition form in which several distinct coalitions can form. We propose a polynomial time distributed coalition formation algorithm based on coalition merging and prove that its solution lies in the coalition structure stable set of our coalition formation game. Simulation results reveal the cooperation gains for different coalition formation complexities and deviation overhead models.Comment: to appear in IEEE Transactions on Signal Processing, 14 pages, 14 figures, 3 table

Bounds for the Nakamura number

The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric quota games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending of invariants on simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts.Comment: 23 pages, 3 tables; a few more references adde

Solving Cooperative Reliability Games

Cooperative games model the allocation of profit from joint actions, following considerations such as stability and fairness. We propose the reliability extension of such games, where agents may fail to participate in the game. In the reliability extension, each agent only "survives" with a certain probability, and a coalition's value is the probability that its surviving members would be a winning coalition in the base game. We study prominent solution concepts in such games, showing how to approximate the Shapley value and how to compute the core in games with few agent types. We also show that applying the reliability extension may stabilize the game, making the core non-empty even when the base game has an empty core