Overlapping Coalition Formation

Cooperative Games in Graph Structure

By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition being a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. In this paper we consider non-transferable utility games in which for any coalition the set of payoffs depends on a permutation or ordering upon any partition of the coalition into subcoalitions. We call such a game a cooperative game in permutational structure or shortly permutational game. Doing so we extend the scope of the standard cooperative game theory in dealing with economic or political problems. Next we define the concept of core for such games. By introducing balancedness for ordered partitions of coalitions, we prove the nonemptiness of the core of a balanced non-transferable utility permutational game. Moreover we show that the core of a permutational game coincides with the core of an induced game in coalitional structure, but that balancedness of the permutational game need not imply balancedness of the corresponding coalitional game. This leads to a weakening of the conditions for the existence of a nonempty core of a game in coalitional structure, induced by a game in permutational structure. Furthermore, we refine the concept of core for the class of permutational games. We call this refinement the balanced-core of the game and show that the balanced-core of a balanced permutational game is a nonempty subset of the core. The proof of the nonemptiness of the core of a permutational game is based on a new intersection theorem on the unit simplex, which generalizes the well-known intersection theorem of Shapley.

Overlapping Coalition Formation (OCF) games [3, 4] are cooperative games where the players can simultaneously participate in several coalitions. Capturing the notion of stability in OCF games is a difficult task: a player may deviate by abandoning some, but not all of the coalitions he is involved in, and the crucial question is whether he then gets to keep his payoff from the unaffected coalitions. In [4] the authors introduce three stability concepts for OCF games---the conservative, refined, and optimistic core---that are based on different answers to this question. In this paper, we propose a unified framework for the study of stability in the OCF setting, which encompasses the concepts considered in [4] as well as a wide variety of alternative stability concepts. Our approach is based on the notion of an arbitrator, which can be thought of as an external party that determines payoff to deviators. We give a complete characterization of outcomes that are stable under arbitration. In particular, our results provide a criterion for the outcome to be in the refined or optimistic core, thus complementing the results in [4] for the conservative core, and answering questions left open in [4]. We also introduce a notion of the nucleolus for arbitrated OCF games, and argue that it is non--empty. Finally, we extend the definition of the Shapley value [12] to the OCF setting, and provide an axiomatic characterization for it.

This paper develops two models to investigate the impact of a tariff on logistics strategies in an international e-commerce trade, where the methods of game theory and agent-based model are utilized. We consider a foreign supplier and a domestic e-tailer. There are two logistics strategies: the supplier undertakes the logistics and the e-tailer shoulders the logistics. First, two strategies are compared under the game theory model. Then, we capture the uncertainties of market demand and tariff. An agent-based model is adopted to further study how the tariff affects the supply chain’s decisions and strategies. Some meaningful results are obtained. (i) The supplier should provide the logistics service when market demand and tariff are certain. (ii) Under demand uncertainty, the supplier and the e-tailer can reach an agreement on logistics strategies when the tariff is small. (iii) When tariff uncertainty is moderate, two players both abandon the logistics intention..

The thesis focuses on cooperative games with transferable utility and incorporates two topics: Solutions for TU-games with restricted coalition formation and the stability of the grand coalition. <br/><br/>Chapters 3 is devoted to a new solution for cooperative games with coalition structures, called the α-egalitarian Owen value, and this coalitional value is characterized by three approaches. Firstly, we provide two axiomatizations by introducing the α-indemnificatory null player axiom, and the (intra) coalitional quasi-balanced contributions axiom. Secondly, we characterize the coalitional value by introducing an α-guarantee potential function. Finally, the coalitional value is implemented by a punishment-reward bidding mechanism. <br/><br/>In Chapter 4, we continue to work with TU-games restricted by coalition structures and propose a coalitional value called the two-step Shapley-solidarity value. A procedural interpretation is provided for this coalitional value, and we introduce a new axiom called the coalitional A-null player axiom to axiomatize the value based on additivity. Moreover, two other axiomatizations on the basis of quasi-balanced contributions for the grand coalition are also provided. <br/><br/>In Chapter 5, we focus on cooperative games with communication structures and provide efficient extensions of the Myerson value (Myerson, 1977). The idea lies in introducing the Shapley payoffs of the underlying game as players' claims to derive a graph-induced bankruptcy problem. Then, two efficient extensions of the Myerson value are achieved through bankruptcy rules, including the CEA rule and the CEL rule (Aumann &amp; Maschler, 1985).<br/>Moreover, corresponding axiomatizations are also provided.<br/><br/>Chapter 6 proceeds with studying the stability of the grand coalition for cost TU-games by addressing an optimization problem to maximize the total shareable cost over what we called the almost core. We analyze the computational complexity of this optimization problem, in relation to the computational complexity of related problems for the core. In particular, we consider a special class of games, i.e., the minimum cost spanning tree games. We show that maximizing the total shareable costs over the (non-negative) almost core is NP-hard for mcst games, and we provide a tight 2-approximation algorithm for this almost core optimization problem with the additional non-negative constraint. <br/>

Discrete time spatial models arising in genetics, evolutionary game theory, and branching processes.

Farsightedness and Cautiousness in Coalition Formation

Chapter 37 Coalition structures

Overlapping coalition formation in game theory: A state-of-the-art review

Based on cooperative games with coalition structure, Owen introduced the modified Banzhaf value as an extension of Banzhaf value. Spired by the conclusion of interaction phenomena among players in cooperative games, In this paper, a concept of derivative to measure interaction for games with coalition structure is established. Furthermore, the Banzhaf index of interation among players in games with coalition structure is proposed. This index can be viewed as the extension of modified Banzhaf value. Some properties of the indices are researched as well.

One of the most difficult and major problems of game theory consists in understanding how players may choose to organize themselves in a game in order to get joint-maximizing profits. The problem raises in fact two different questions: which organizations are likely to emerge from a game (first problem) and how are members of these organizations going to share its gains (second problem)? Hence, if these organizations are coalitions of players, a solution to the game shall specify those coalitions that are likely to form (ex post stable coalitions and/or coalitions that could form during the process at an intermediary step) and the way players, in each coalition, shall share common utility they get through their coalition. In Theory of Games and Economic Behavior (TGEB) Von Neumann and Morgenstern proposed developing a theory of n-person games in the framework of the characteristic function model in order to solve the second problem. Later, Shapley (1953), Aumann and Maschler (1964), and many other authors have responded to the question by proposing various solution concepts: respectively, the Shapley value, the bargaining set, the core, the kernel, the nucleolus, etc. (Although the core is a major solution concept in cooperative games, we will not tackle it here, except in comparing it to bargaining sets, because we want to focus our attention on coalition structure. We will only give a brief definition and speak of the “core of a coalition structure.” We refer to Herve Moulin’s chapter in this book for a presentation of the core.)In cooperative game theory it seems hard to disentangle the two problems: the way players share the worth of the coalition depends on which coalitions are likely to emerge from the game. It is this link between the two problems that research on coalition structure games tries to elucidate, seeking a better understanding of how a given coalition structure affects each player’s individual payoff. Once this question is answered (and there are many ways to answer it), one can face the first problem (how players choose to organize themselves) from different viewpoints. Either one links explicitly the two problems considering that the “sharing” game within each coalition and the game between coalitions that determines stable coalitions should havecommon features,1 or one separates them. A way of avoiding arbitrary or ad hoc aspects of a specific choice of a solution concept may consist in introducing first a non-cooperative step into the game, which describes the whole set of possible links among players. Starting from a graph-theoretic extension of the Shapley value, Aumann and Myerson (1988) propose for instance justifying the existence of a cooperation structure in a cooperative game by a non-cooperative “linking” game, the issue of which is a subgame perfect Nash equilibrium.

An evolutionary game theory model of binary opinion formation

Throughout this study, an adaptive competitive second-price pay-to-bid sealed auction game is presented as a solution to the fairness problem of spectrum sharing among a number of users in a cognitive radio environment. First, a comparison among three main spectrum-sharing game models; optimal, cooperative and competitive game models is discussed to highlight the advantages and disadvantages of each model. The results prove that cooperative games aim to achieve Nash equilibrium between primary and secondary players and provide better revenue to them. Also, it proves that cooperative games are best when the number of secondary users changes dynamically, but only when the number is low. As in practical situations, the number of secondary users might increase dramatically and cooperative games will lose their powerful advantage once that happens. As a result, the proposed mechanism creates a competition between the bidders and offers better revenue to all players in terms of fairness. The proposed model ensures that users with better channel quality, higher traffic priority and fairer bids will get a better chance to share the offered spectrum. It is shown by numerical results that the proposed mechanism could reach the maximum total profit for all users with better fairness.

In many respects, water resources allocation problems have fuzzy characteristics owing to uncertainty and imprecision not only as numerical data but also as linguistic data, in addition to the political nature of resource allocation. All classical techniques require numerical data and cannot treat linguistic data, but fuzzy inference systems (FIS) can deal with both types of data. Hence, it is suitable to reflect the dynamic nature of country benefits, reflective variations in the main drainage and subdrainage basin flows, and various system parameters in search of a coalition among a few riparian countries. The size of core, which is generated from a series of linear programming (LP) optimization models and game theory concepts, shows the potential extra benefit advantage of being in grand coalition to all parties, but it does not provide a sufficient condition to assemble a robust coalition unless an agreeable allocation scheme or principle is provided. In this respect, there is a fuzzy link between being in the grand coalition and percentage share received from a core. Hence, a proper FIS is proposed, and the necessary steps are developed with the Shapley value and the core as game theory methods, which provide a methodological base in the evaluation of possible allocation schemes among the parties. The application of the methodology is presented to search for coalition possibilities among the riparian countries: Turkey, Syria, and Iraq. Basic results are generated over the application of the Inter‐Temporal Euphrates and Tigris River Basin Model, which is an optimization model for the allocation of scarce water resources considering agricultural and urban uses, energy generation, and conveyance costs in the Euphrates‐Tigris basin.

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.