DYNAMIC COOPERATIVE GAMES

Methods for comparison of coalition influence on games in characteristic function form and their interrelationships

Additional results are presented in this paper which further extend and modify the theory of cooperative games in characteristic function form so as to encompass situations where the values of the coalitions are random variables with given distribution functions. We reintroduce the prior nucleolus and provide a new characterization of it in terms of an optimal solution to a finite sequence of consistent and bounded NLP problems. From this new characterization it follows immediately that the prior nucleolus is unique for strictly monotone increasing distribution functions. We also extend the notions of objections and counterobjections to these games and construct and study the prior kernel and prior bargaining set 𝒫ℛ1(i). A new notion of solution for the second part of the play of these games is suggested.

The paper examines effects of long-term international integration in the field of electric power industry. An advantage of integration is provoked by the fact that the cost of introducing new generation capacities significantly exceeds the cost of new power lines and transmission of the energy from existing power station of another country. When countries form a coalition, the problem is to allocate the coalition’s surplus over its participants. It can be solved by notions of cooperative game theory. The present investigation is based on the real data on six countries of the Northeast Asian region: Russia, Mongolia, China, North Korea, South Korea and Japan. Electric power system is described by the ORIRES model. This model optimizes power generation, power flows, and the development of generation capacities and power lines of the system. We formulate corresponding cooperative game in characteristic function form, and specify the Core and the Shapley value. Effects of international integration are discussed.

This paper proposes a binary relation to compare coalition influence and a function which assigns a real number which expresses potential profitability that a coalition has to the coalition in the framework of games in characteristic function form. Some examples how the proposed binary relation and the function in games in characteristic function form are provided. The authors give some propositions which the proposed method and the function satisfy.

Centrality and power in social networks: a game theoretic approach

NPER II is an on-line PDP-11/45 computer program for studying bargaining and coalition formation processes within the framework of n-person games in characteristic function form. The extensive vocabulary of the program and its mode of operation are presented nontechnically, followed by an examination of a protocol from a sample four-person game. Improvements over a previous version of the program and applications to other experimental paradigms investigating coalition formation and bargaining are discussed in the last two sections of the paper.KeywordsCoalition FormationGrand CoalitionLinked ListCharacteristic Function FormCoalition Formation ProcessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Big Boss Games, Clan Games, and Information Market Games

The Negotiation Agreement Area (NAA) is a descriptive area theory for two-person bargaining games in characteristic function form. Laboratory experiments show that the NAA is significantly more successful in prediction than other theories. The experiments conducted in this study as well as previous experiments (Horowitz 1977) indicate that subjects tend to have a “behavioral threshold” at zero if the coalition shares can be negative. Respecting this fact, the NAA differentiates between games with positive and those with negative alternatives. Moreover, the paper investigates opening demands of negotiators and the negotiation process.

We consider the problem of cost allocation among users of a minimum cost spanning tree network. It is formulated as a cooperative game in characteristic function form, referred to as a minimum cost spanning tree (m.c.s.t.) game. We show that the core of a m.c.s.t. game is never empty. In fact, a point in the core can be read directly from any minimum cost spanning tree graph associated with the problem. For m.c.s.t. games with efficient coalition structures we define and construct m.c.s.t. games on the components of the structure. We show that the core and the nucleolus of the original game are the cartesian products of the cores and the nucleoli, respectively, of the induced games on the components of the efficient coalition structure.

A hybrid exact algorithm for complete set partitioning

This paper concerns cores of economies with asymmetric information. Alternative definitions of the information available to traders in coalitions and the cooperative games they generate are analyzed. An important technical result states that such NTU games in characteristic function form are well defined. Properties of various cores with asymmetric information are examined. Sufficient conditions on information sharing rules are provided for the induced games to be totally balanced or balanced, so that their cores are nonempty. Incentive compatibility issues are considered. Finally, a perspective on this research area is provided.

Games where players' payoffs are given by intervals, instead of scalars, provide a conceptually attractive framework to account for uncertainty and vagueness in decision-making processes. In a cooperative game, one needs to determine the characteristic function values for all possible coalitions. In this paper, we extend the classical $ \alpha $ and $ \beta $ characteristic functions (CFs), initially defined for scalar payoffs, to cooperative interval games. Both characteristic functions are based on a solution of zero-sum interval games with the coalition being the maximizer player and the anti-coalition the minimizer player. We propose an algorithm to define the interval values of a cooperative game in the form of $ \alpha $ and $ \beta $ CFs and illustrate its use on an example with three players. Further, we discuss some properties of cooperative interval games and calculate an interval Shapley value. As expected, different characteristic functions lead to different Shapley values.

A New Approximation Method for the Shapley Value Applied to the WTC 9/11 Attack

This paper proposes a new method for assigning distribution system (DS) losses to consumers and distributed generations in the framework of a liberalized energy market. The architecture of the proposed method is based on a synthesis of electrical network analytical theory and cooperative game theory (CGT). Weighted shapely value (WSV) is a solution idea that is regarded to be efficient in CGT for n-player games with characteristic function form. By retaining the notion of electrical networks, the axioms of game theory will be preserved while employing the WSV in loss allocation theory. To allocate system losses, the proposed method estimates the average marginal contribution of each participant to all conceivable coalitions and also determines the weight of the marginal contribution of each participant based on its power rating. Rather than a sequential branch-oriented approach, the proposed methodology is node-oriented. Additionally, a majority rule game is used to constrain the population of a coalition without jeopardizing the efficacy of the solutions to achieve a fair allocation of losses. The results of the proposed method are presented and contrasted with those obtained using the conventional pro-rata method and the well-known Shapley value solution concept in CGT.