Monotonicity and dummy free property for multi-choice cooperative games

Given a coalition of ann-person cooperative game in characteristic function form, we can associate a zero-one vector whose non-zero coordinates identify the players in the given coalition. The cooperative game with this identification is just a map on such vectors. By allowing each coordinate to take finitely many values we can define multi-choice cooperative games. In such multi-choice games we can also define Shapley value axiomatically. We show that this multi-choice Shapley value is dummy free of actions, dummy free of players, non-decreasing for non-decreasing multi-choice games, and strictly increasing for strictly increasing cooperative games. Some of these properties are closely related to some properties of independent exponentially distributed random variables. An advantage of multi-choice formulation is that it allows to model strategic behavior of players within the context of cooperation.Partially funded by the NSF grant DMS-9024408     

The multichoice consistent value

We consider multichoice NTU games, i.e., cooperative NTU games in which players can participate in the game with several levels of activity. For these games, we define and characterize axiomatically the multichoice consistent value, which is a generalization of the consistent NTU value for NTU games and of the multichoice value for multichoice TU games. Moreover, we show that this value coincides with the consistent NTU value of a replicated NTU game and we provide a probabilistic interpretation. Received: May 1998/Final version: January 2000     

A polynomial time algorithm for computing the nucleolus for a class of disjunctive games with a permission structure

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many such allocation problems there is some hierarchical ordering of the players. In this paper we consider a class of games with a permission structure describing situations in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. The corresponding restricted game takes account of the limited cooperation possibilities by assigning to every coalition the worth of its largest feasible subset. In this paper we provide a polynomial time algorithm for computing the nucleolus of the restricted games corresponding to a class of games with a permission structure which economic applications include auction games, dual airport games, dual polluted river games and information market games.     

Condorcet winner configurations of linear networks

We consider the problem of locating a fixed number of facilities along a line to serve n players. We model this problem as a cooperative game and assume that any locational configuration can be eventually disrupted through a strict majority of players voting for an alternative configuration. A solution of such a voting location problem is called a Condorcet winner configuration. In this article, we state three necessary and one sufficient condition for a configuration to be a Condorcet winner. Consequently, we propose a fast algorithm which enables us to verify whether a given configuration is a Condorcet winner, and can be efficiently used also for computing the (potentially empty) set of all Condorcet winner configurations.     

Axiomatizations of Banzhaf permission values for games with a permission structure

In games with a permission structure it is assumed that players in a cooperative transferable utility game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. We provide axiomatic characterizations of Banzhaf permission values being solutions that are obtained by applying the Banzhaf value to modified TU-games. In these characterizations we use power- and player split neutrality properties. These properties state that splitting a player’s authority and/or contribution over two players does not change the sum of their payoffs.     

Cooperative Markov decision processes: time consistency, greedy players satisfaction, and cooperation maintenance

We deal with multi-agent Markov decision processes (MDPs) in which cooperation among players is allowed. We find a cooperative payoff distribution procedure (MDP-CPDP) that distributes in the course of the game the payoff that players would earn in the long run game. We show under which conditions such a MDP-CPDP fulfills a time consistency property, contents greedy players, and strengthen the coalition cohesiveness throughout the game. Finally we refine the concept of Core for Cooperative MDPs.     

An algorithm for computing the nucleolus of disjunctive non-negative additive games with an acyclic permission structure

A cooperative game with a permission structure describes a situation in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. In this paper we consider non-negative additive games with an acyclic permission structure. For such a game we provide a polynomial time algorithm for computing the nucleolus of the induced restricted game. The algorithm is applied to a market situation where sellers can sell objects to buyers through a directed network of intermediaries.     

On the rate of taxation in a cooperative bin packing game

We investigate a cooperative game with two types of players envolved: Every player of the first type owns a unit size bin, and every player of the second type owns an item of size at most one. The value of a coalition of players is defined to be the maximum overall size of packed items over all packings of the items owned by the coalition into the bins owned by the coalition.We prove that for=1/3 this cooperative bin packing game is-balanced in the taxation model of Faigle and Kern (1993).This research was supported by the Christian Doppler Laboratorium für Diskrete Optimierung.     

Convexity properties for interior operator games

Interior operator games arose by abstracting some properties of several types of cooperative games (for instance: peer group games, big boss games, clan games and information market games). This reason allow us to focus on different problems in the same way. We introduced these games in Bilbao et al. (Ann. Oper. Res. 137:141–160, 2005) by a set system with structure of antimatroid, that determines the feasible coalitions, and a non-negative vector, that represents a payoff distribution over the players. These games, in general, are not convex games. The main goal of this paper is to study under which conditions an interior operator game verifies other convexity properties: 1-convexity, k-convexity (k≥2 ) or semiconvexity. But, we will study these properties over structures more general than antimatroids: the interior operator structures. In every case, several characterizations in terms of the gap function and the initial vector are obtained. We also find the family of interior operator structures (particularly antimatroids) where every interior operator game satisfies one of these properties.     

On computational complexity of membership test in flow games and linear production games

Let Γ≡(N,v) be a cooperative game with the player set N and characteristic function v: 2^N→R. An imputation of the game is in the core if no subset of players could gain advantage by splitting from the grand coalition of all players. It is well known that, for the flow game (and equivalently, for the linear production game), the core is always non-empty and a solution in the core can be found in polynomial time. In this paper, we show that, given an imputation x, it is NP-complete to decide x is not a member of the core, for the flow game. And because of the specific reduction we constructed, the result also holds for the linear production game. Received: October 2000/Final version: March 2002     

Cooperative games of choosing partners and forming coalitions in the marketplace

Two games of interacting between a coalition of players in a marketplace and the residual players acting there are discussed, along with two approaches to fair imputation of gains of coalitions in cooperative games that are based on the concepts of the Shapley vector and core of a cooperative game. In the first game, which is an antagonistic one, the residual players try to minimize the coalition's gain, whereas in the second game, which is a noncooperative one, they try to maximize their own gain as a coalition. A meaningful interpretation of possible relations between gains and Nash equilibrium strategies in both games considered as those played between a coalition of firms and its surrounding in a particular marketplace in the framework of two classes of n-person games is presented. A particular class of games of choosing partners and forming coalitions in which models of firms operating in the marketplace are those with linear constraints and utility functions being sums of linear and bilinear functions of two corresponding vector arguments is analyzed, and a set of maximin problems on polyhedral sets of connected strategies which the problem of choosing a coalition for a particular firm is reducible to are formulated based on the firm models of the considered kind.     

Dynamic resource allocation in fuzzy coalitions: a game theoretic model

We introduce an efficient and dynamic resource allocation mechanism within the framework of a cooperative game with fuzzy coalitions (cooperative fuzzy game). A fuzzy coalition in a resource allocation problem can be so defined that membership grades of the players in it are proportional to the fractions of their total resources. We call any distribution of the resources possessed by the players, among a prescribed number of coalitions, a fuzzy coalition structure and every membership grade (equivalently fraction of the total resources), a resource investment. It is shown that this resource investment is influenced by the satisfaction of the players in regard to better performance under a cooperative setup. Our model is based on the real life situations, where possibly one or more players compromise on their resource investments in order to help forming coalitions.     

<Emphasis Type="Italic">n</Emphasis>-Person Second-Order Games: A Paradigm Shift in Game Theory

In this paper, we present a new approach to n-person games based on the Habitual domain theory. Unlike the traditional game theory models, the constructed model captures the fact that the underlying changes in the psychological aspects and mind states of the players over the arriving events are the key factors, which determine the dynamic process of coalition formation. We introduce two new concepts of solution for games: strategically stable mind profile and structurally stable mind profile. The theory introduced in this paper overcomes the dichotomy of non-cooperative/cooperative games, prevailing in the existing game theory, which makes game theory more applicable to real-world game situations.     

Link formation in cooperative situations

In this paper we study the endogenous formation of cooperation structures or communication graphs between players in a superadditive TU game. For each cooperation structure that is formed, the payoffs to the players are determined by an exogenously given solution. We model the process of cooperation structure formation as a game in strategic form. It is shown that several equilibrium refinements predict the formation of the complete cooperation structure or some structure which is payoff-equivalent to the complete structure. These results are obtained for a large class of solutions for cooperative games with cooperation structures. Received September 1995/Revised version I October 1996/Revised version II April 1997/Final version September 1997     

The core of a game with a continuum of players and finite coalitions: The model and some results

In this paper we develop a new model of a cooperative game with a continuum of players. In our model, only finite coalitions - ones containing only finite numbers of players - are permitted to form. Outcomes of cooperative behavior are attainable by partitions of the players into finite coalitions: this is appropriate in view of our restrictions on coalition formation. Once feasible outcomes are properly defined, the core concept is standard - no permissible coalition can improve upon its outcome. We provide a sufficient condition for the nonemptiness of the core in the case where the players can be divided into a finite number of types. This result is applied to a market game and the nonemptiness of the core of the market game is stated under considerably weak conditions (but with finite types). In addition, it is illustrated that the framework applies to assignment games with a continuum of players.     

Absolutely cooperative solution for a linear,multiplayer differential game

If the players of ann-player differential game agree to cooperate, then the solutions to the game should be confined to undominated ones. A property of an undominated or Pareto-optimal solution is that, when compared locally with any other solution, at least one player must do worse or all do the same if they use a solution other than the Pareto-optimal one.Closely related to the concept of a Pareto-optimal solution is the concept of an absolutely cooperative solution. The absolutely cooperative solution is given the property that, when compared locally with any other solution, every player will do no better if a solution other than the absolutely cooperative one is used.A set of necessary and sufficient conditions for an absolutely cooperative solution is presented in this paper. The circumstance under which the control variables may take on interior values is also included.This research was supported in part by NASA Grant No. NGR-03-002-011.     

全对策的边际贡献值

全对策是定义在局中人集合的所有分划集上的一类特殊合作对策.本文在效用可转移情形下研究全对策的"值"问题.定义了全对策的边际贡献值,得出全对策的Shapley值,以及具有某些性质的值是边际贡献值,并给出两种边际贡献值的具体表达式,及其一些性质.     

Inducing cooperation by reciprocative strategy in non-zero-sum games

In this paper, we investigate the use of reciprocative strategy to induce cooperative behavior in non-zero-sum games. Reciprocative behavior is defined mathematically in the context of a two-person non-zero-sum game in which both the players have a common set of pure strategies. Conditions under which mutual cooperative behavior results when one of the players responds optimally to reciprocative behavior by the other player are described. Also, the desirability of playing the reciprocative strategy is investigated by stating conditions under which reciprocative strategy by one of the players or by both the players leading to mutual cooperative behavior is a Nash equilibrium outcome.     

Random reduction consistency of the Weber set,the core and the anti-core

In this paper we introduce a new consistency condition and provide characterizations for several solution concepts in TU cooperative game theory. Our new consistency condition, which we call the random reduction consistency, requires the consistency of payoff vectors assigned by a solution concept when one of the players is removed with some probability. We show that the random reduction consistency and other standard properties characterize the Weber set, the convex hull of the marginal contribution vectors. Another salient feature of random reduction consistency is that, by slightly changing its definition, we can characterize the core and the anti-core in a parallel manner. Our result enables us to compare the difference between the three solution concepts from the viewpoint of consistency.     

A <Emphasis Type="Italic">q</Emphasis>-player impartial avoidance game for generating finite groups

We study a q-player variation of the impartial avoidance game introduced by Anderson and Harary, where q is a prime. The game is played by the q players taking turns selecting previously-unselected elements of a finite group. The losing player is the one who selects an element that causes the set of jointly-selected elements to be a generating set for the group, with the previous player winning. We introduce a ranking system for the other players to prevent coalitions. We describe the winning strategy for these games on cyclic, nilpotent, dihedral, and dicyclic groups.