Neighbor games and the leximax solution

Neighbor games arise from certain matching or sequencing situations in which only some specific pairs of players can obtain a positive gain. As a consequence, the class of neighbor games is the intersection of the class of assignment games (Shapley and Shubik (1972)) and the class of component additive games (Curiel et al. (1994)). We first present some elementary features of neighbor games. After that we provide a polynomially bounded algorithm of order p ³ for calculating the leximax solution (cf. Arin and Iñarra (1997)) of neighbor games, where p is the number of players. This authors work has been supported by CentER and the Department of Econometrics, Tilburg University and by the Foundation for the Hungarian Higher Education and Research (AMFK).     

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.     

The consistent Shapley value for hyperplane games

A new value is defined for n-person hyperplane games, i.e., non-sidepayment cooperative games, such that for each coalition, the Pareto optimal set is linear. This is a generalization of the Shapley value for side-payment games. It is shown that this value is consistent in the sense that the payoff in a given game is related to payoffs in reduced games (obtained by excluding some players) in such a way that corrections demanded by coalitions of a fixed size are cancelled out. Moreover, this is the only consistent value which satisfies Pareto optimality (for the grand coalition), symmetry and covariancy with respect to utility changes of scales. It can be reached by players who start from an arbitrary Pareto optimal payoff vector and make successive adjustments.     

Games with permission structures: The conjunctive approach

This paper is devoted to the game theoretic analysis of decision situations, in which the players have veto power over the actions undertaken by certain other players. We give a full characterization of the dividends in these games with a permission structure. We find that the collection of these games forms a subspace of the vector space of all games with side payments on a specified player set.Two applications of these results are provided. The first one deals with the projection of additive games on a permission structure. It is shown that the Shapley value of these projected games can be interpreted as an index that measures the power of the players in the permission structure. The second application applies the derived results on games, where the organization structure can be analysed separately from the production capacities of the participating players.     

Rankings and values for team games

In this article we attack several problems that arise when a group of individuals is organized in several teams with equal number of players in each one (e.g., for company work, in sports leagues, etc). We define a team game as a cooperative game v that can have non-zero values only on coalitions of a given cardinality; it is further shown that, for such games, there is essentially a unique ranking among the players. We also study the way the ranking changes after one or more players retire. Also, we characterize axiomatically different ways of ranking the players that intervene in a cooperative game.     

Games with Ordered Outcomes

We present a brief review of the most important concepts and results concerning games in which the goal structure is formalized by binary relations called preference relations. The main part of the work is devoted to games with ordered outcomes, i.e., game-theoretic models in which preference relations of players are given by partial orders on the set of outcomes. We discuss both antagonistic games and n-person games with ordered outcomes. Optimal solutions in games with ordered outcomes are strategies of players, situations, or outcomes of the game. In the paper, we consider noncooperative and certain cooperative solutions. Special attention is paid to an extension of the order on the set of probabilistic measures since this question is substantial for constructing the mixed extension of the game with ordered outcomes. The review covers works published from 1953 until now.     

Preferred coalitions in cooperative differential games

There are many interesting situations which can be described by anN-person general-sum differential game. Such games are characterized by the fact that the strategy of each player depends upon reasonable assumptions about the strategies of the remaining players; and, thus, these games cannot be considered asN uncoupled optimal control problems. In such cases, we say that the game is not strictly competitive, but involves a mutual interest which makes it possible for all of the players to reduce their costs by cooperating with one another, provided the resulting agreement can be enforced. When cooperation is allowed and there are more than two players, there is always the question of whether all possible subcoalitions will be formed with equal ease. This work considers the situation in which a particular subcoalition is preferred. A theory of general-sum games with preferred coalitions is presented, together with constructive examples of alternative approaches which are unsatisfactory.     

Non-cooperative monopolistic games and monopolistic market games

A class of non-cooperative games is discussed in which one player (“the monopolist”) by choosing his strategy restricts the other players to subsets of their strategy sets. Examples of such games in various fields are given. In particular it is shown that some very important economic situations fall within this class of games. A solution concept is defined and sufficient conditions for its existence are derived. The question of the advantages a player derives from being a monopolist is raised and conditions are derived for him to benefit from being a monopolist.     

On weighted Shapley values

Nonsymmetric Shapley values for coalitional form games with transferable utility are studied. The nonsymmetries are modeled through nonsymmetric weight systems defined on the players of the games. It is shown axiomatically that two families of solutions of this type are possible. These families are strongly related to each other through the duality relationship on games. While the first family lends itself to applications of nonsymmetric revenue sharing problems the second family is suitable for applications of cost allocation problems. The intersection of these two families consists essentially of the symmetric Shapley value. These families are also characterized by a probabilistic arrival time to the game approach. It is also demonstrated that lack of symmetries may arise naturally when players in a game represent nonequal size constituencies.     

Games with a permission structure - A survey on generalizations and applications

In the field of cooperative games with restricted cooperation, various restrictions on coalition formation are studied. The most studied restrictions are those that arise from restricted communication and hierarchies. This survey discusses several models of hierarchy restrictions and their relation with communication restrictions. In the literature, there are results on game properties, Harsanyi dividends, core stability, and various solutions that generalize existing solutions for TU-games. In this survey, we mainly focus on axiomatizations of the Shapley value in different models of games with a hierarchically structured player set, and their applications. Not only do these axiomatizations provide insight in the Shapley value for these models, but also by considering the types of axioms that characterize the Shapley value, we learn more about different network structures. A central model of games with hierarchies is that of games with a permission structure where players in a cooperative transferable utility game are part of a permission structure in the sense that there are players that need permission from other players before they are allowed to cooperate. This permission structure is represented by a directed graph. Generalizations of this model are, for example, games on antimatroids, and games with a local permission structure. Besides discussing these generalizations, we briefly discuss some applications, in particular auction games and hierarchically structured firms.     

Splitting graphs when calculating Myerson value for pure overhead games

A communication situation consists of a coalitional game and a graph, the nodes of the graph corresponding to the players of the game. To calculate the Myerson value for such situations, we obtain results which extend those well known for trees and cycle-complete graphs. On the other hand, in order to reduce the associated calculus for communication situations with a pure overhead game, the possibility of splitting the graph in several subgraphs is analyzed. For each fixed decomposition of the graph, a subspace of games compatible with this decomposition is given.     

A link between sequential semi-anonymous nonatomic games and their large finite counterparts

We show that obtainable equilibria of a multi-period nonatomic game can be used by players in its large finite counterparts to achieve near-equilibrium payoffs. Such equilibria in the form of random state-to-action rules are parsimonious in form and easy to execute, as they are both oblivious of past history and blind to other players’ present states. Our transient results can be extended to a stationary case, where the finite multi-period games are special discounted stochastic games. In both nonatomic and finite games, players’ states influence their payoffs along with actions they take; also, the random evolution of one particular player’s state is driven by all players’ states as well as actions. The finite games can model diverse situations such as dynamic price competition. But they are notoriously difficult to analyze. Our results thus suggest ways to tackle these problems approximately.     

Dynamic linear programming games with risk-averse players

Motivated by situations in which independent agents wish to cooperate in some uncertain endeavor over time, we study dynamic linear programming games, which generalize classical linear production games to multi-period settings under uncertainty. We specifically consider that players may have risk-averse attitudes towards uncertainty, and model this risk aversion using coherent conditional risk measures. For this setting, we study the strong sequential core, a natural extension of the core to dynamic settings. We characterize the strong sequential core as the set of allocations that satisfy a particular finite set of inequalities that depend on an auxiliary optimization model, and then leverage this characterization to establish sufficient conditions for emptiness and non-emptiness. Qualitatively, whereas the strong sequential core is always non-empty when players are risk-neutral, our results indicate that cooperation in the presence of risk aversion is much more difficult. We illustrate this with an application to cooperative newsvendor games, where we find that cooperation is possible when it least benefits players, and may be impossible when it offers more benefit.     

The Compromise Value for Cooperative Games with Random Payoffs

This paper introduces and studies the compromise value for cooperative games with random payoffs, that is, for cooperative games where the payoff to a coalition of players is a random variable. This value is a compromise between utopia payoffs and minimal rights and its definition is based on the compromise value for NTU games and the τ-value for TU games. It is shown that the nonempty core of a cooperative game with random payoffs is bounded by the utopia payoffs and the minimal rights. Consequently, for such games the compromise value exists. Further, we show that the compromise value of a cooperative game with random payoffs coincides with the τ-value of a related TU game if the players have a certain type of preferences. Finally, the compromise value and the marginal value, which is defined as the average of the marginal vectors, coincide on the class of two-person games. This results in a characterization of the compromise value for two-person games.I thank Peter Borm, Ruud Hendrickx and two anonymous referees for their valuable comments.     

Value of games with two-layered hypergraphs

This paper studies cooperative games with restricted cooperation among players. We define situations in which a priori unions and hypergraphs coexist simultaneously and mutually depend on each other. We call such structures two-layered hypergraphs. Using a two-step approach, we define a value of the games with two-layered hypergraphs. The value is characterized by Owen’s coalitional value of hypergraph-restricted games and in terms of weighted Myerson value. Further, our value is axiomatically characterized by component efficiency and a coalition size normalized balanced contributions property.     

Common pool resources with support

We examine the role of support for coalition stability in common pool resource games such as fisheries games. Some players may not want to join a coalition that jointly manages a resource. Still, because they benefit from spillovers, they may want to support the coalition with a transfer payment to set incentives for others to join. We find that the impact of support on equilibria of this game is limited to games with three or five players. Recommendations for Resource Managers

• Coalitions may be able to effectively manage common pool resources such as fisheries but such coalitions are often not stable due to free-rider incentives.
• We explore the impact of a transfer scheme that can improve this coalition stability which would lead to larger and more effective coalitions.
• Our results show that this new transfer scheme works only for cases where the number of players is small.

     

Two Game Models for Cooperation with Implicit Noncooperation

We propose two flexible game models to represent and analyze cases that cannot be modeled by current game models. One is called sharing creditability game (SCG) and the other is called bottomline game (BLG). The new models transform cooperative games into new games that incorporate auxiliary information (noncooperative in nature) usually neglected in previous theories. The new games will be solved only by traditional noncooperative game theory. When the new solutions are applied to the original games, the solutions can reflect the auxiliary information in addition to the original objectives of the decision makers or players. Generally, the new solutions are different from the cooperative and the noncooperative solutions of the original games. Existing transferable utility (TU) games and noncooperative games will coincide with special cases of the two new game models. Using SCG and BLG, the prisoner’s dilemma can be reformulated and a richer set of decisions can be considered for the players. The two new game models have potential applications in military and socioeconomic situations.This research was partly funded by the College Engineering, Ohio State University.     

Games of timing related to distribution of resources

The paper considers a class of zero-sum, two-person games which are related to distribution of resources. Each of the players is in possession of an amount of resource, to be distributed by him in the time interval [0, 1] according to an arbitrary measure. The payoff function is defined in such a manner that the games are a generalization of the so-called silent, nondiscrete duels. It is proven that these games have a value, and the optimal strategies for the players are found. The results of the paper bring to light new, essential elements, common to almost all games of timing on [0, 1].     

Single-payoff farsighted stable sets in strategic games with dominant punishment strategies

We investigate farsighted stable sets in a class of strategic games with dominant punishment strategies. In this class of games, each player has a strategy that uniformly minimizes the other players’ payoffs for any given strategies chosen by these other players. We particularly investigate a special class of farsighted stable sets, each of which consists of strategy profiles yielding a single payoff vector. We call such a farsighted stable set as a single-payoff farsighted stable set. We propose a concept called an inclusive set that completely characterizes single-payoff farsighted stable sets in strategic games with dominant punishment strategies. We also show that the set of payoff vectors yielded by single-payoff farsighted stable sets is closely related to the strict \(\alpha \)-core in a strategic game. Furthermore, we apply the results to strategic games where each player has two strategies and strategic games associated with some market models.