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.     

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.     

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.     

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.     

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.     

Stochastic games with metric state space

In this paper the stochastic two-person zero-sum game of Shapley is considered, with metric state space and compact action spaces. It is proved that both players have stationary optimal strategies, under conditions which are weaker than those ofMaitra/Parthasarathy (a.o. no compactness of the state space). This is done in the following way: we show the existence of optimal strategies first for the one-period game with general terminal reward, then for then-period games (n=1,2,...); further we prove that the game over the infinite horizon has a valuev, which is the limit of then-period game values. Finally the stationary optimal strategies are found as optimal strategies in the one-period game with terminal rewardv.     

Share functions for cooperative games with levels structure of cooperation

In a standard TU-game it is assumed that every subset of the player set N can form a coalition and earn its worth. One of the first models where restrictions in cooperation are considered is the one of games with coalition structure of Aumann and Drèze (1974). They assumed that the player set is partitioned into unions and that players can only cooperate within their own union. Owen (1977) introduced a value for games with coalition structure under the assumption that also the unions can cooperate among them. Winter (1989) extended this value to games with levels structure of cooperation, which consists of a game and a finite sequence of partitions defined on the player set, each of them being coarser than the previous one.     

Single-suit two-person card play

We consider a two-person constant sum perfect information game, which we call theEnd Play Game, which arises from an abstraction of simple end play positions in card games of the whist family, including bridge. This game was described in 1929 by Emanuel Lasker, the mathematician and world chess champion, who called itwhistette. The game uses a deck of cards that consists of a single totally ordered suit of 2n cards. To begin play the deck is divided into two handsA andB ofn cards each, held by players Left and Right, and one player is designated as having thelead. The player on lead chooses one of his cards, and the other player after seeing this card selects one of his own to play. The player with the higher card wins a “trick” and obtains the lead. The cards in the trick are removed from each hand, and play then continues until all cards are exhausted. Each player strives to maximize his trick total, and thevalue of the game to each player is the number of tricks he takes. Despite its simple appearance, this game is quite complicated, and finding an optimal strategy seems difficult. This paper derives basic properties of the game, gives some criteria under which one hand is guaranteed to be better than another, and determines the optimal strategies and value functions for the game in several special cases.     

Harsanyi power solutions for graph-restricted games

We consider cooperative transferable utility games, or simply TU-games, with limited communication structure in which players can cooperate if and only if they are connected in the communication graph. Solutions for such graph games can be obtained by applying standard solutions to a modified or restricted game that takes account of the cooperation restrictions. We discuss Harsanyi solutions which distribute dividends such that the dividend shares of players in a coalition are based on power measures for nodes in corresponding communication graphs. We provide axiomatic characterizations of the Harsanyi power solutions on the class of cycle-free graph games and on the class of all graph games. Special attention is given to the Harsanyi degree solution which equals the Shapley value on the class of complete graph games and equals the position value on the class of cycle-free graph games. The Myerson value is the Harsanyi power solution that is based on the equal power measure. Finally, various applications are discussed.     

Values and potential of games with cooperation structure

Games with cooperation structure are cooperative games with a family offeasible coalitions, that describes which coalitions can negotiate in the game. We study a model ofcooperation structure and the corresponding restricted game, in which the feasible coalitions are those belonging to apartition system. First, we study a recursive procedure for computing the Hart and Mas-Colell potential of these games and we develop the relation between the dividends of Harsanyi in the restricted game and the worths in the original game. The properties ofpartition convex geometries are used to obtain formulas for theShapley andBanzhaf values of the players in the restricted game in terms of the original gamev. Finally, we consider the Owen multilinear extension for the restricted game.The author is grateful to Paul Edelman, Ulrich Faigle and the referees for their comments and suggestions. The proof of Theorem 1 was proposed by the associate editor's referee.     

Local strategies and equilibria with an application to the committee decision process

In this paper, a formalization of the idea of “lookingk turns ahead” in the context of an-person non-cooperative game, in which the rules specify a sequence of turns, is considered. At thei-th move, there is a game ofk turns an equilibrium of which gives a behavioral strategy for thei-the move. In general, this strategy is not an equilibrium but is attractive as an explanation of the behavior of experienced players playing a complicated game. As an application of this idea, a committee voting game is analyzed.     

The core and the Weber set of games on augmenting systems

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed a general model for cooperative games defined on lattice structures. In this paper, the restrictions to the cooperation are given by a combinatorial structure called augmenting system which generalizes antimatroid structure and the system of connected subgraphs of a graph. In this framework, the core and the Weber set of games on augmenting systems are introduced and it is proved that monotone convex games have a non-empty core. Moreover, we obtain a characterization of the convexity of these games in terms of the core of the game and the Weber set of the extended game.     

Impact of neighborhood separation on the spatial reciprocity in the prisoner’s dilemma game

The evolutionary game theory is a very powerful tool to understand the collective cooperation behavior in many real-world systems. In the spatial game model, the payoff is often first obtained within a specific neighborhood (i.e., interaction neighborhood) and then the focal player imitates or learns the behavior of a randomly selected one inside another neighborhood which is named after the learning neighborhood. However, most studies often assume that the interaction neighborhood is identical with the learning neighborhood. Beyond this assumption, we present a spatial prisoner’s dilemma game model to discuss the impact of separation between interaction neighborhood and learning neighborhood on the cooperative behaviors among players on the square lattice. Extensive numerical simulations demonstrate that separating the interaction neighborhood from the learning neighborhood can dramatically affect the density of cooperators (ρ_C) in the population at the stationary state. In particular, compared to the standard case, we find that the medium-sized learning (interaction) neighborhood allows the cooperators to thrive and substantially favors the evolution of cooperation and ρ_C can be greatly elevated when the interaction (learning) neighborhood is fixed, that is, too little or much information is not beneficial for players to make the contributions for the collective cooperation. Current results are conducive to further analyzing and understanding the emergence of cooperation in many natural, economic and social systems.     

Non-cooperative stochastic dominance games

A non-cooperative stochastic dominance game is a non-cooperative game in which the only knowledge about the players' preferences and risk attitudes is presumed to be their preference orders on the set ofn-tuples of pure strategies. Stochastic dominance equilibria are defined in terms of mixed strategies for the players that are efficient in the stochastic dominance sense against the strategies of the other players. It is shown that the set of SD equilibria equals all Nash equilibria that can be obtained from combinations of utility functions that are consistent with the players' known preference orders. The latter part of the paper looks at antagonistic stochastic dominance games in which some combination of consistent utility functions is zero-sum over then-tuples of pure strategies.     

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     

Inheritance of properties in communication situations

In this paper we consider cooperative games in which the possibilities for cooperation between the players are restricted because communication between the players is restricted. The bilateral communication possibilities are modeled by means of a (communication) graph. We are interested in how the communication restrictions influence the game. In particular, we investigate what conditions on the communication graph guarantee that certain appealing properties of the original game are inherited by the graph-restricted game, the game that arises once the communication restrictions are taken into account. We study inheritance of the following properties: average convexity, inclusion of the Shapley value in the core, inclusion of the Shapley values of a game and all its subgames in the corresponding cores, existence of a population monotonic allocation scheme, and the property that the extended Shapley value is a population monotonic allocation scheme. Received May 1998/Revised version January 2000     

The assignment game I: The core

The assignment game is a model for a two-sided market in which a product that comes in large, indivisible units (e.g., houses, cars, etc.) is exchanged for money, and in which each participant either supplies or demands exactly one unit. The units need not be alike, and the same unit may have different values to different participants. It is shown here that the outcomes in thecore of such a game — i.e., those that cannot be improved upon by any subset of players — are the solutions of a certain linear programming problem dual to the optimal assignment problem, and that these outcomes correspond exactly to the price-lists that competitively balance supply and demand. The geometric structure of the core is then described and interpreted in economic terms, with explicit attention given to the special case (familiar in the classic literature) in which there is no product differentiation — i.e., in which the units are interchangeable. Finally, a critique of the core solution reveals an insensitivity to some of the bargaining possibilities inherent in the situation, and indicates that further analysis would be desirable using other game-theoretic solution concepts.     

On the communication complexity oft-intersection problems in generalized Boolean algebras

We consider the following game: Two players independently choose a chain in a partially ordered set. How many bits of information have to be communicated until at least one of the players knows whether the chains have exactlyt elements in common? This model generalizes thet-intersection problem for subsets of a finite set. We establish the deterministic communication complexity in general. For the special cases of generalized Boolean algebras, we present improved nondeterministic and probabilistic protocols that are of optimal order of complexity for classes with fixed widthq.     

The game of two identical cars

This paper describes a third-order pursuit—evasion game in which both players have the same speed and minimum turn radius. The game of kind is first solved for thebarrier or envelope of capturable states. When capture is possible, the game of degree is then solved for the optimal controls of the two players as functions of the relative position. The solution is found to include a universal surface for the pursuer and a dispersal surface for the evader.