A Banzhaf share function for cooperative games in coalition structure

A cooperative game with transferable utility–or simply a TU-game– describes a situation in which players can obtain certain payoffs by cooperation. A value function for these games assigns to every TU-game a distribution of payoffs over the players. Well-known solutions for TU-games are the Shapley and the Banzhaf value. An alternative type of solution is the concept of share function, which assigns to every player in a TU-game its share in the worth of the grand coalition. In this paper we consider TU-games in which the players are organized into a coalition structure being a finite partition of the set of players. The Shapley value has been generalized by Owen to TU-games in coalition structure. We redefine this value function as a share function and show that this solution satisfies the multiplication property that the share of a player in some coalition is equal to the product of the Shapley share of the coalition in a game between the coalitions and the Shapley share of the player in a game between the players within the coalition. Analogously we introduce a Banzhaf coalition structure share function. Application of these share functions to simple majority games show some appealing properties.     

Who gains from information asymmetry?

This article considers an asymmetric contest with incomplete information. There are two types of players: informed and uninformed. Each player has a different ability to translate effort into performance in terms of the contest success function. While one player’s type is known to both players, the other is private information and known only to the player himself. We compare the Bayesian Nash equilibrium outcome of a one-sided private information contest to the Nash equilibrium with no private information, in which both players know the type of the other player. We show conditions under which uncertainty increases the investment of the uninformed player and the rent dissipation of the contest, while decreasing the expected net payoff of the informed player. In addition, we consider conditions under which the informed player—before knowing his own type—prefers that the uninformed player knows his type. Moreover, we show conditions for the existence/non-existence of equilibrium in a two-stage contest in which the informed player declares his type (or does not declare) in the first stage and in the second stage the two players play according to the information available to them.     

Stable coalition structures in symmetric majority games: a coincidence between myopia and farsightedness

The objective of this paper is to study stable coalition structures in symmetric majority games. We assume that players deviate from a coalition structure to another to maximize their power given by the Owen power index. We introduce three myopic core concepts and one farsighted stability concept, the farsighted vNM stable set. Our main result is that the pessimistic core, the largest myopic core, coincides with some farsighted vNM stable set for any number of players. Moreover, we show that a coalition structure belongs to the pessimistic core and the farsighted vNM stable set if and only if it contains an exact majority coalition.     

Partial cooperation in strategic multi-sided decision situations

We consider a normal-form game in which there is a single exogenously given coalition of cooperating players that can write a binding agreement on pre-selected actions. The actions representing other dimensions of the strategy space remain under the sovereign, individual control of the players. We consider a standard extension of the Nash equilibrium concept denoted as a partial cooperative equilibrium as well as an equilibrium concept in which the coalition of cooperators has a leadership position. Existence results are stated and we identify conditions under which the various equilibrium concepts are equivalent. We apply this framework to existing models of multi-market oligopolies and international pollution abatement. In a multi-market oligopoly, typically, a merger paradox emerges in the partial cooperative equilibrium. The paradox vanishes if the cartel attains a leadership position. For international pollution abatement treaties, cooperation by a sufficiently large group of countries results in a Pareto improvement over the standard tragedy of the commons outcome described by the Nash equilibrium.     

Sequential strategies in dual control problems

Sequential decision problems are studied in which there are two decision-makers, or players, who are trying to control the same stochastic system in order to minimize their own individual expected losses. The standard linear model for the stochastic control process and standard quadratic loss functions are assumed. The players take turns choosing the value of the control variable, and each player has his own sequence of targets and his own loss function. Optimal pure competitive strategies are derived by backward induction, and limiting strategies and equilibria are determined. Myopic strategies are introduced and shown to yield smaller losses for each player than the pure competitive strategies. Coefficients of cooperation are then defined, and various cooperative sequential strategies based on them are shown to be mutually beneficial to both players.     

Farsightedness and Cautiousness in Coalition Formation Games with Positive Spillovers

We adopt the largest consistent set defined by Chwe (1994; J. Econ. Theory 63: 299–325) to predict which coalition structures are possibly stable when players are farsighted. We also introduce a refinement, the largest cautious consistent set, based on the assumption that players are cautious. For games with positive spillovers, many coalition structures may belong to the largest consistent set. The grand coalition, which is the efficient coalition structure, always belongs to the largest consistent set and is the unique one to belong to the largest cautious consistent set.     

The Right to Remain Silent

The paper points out that in dynamic games a player may be better-off if other players do not know his choice of strategy. That is, a player may benefit by not revealing (or not pre-determining) the choice of his action in an information set he (thereby) hopes will not be reached. He would be better-off by exercising his ``right to remain silent' if he believes –- as the empirical evidence shows –- that players display aversion to ``Knightian uncertainty'. In this case, a player who behaves strategically, may wish to avoid revealing his strategy. This is true under various interpretations of the notion of ``strategy profiles'.     

Values with exogenous payments

The aim of cooperative game theory is to suggest and defend payoffs for the players that depend on a coalition function (characteristic function) describing the economic, social, or political situation. We consider situations where the payoffs for some players are determined exogenously. For example, in many countries, lawyers or real-estate agents obtain a regulated fee or a regulated percentage of the business involved. The aim of this article is to suggest and axiomatize two values with exogenous payments, an unweighted one and a weighted one.     

Representability of Ordinal Relations on a Set of Conditional Events

Any dynamic decision model should be based on conditional objects and must refer to (not necessarily structured) domains containing only the elements and the information of interest. We characterize binary relations, defined on an arbitrary set of conditional events, which are representable by a coherent generalized decomposable conditional measure and we study, in particular, the case of binary relations representable by a coherent conditional probability.     

Game-theoretic analyses of coalition behavior

Coalitions are frequently more visible than payoffs. The theory of n-person games seeks primarily to identify stable allocations of valued resources; consequently, it gives inadequate attention to predicting which coalitions form. This paper explores a way of correcting this deficiency of game-theoretic reasoning by extending the theory of two-person cooperative games to predict both coalitions and payoffs in a three-person game of status in which each player seeks to maximize the rank of his total score. To accomplish this, we analyze the negotiations within each potential two-person coalition from the perspective of Nash's procedure for arbitrating two-person bargaining games, then assume that players expect to achieve the arbitrated outcome selected by this procedure and use these expectations to predict achieved ranks and to identify players' preferences between alternative coalition partners in order to predict the probability that each coalition forms. We test these payoff and coalition predictions with data from three laboratory studies, and compare the results with those attained in the same data by von Neumann and Morgenstern's solution of two-person cooperative games, Aumann and Maschler's bargaining set solution for cooperative n-person games, and an alternative model of coalition behavior in three-person sequential games of status.

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Socially Structured Games

We generalize the concept of a cooperative non-transferable utility game by introducing a socially structured game. In a socially structured game every coalition of players can organize themselves according to one or more internal organizations to generate payoffs. Each admissible internal organization on a coalition yields a set of payoffs attainable by the members of this coalition. The strengths of the players within an internal organization depend on the structure of the internal organization and are represented by an exogenously given power vector. More powerful players have the power to take away payoffs of the less powerful players as long as those latter players are not able to guarantee their payoffs by forming a different internal organization within some coalition in which they have more power. We introduce the socially stable core as a solution concept that contains those payoffs that are both stable in an economic sense, i.e., belong to the core of the underlying cooperative game, and stable in a social sense, i.e., payoffs are sustained by a collection of internal organizations of coalitions for which power is distributed over all players in a balanced way. The socially stable core is a subset and therefore a refinement of the core. We show by means of examples that in many cases the socially stable core is a very small subset of the core. We will state conditions for which the socially stable core is non-empty. In order to derive this result, we formulate a new intersection theorem that generalizes the KKMS intersection theorem. We also discuss the relationship between social stability and the wellknown concept of balancedness for NTU-games, a sufficient condition for non-emptiness of the core. In particular we give an example of a socially structured game that satisfies social stability and therefore has a non-empty core, but whose induced NTU-game does not satisfy balancedness in the general sense of Billera.      

Perceptron versus automaton in the finitely repeated prisoner’s dilemma

We study the finitely repeated prisoner’s dilemma in which the players are restricted to choosing strategies which are implementable by a machine with a bound on its complexity. One player has to use a finite automaton while the other player has to use a finite perceptron. Some examples illustrate that the sets of strategies which are induced by these two types of machines are different and not ordered by set inclusion. Repeated game payoffs are evaluated according to the limit of means. The main result establishes that a cooperation at almost all stages of the game is an equilibrium outcome if the complexity of the machines the players may use is limited enough and if the length T of the repeated game is sufficiently large. This result persists when more than T states are allowed in the player’s automaton. We further consider a variant of the model in which the two players are restricted to choosing strategies which are implementable by perceptrons and prove that the players can cooperate at most of the stages provided that the complexity of their perceptrons is sufficiently reduced.     

FORWARD INDUCTION IN GAMES WITH AN OUTSIDE OPTION

We provide eductive foundations for the concept of forward induction, in the class of games with an outside option. The formulation presented tries to capture in a static notion the rest point of an introspective process, achievable from some restricted preliminary beliefs. The former requisite is met by requiring the rest point to be a Nash equilibrium that yields a higher payoff than the outside option. With respect to the beliefs, we propose the Incentive Dominance Criterion. Players should consider one action more likely than another whenever the former is better than getting the outside option for more conjectures over his rival's actions. We apply this model to the case where the subgame is a coordination game with a conflict between payoff dominance and risk dominance. Our results provide support for dominance solvability, but not for Van Damme's notion of forward induction. We show how the forward induction logic helps to select the Pareto dominant equilibrium. This is the case whenever player 1's act of giving up the outside option reverses the incentive dominance relations among 1's pure actions in the subgame.     

Nullified equal loss property and equal division values

We provide characterizations of the equal division values and their convex mixtures, using a new axiom on a fixed player set based on player nullification which requires that if a player becomes null, then any two other players are equally affected. Two economic applications are also introduced concerning bargaining under risk and common-pool resource appropriation.     

Double deception: Two against one in three-person games

This article examines deception possibilities for two players in simple three-person voting games. An example of one game vulnerable to (tacit) deception by two players is given and its implications discussed. The most unexpected findings of this study is that in those games vulnerable to deception by two players, the optimal strategy of one of them is always to announce his (true) preference order. Moreover, since the player whose optimal announcement is his true one is unable to induce a better outcome for himself by misrepresenting his preference, while his partner can, this player will find that possessing a monopoly of information will not give him any special advantage. In fact, this analysis demonstrates that he may have incentives to share his information selectively with one or another of his opponents should he alone possess complete information at the outset.     

Monotonicity of power in games with a priori unions

Power indices are commonly required to assign at least as much power to a player endowed with some given voting weight as to any player of the same game with smaller weight. This local monotonicity and a related global property however are frequently and for good reasons violated when indices take account of a priori unions amongst subsets of players (reflecting, e.g., ideological proximity). This paper introduces adaptations of the conventional monotonicity notions that are suitable for voting games with an exogenous coalition structure. A taxonomy of old and new monotonicity concepts is provided, and different coalitional versions of the Banzhaf and Shapley–Shubik power indices are compared accordingly.      

A test of the Core,Bargaining Set,Kernel and Shapley models in n-person quota games with one weak player

n-person (n – 1)-quota-games where the quotas are positive for certain n – 1 (strong) players and negative for the remaining (weak) player, are discussed. Normative solutions predicted by the Core,the Kernel, the Bargaining Set, the Competitive Bargaining Set, and by the Shapley Value are presented and exemplified.Each of twelve groups of subjects participated in a four-person and a five-person (n – 1)-quota games with one weak player. The weak player was always excluded from the ratified coalition. The division of payoffs among the strong players was more egalitarian than the Kernel solution but less egalitarian than the Shapley value. The Core and the Bargaining Sets were fully supported for the two strongest players, but less supported for the other players. Analyses of the bargaining process confirmed a dynamic interpretation of the Bargaining Set Theory.This research was performed while the author was at the University of North Carolina. The research was partially supported by a PHS Research Grant No. MH-10006 from the National Institute of Mental Health. The author thanks Professor Amnon Rapoport for helpful advice in the design of this study.     

Axiomatization of a class of share functions for n-person games

The Shapley value is the unique value defined on the class of cooperative games in characteristic function form which satisfies certain intuitively reasonable axioms. Alternatively, the Banzhaf value is the unique value satisfying a different set of axioms. The main drawback of the latter value is that it does not satisfy the efficiency axiom, so that the sum of the values assigned to the players does not need to be equal to the worth of the grand coalition. By definition, the normalized Banzhaf value satisfies the efficiency axiom, but not the usual axiom of additivity.In this paper we generalize the axiom of additivity by introducing a positive real valued function on the class of cooperative games in characteristic function form. The so-called axiom of -additivity generalizes the classical axiom of additivity by putting the weight (v) on the value of the gamev . We show that any additive function determines a unique share function satisfying the axioms of efficient shares, null player property, symmetry and -additivity on the subclass of games on which is positive and which contains all positively scaled unanimity games. The axiom of efficient shares means that the sum of the values equals one. Hence the share function gives the shares of the players in the worth of the grand coalition. The corresponding value function is obtained by multiplying the shares with the worth of the grand coalition. By defining the function appropiately we get the share functions corresponding to the Shapley value and the Banzhaf value. So, for both values we have that the corresponding share functions belong to this class of share functions. Moreover, it shows that our approach provides an axiomatization of the normalized Banzhaf value. We also discuss some other choices of the function and the corresponding share functions. Furthermore we consider the axiomatization on the subclass of monotone simple games.     

Divide-the-Dollar Game Revisited

In the Divide-the-Dollar (DD) game, two players simultaneously make demands to divide a dollar. Each player receives his demand if the sum of the demands does not exceed one, a payoff of zero otherwise. Note that, in the latter case, both parties are punished severely. A major setback of DD is that each division of the dollar is a Nash equilibrium outcome. Observe that, when the sum of the two demands x and y exceeds one, it is as if Player 1's demand x (or his offer (1−x) to Player 2) suggests that Player 2 agrees to λ_x < 1 times his demand y so that Player 1's demand and Player 2's modified demand add up to exactly one; similarly, Player 2's demand y (or his offer (1−y) to Player 1) suggests that Player 1 agrees to λ_yx so that λ_yx+y = 1. Considering this fact, we change DD's payoff assignment rule when the sum of the demands exceeds one; here in this case, each player's payoff becomes his demand times his λ; i.e., each player has to make the sacrifice that he asks his opponent to make. We show that this modified version of DD has an iterated strict dominant strategy equilibrium in which each player makes the egalitarian demand 1/2. We also provide a natural N-person generalization of this procedure. This revised version was published online in June 2006 with corrections to the Cover Date.