René van den Brink

Measuring Domination in Directed Networks

Abstract Dominance relations between individuals can be represented by a directed social network . A relational power measure is a function that assigns to each position in a directed network a value representing the relational power of that position in the network. We axiomatically characterize two such power measures, the β - measure and the score-measure . We also apply these measures to weighted directed networks that can be interpreted as dominance structures which dominance relations are assigned weights representing the ‘importance’ of each relation.

An axiomatization of the disjunctive permission value for games with a permission structure

Players that participate in acooperative game with transferable utilities are assumed to be part of apermission structure being a hierarchical organization in which there are players that need permission from other players before they can cooperate. Thus a permission structure limits the possibilities of coalition formation.Various assumptions can be made about how a permission structure affects the cooperation possibilities. In this paper we consider thedisjunctive approach in which it is assumed that each player needs permission from at least one of his predecessors before he can act. We provide an axiomatic characterization of thedisjunctive permission value being theShapley value of a modified game in which we take account of the limited cooperation possibilities.

Null or nullifying players: The difference between the Shapley value and equal division solutions

A famous solution for cooperative transferable utility games is the Shapley value. Most axiomatic characterizations of this value use some axiom related to null players, i.e. players who contribute zero to any coalition. We show that replacing null players with nullifying players characterizes the equal division solution distributing the worth of the ‘grand coalition’ equally among all players. A player is nullifying if every coalition containing this player earns zero worth. Using invariance we provide similar characterizations of the equal surplus division solution assigning to every player its own worth, and distributing the remaining surplus equally among all players.

An Axiomatization of the Shapley Value Using a Fairness Property

Abstract. In this paper we provide an axiomatization of the Shapley value for TU-games using a fairness property. This property states that if to a game we add another game in which two players are symmetric then their payoffs change by the same amount. We show that the Shapley value is characterized by this fairness property, efficiency and the null player property. These three axioms also characterize the Shapley value on the class of simple games.In this paper we provide an axiomatization of the Shapley value for TU-games using a fairness property. This property states that if to a game we add another game in which two players are symmetric then their payoffs change by the same amount. We show that the Shapley value is characterized by this fairness property, efficiency and the null player property. These three axioms also characterize the Shapley value on the class of simple games.

Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values

One of the main issues in economic allocation problems is the trade-off between marginalism and egalitarianism. In the context of cooperative games this trade-off can be framed as one of choosing to allocate according to the Shapley value or the equal division solution. In this paper we provide three different characterizations of egalitarian Shapley values being convex combinations of the Shapley value and the equal division solution. First, from the perspective of a variable player set, we show that all these solutions satisfy the same reduced game consistency. Second, on a fixed player set, we characterize this class of solutions using monotonicity properties. Finally, towards a strategic foundation, we provide a non-cooperative implementation for these solutions which only differ in the probability of breakdown at a certain stage of the game. These characterizations discover fundamental differences as well as intriguing connections between marginalism and egalitarianism.

Digraph competitions and cooperative games

Digraph games are cooperative TU-games associated to domination structures which can be modeled by directed graphs. Examples come from sports competitions or from simple majority win digraphs corresponding to preference profiles in social choice theory. The Shapley value, core, marginal vectors and selectope vectors of digraph games are characterized in terms of so-called simple score vectors. A general characterization of the class of (almost positive) TU-games where each selectope vector is a marginal vector is provided in terms of game semi-circuits. Finally, applications to the ranking of teams in sports competitions and of alternatives in social choice theory are discussed.

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.

A class of consistent share functions for 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 is a function which assigns to every such a game a distribution of the payoffs over the players in the game.An alternative type of solutions are share functions which assign to every player in a TU-game its share in the payoffs to be distributed.In this paper we consider cooperative games in which the players are organized into an a priori coalition structure being a finite partition of the set of players.We introduce a general method for defining a class of share functions for such games in coalition structure using a multiplication property that states that the share of player i in the total payoff is equal to the share of player i in some internal game within i s a priori coalition, multiplied by the share of this coalition in an external game between the a priori given coalitions.We show that these coalition structure share functions satisfy certain consistency properties.We provide axiomatizations of this class of coalition structure share functions using these consistency and multiplication properties.

An Iterative Procedure for Evaluating Digraph Competitions

A competition which is based on the results of (partial) pairwise comparisons can be modelled by means of a directed graph. Given initial weights on the nodes in such digraph competitions, we view the measurement of the importance (i.e., the cardinal ranking) of the nodes as an allocation problem where we redistribute the initial weights on the basis of insights from cooperative game theory. After describing the resulting procedure of redistributing the initial weights, an iterative process is described that repeats this procedure: at each step the allocation obtained in the previous step determines the new input weights. Existence and uniqueness of the limit is established for arbitrary digraphs. Applications to the evaluation of, e.g., sport competitions and paired comparison experiments are discussed.

A Social Power Index for Hierarchically Structured Populations of Economic Agents

This paper presents a model of a finite collection of socially related economic agents. We assume that an agent in an economy is part of some social structure in which he might dominate some agents while he himself is dominated by other agents. We consider structures in which these social relations between the agents have some special features. Such a group of agents endowed with a social structure is called a hierarchically structured population. We identify two types of social differences between economic agents in a hierarchically structured population. Firstly we show that the agents can be subdivided into groups that can be ordered such that agents in ‘higher’ groups dominate agents in ‘lower’ groups. Secondly we show that the communication structure between the agents, in general, will be incomplete.