Conrado Manuel

Centrality and power in social networks: a game theoretic approach

Abstract A new family of centrality measures, based on game theoretical concepts, is proposed for social networks. To reflect the interests that motivate the interactions among individuals in a network, a cooperative game in characteristic function form is considered. From the graph and the game, the graph-restricted game is obtained. Shapley value in a game is considered as actor’s power. The difference between actor’s power in the new game and his/her power in the original one is proposed as a centrality measure. Conditions are given to reach some desirable properties. Finally, a decomposition is proposed.

A probabilistic position value

In this article, we generalize the position value, defined by Meessen (Master’s thesis, 1988) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations (Gómez et al. in European Journal of Operational Research 190:539–556, 2008). We provide two characterizations of this new allocation rule. Following in Slikker’s (International Journal of Game Theory 33:505–514, 2005a) footsteps, we characterize the probabilistic position value using probabilistic versions of component efficiency and balanced link contributions. Then we generalize the notion of link potential, defined by Slikker (International Game Theory Review 7:473–489, 2005b) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations, and use it to characterize our allocation rule.

A value for generalized probabilistic communication situations

We introduce in this work an extension of the model of games with probabilistic graphs arising in Calvo et al. (1999, Math. Soc. Sci. 37, 79), which itself generalizes the one developed by Myerson (1977, Math. of Oper. Res. 2, 225) for games with communications restrictions. In the first of these models, each pair of nodes has a given probability of direct communication. In this paper a more general setting is considered: we suppose that a probability distribution over the set of all possible communication networks among the players is given. A generalization of the Myerson value is defined and characterized in this context.

Splitting graphs when calculating Myerson value for pure overhead games

Abstract.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 Value for Directed Communication Situations

In this paper we introduce an extension of the model of restricted communication in cooperative games as introduced in Myerson (1977) by allowing communication links to be directed and the worth of a coalition to depend on the order in which the players enter the coalition. Therefore, we model the communication network by a directed graph and the cooperative game by a generalized characteristic function as introduced in Nowak and Radzik (1994). We generalize the Myerson value for undirected (or standard) communication situations to the context of directed communication and provide two axiomatizations of this digraph Myerson value using component efficiency and either fairness or the balanced contributions property.

A Unified Approach to the Myerson Value and the Position Value

We reconsider the Myerson value and the position value for communication situations. In case the underlying game is a unanimity game, we show that each of these values can be computed using the inclusion--exclusion principle. Linearity of both values permits us to calculate them without needing the dividends of the induced games (graph-restricted game and link game). The expression of these dividends is only derived in the existing literature for special communication situations. Moreover, the associated inclusion--exclusion decomposability property depends on what we have called the graph allocation rule. This rule is the relative degree (relative indicator) for the position value (Myerson value). Copyright Kluwer Academic Publishers 2004

A social capital index

In this paper we propose a social capital measure for individuals belonging to a social network. To do this, we use a game theoretical approach and so we suppose that these individuals are also involved in a cooperative TU-game modelling the economic or social interests that motivate their interactions. We propose as a measure of individual social capital the difference between the Myerson and the Shapley values of actors in the social network and explore the properties of such a measure. This definition is close to our previous measure of centrality (Gomez et al., 2003) and so in this paper we also study the relation between social capital and centrality, finding that this social capital measure can be considered as a vector magnitude with two additive components: centrality and positional externalities. Finally, several real political examples are used to show the agreement of our conclusions with the reality in these situations.

Order Monotonic Solutions for Generalized Characteristic Functions

Generalized characteristic functions extend characteristic functions of ‘classical’ TU-games by assigning a real number to every ordered coalition being a permutation of any subset of the player set. Such generalized characteristic functions can be applied when the earnings or costs of cooperation among a set of players depend on the order in which the players enter a coalition.

Values of games with weighted graphs

In this paper we deal with TU games in which cooperation is restricted by means of a weighted network. We admit several interpretations for the weight of a link: capacity of the communication channel, flow across it, intimacy or intensity in the relation, distance between both incident nodes/players, cost of building or maintaining the communication link or even probability of the relation (as in Calvo, Lasaga, and van den Noweland, 1999). Then, according to the different interpretations, we introduce several point solutions for these restricted games in a way parallel to the familiar environment of Myerson. Finally, we characterize these values in terms of the (adapted) component efficiency, fairness and balanced contributions properties and we analyze the extent to which they satisfy a link/weight monotonicity property.

Players Indifferent to Cooperate and Characterizations of the Shapley Value

In this paper we provide new axiomatizations of the Shapley value for TU-games using axioms that are based on relational aspects in the interactions among players. Some of these relational aspects, in particular the economic or social interest of each player in cooperating with each other, can be found embedded in the characteristic function. We define a particular relation among the players that it is based on mutual indifference. The first new axiom expresses that the payoffs of two players who are not indifferent to each other are affected in the same way if they become enemies and do not cooperate with each other anymore. The second new axiom expresses that the payoff of a player is not affected if players to whom it is indifferent leave the game. We show that the Shapley value is characterized by these two axioms together with the well-known efficiency axiom. Further, we show that another axiomatization of the Shapley value is obtained if we replace the second axiom and efficiency by the axiom which applies the efficiency condition to every class of indifferent players. Finally, we extend the previous results to the case of weighted Shapley values.