Influence functions, followers and command games

We study and compare two frameworks: a model of influence, and command games. In the influence model, in which players are to make a certain acceptance/rejection decision, due to influence of other players, the decision of a player may be different from his inclination. We study a relation between two central concepts of this model: influence function, and follower function. We deliver sufficient and necessary conditions for a function to be a follower function, and we describe the structure of the set of all influence functions that lead to a given follower function. In the command structure introduced by Hu and Shapley, for each player a simple game called the command game is built. One of the central concepts of this model is the concept of command function. We deliver sufficient and necessary conditions for a function to be a command function,and describe the minimal sets generating a normal command game. We also study the relation between command games and influence functions. A sufficient and necessary condition for the equivalence between an influence function and a normal command game is delivered.influence function, follower function, lower and upper inverses, kernel, command game, command function, minimal sets generating a command game

Influence functions, followers and command games

We study and compare two frameworks: a model of influence, and command games. In the influence model, in which players are to make a certain acceptance/rejection decision, due to influence of other players, the decision of a player may be different from his inclination. We study a relation between two central concepts of this model: influence function, and follower function. We deliver sufficient and necessary conditions for a function to be a follower function, and we describe the structure of the set of all influence functions that lead to a given follower function. In the command structure introduced by Hu and Shapley, for each player a simple game called the command game is built. One of the central concepts of this model is the concept of command function. We deliver sufficient and necessary conditions for a function to be a command function, and describe the minimal sets generating a normal command game. We also study the relation between command games and influence functions. A sufficient and necessary condition for the equivalence between an influence function and a normal command game is delivered.influence function;follower function;lower and upper inverses;kernel;command game;command function;minimal sets generating a command game

Measuring influence in command games

In the paper, we study a relation between command games proposed by Hu and Shapley and an influence model. We show that our framework of influence is more general than the framework of the command games. We define several influence functions which capture the command structure. These functions are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. For some influence functions we define the command games such that the influence functions are compatible with these games. We show that not for all influence functions such command games exist. Moreover, we propose a more general definition of the influence index and show that some power indices, which can be used in the command games, coincide with some expressions of the weighted influence indices. We show exact relations between an influence function and a follower function, between a command game and commandable players, and between influence functions and command games. An example of the Confucian model of society is broadly examined.Banzhaf index ; Coleman indices ; command game ; follower of a coalition ; influence function ; influence indices ; Shapley-Shubik index

A study of the dynamic of influence through differential equations

URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2011.22 - ISSN : 1955-611XThe paper concerns a model of influence in which agents make their decisions on a certain issue. It is assumed that each agent is inclined to make a particular decision, but due to a possible influence of the others, his final decision may be different from his initial inclination. Since in reality the influence does not necessarily stop after one step, but may iterate, we present a model which allows us to study the dynamic of influence. The use of continuous variable permits the application of differential equations systems to the analysis of the convergence of agents' decisions in long-time. In particular, by applying the approach based on differential equations of the influence model, we recover the results of the discrete model on classical influence functions and the results on the boss and approval sets for the command games equivalent to some influence functions.Le papier concerne un modèle d'influence dans lequel les agents prennent des décisions sur une proposition. On suppose que chaque agent est incliné à prendre une certaine décision, mais que dû à une influence possible des autres agents, sa décision finale peut être différente de son inclination initiale. Comme dans la réalité l'influence ne s'arrête pas nécessairement après une étape mais peut itérer, nous présentons un modèle qui nous permet d'étudier la dynamique de l'influence. L'utilisation de variables continues permet l'application des systèmes d'équations différentielles à l'analyse de la convergence des décisions des agents. En particulier, en appliquant l'approche basée sur les équations différentielles au modèle d'influence, nous retrouvons les résultats du modèle discret sur les fonctions d'influence classiques, ainsi que les résultats sur les "boss sets" et les ensembles d'approbation pour les jeux de commande équivalents aux fonctions d'influence

Measuring influence in command games

URL des Documents de travail : http://ces.univ-paris1.fr/cesdp/CESFramDP2008.htmClassification JEL : C7, D7.Documents de travail du Centre d'Economie de la Sorbonne 2008.78 - ISSN : 1955-611XIn the paper, we study a relation between command games proposed by Hu and Shapley and an influence model. We show that our framework of influence is more general than the framework of the command games. We define several influence functions which capture the command structure. These functions are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. Some of the presented influence functions are equivalent to the command games, that is, they are compatible with the command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games. For some influence functions, we define the equivalent command games. We show that not for all influence functions the compatible command games exist. Moreover, we propose a more general definition of the influence index and show that under some assumptions, some power indices, which can be used in the command games, coincide with some expressions of the weighted influence indices. Both the Shapley-Shubik index and the Banzhaf index are equal to a difference between the weighted influence indices under some influence functions, and the only difference between thes two power indices lies in the weights for the influence indices. An example of the Confucian model od society is broadly examined.Nous étudions la relation entre un jeu de commande de Hu et Shapley et un modèle d'influence. Nous montrons que notre modèle d'influence est plus général. Nous définissons plusieurs fonctions d'influence qui modélisent un jeu de commande. Pour certaines fonctions d'influence, nous définissons les jeux de commande équivalents. Nous proposons également une forme générale de l'indice d'influence et montrons qu'ils permettent de retrouver des indices de pouvoir liés aux jeux de commande

A model of influence with an ordered set of possible actions

International audienceIn the paper, a yes-no model of influence is generalized to a multi-choice framework. We introduce and study weighted influence indices of a coalition on a player in a social network, where players have an ordered set of possible actions. Each player has an inclination to choose one of the actions. Due to mutual influence among players, the final decision of each player may be different from his original inclination. In a particular case, the decision of the player is closer to the inclination of the influencing coalition than his inclination was, i.e., the distance between the inclinations of the player and of the coalition is greater than the distance between the decision of the player and the inclination of the coalition in question. The weighted influence index which captures such a case is called the weighted positive influence index. We also consider the weighted negative influence index, where the final decision of the player goes farther away from the inclination of the coalition. We consider several influence functions defined in the generalized model of influence and study their properties. The concept of a follower of a given coalition, and its particular case, a perfect follower, are defined. The properties of the set of followers are analyzed

A model of influence with a continuum of actions

Working Paper GATE 2010-04In the paper, we generalize a two-action (yes-no) model of influence to a framework in which every player has a continuum of actions and he has to choose one of them. We assume the set of actions to be an interval. Each player has an inclination to choose one of the actions. Due to influence among players, the final decision of a player, i.e., his choice of one action, may be different from his original inclination. In particular, a coalition of players with the same inclination may influence another player with different inclination, and as a result of this influence, the decision of the player is closer to the inclination of the influencing coalition than his inclination was. We introduce and study a measure of such a positive influence of a coalition on a player. Several unanimous influence functions in this generalized framework are considered. Moreover, we investigate other tools for analyzing influence, like the concept of a follower of a given coalition, its particular case - a perfect follower, and the kernel of an influence function. We study properties of these concepts. Also the set of fixed points under a given influence function is analyzed. Furthermore, we study linear influence functions. We also introduce a measure of a negative influence of a coalition on a player

Social networks: Prestige, centrality, and influence (Invited paper)

We deliver a short overview of di erent centrality measures and influence concepts in social networks, and present the relation-algebraic approach to the concepts of power and influence. First, we briefly discuss four kinds of measures of centrality: the ones based on degree, closeness, betweenness, and the eigenvector-related measures. We consider centrality of a node and of a network. Moreover, we give a classi cation of the centrality measures based on a topology of network flows. Furthermore, we present a certain model of influence in a social network and discuss some applications of relation algebra and RelView to this model

Difference games and policy evaluation: A conceptual framework

Economics