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Influence functions, followers and command games. (English) Zbl 1236.91021

Summary: 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 X. Hu and L. S. Shapley [Games Econ. Behav. 45, No. 1, 132–152 (2003; Zbl 1054.91011); ibid. 45, No. 1, 153–170 (2003; Zbl 1071.91006)], 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.

MSC:

91A12 Cooperative games
91A65 Hierarchical games (including Stackelberg games)
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