On a Network Centrality Maximization Game
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We study a network formation game whereby n players, identified with the nodes of the network to be formed, chose where to wire their outgoing links in order to maximize their Bonacich centrality. Specifically, the action of every player i consists in the wiring of a predetermined number d_i of directed out-links, and her utility is her own Bonacich centrality in the network resulting from the actions of all players. We show that this is an ordinal potential game and that the best response correspondence always exhibits a local structure in that it is never convenient for a node to link to other nodes that are at incoming distance more than d_i from her. We then study the equilibria of this game determining necessary conditions for a graph to be a (strict, recurrent) Nash equilibrium. Moreover, in the special cases where for all players d_i=1 or d_i=2, we provide a complete classification of the set of (strict, recurrent) Nash equilibria. Our analysis shows in particular that the considered formation mechanism leads to the emergence of undirected and disconnected or loosely connected networks.
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