The equilibrium existence of strategic influence games in social networks

We study the game model proposed by Grabisch et al. in 2010. Given a set of (nonstrategic) agents in a strongly connected network each of whom has an initial opinion expressed by a number in [0,~1], two (strategic) players with fixed opinions 0 and 1, respectively, compete for the support of the agents. Each player chooses one agent and influences the agent's opinion with an influence factor. Such influence may spread due to the interactions among agents in the network. The objective of each player is to make the sum of the resulting agents' opinions as close to their own (0 or 1) as possible. We develop the model by introducing a new renormalization matrix to evaluate the influence strength of players and exemplify that it relaxes the conditions for the existence of Nash equilibrium (NE) in a range of networks. For the new setting, when the two players have equal influence strength, we give a sufficient condition for NE existence and prove that there exists at least one equilibrium in bidirectional tree networks with uniform weights.