Discrete Preference Games in Heterogeneous Social Networks: Subverted Majorities and the Swing Player

We study discrete preference games in heterogeneous social networks. These games model the interplay between a player's private belief and his/her publicly stated opinion (which could be different from the player's belief) as a strategic game in which the players' strategies are the opinions and the cost of an opinion in a state is a convex combination through a parameter $\alpha\in[0,1]$ of two factors: the disagreement between the player's opinion and his/her internal belief and the number of neighbors whose opinions differ from the one of the player. The parameter $\alpha$ models how stubborn a player is: players with large $\alpha$ change their opinion only if many neighbors disagree with his/her belief. We consider social networks that are heterogeneous in the sense that the parameter $\alpha$ can vary from player to player. We ask if it is possible that the belief shared by the majority of the players does not coincide with the opinion that is publicly announced by the majority of the players in an equilibrium state. Our main result is a characterization of the social networks that admit an initial belief assignment for which there exists a sequence of best response moves that reach an equilibrium in which the initial majority is subverted. Our characterization is effective in the sense that can be tested efficiently and the initial belief assignment that can be subverted can be computed in time polynomial in the number of players. Our result is actually stronger as we show that in each initial belief assignment that can be subverted, subversion is actually obtained in a very strong way: it only takes one move of a single player, the swing player, to lead the social network to a point of no return in which any rational move from any player leads to a subverted majority.