General Opinion Formation Games with Social Group Membership (Discussion Paper)

Modeling how agents form their opinions is of paramount importance for designing marketing and electoral campaigns. In thiswork, we present a newframework for opinion formation which generalizes the well-known Friedkin-Johnsen model by incorporating three important features: (i) social group membership, that limits the amount of influence that people not belonging to the same group may lead on a given agent; (ii) both attraction among friends, and repulsion among enemies; (iii) different strengths of influence lead from different people on a given agent, even if the social relationships among them are the same. We show that, despite its generality, our model always admits a pure Nash equilibrium which, under opportune mild conditions, is even unique. Next, we analyze the performance of these equilibria with respect to a social objective function defined as a convex combination, parametrized by a value λ ϵ [0, 1], of the costs yielded by the untruthfulness of the declared opinions and the total cost of social pressure. We prove bounds on both the price of anarchy and the price of stability which show that, for not-tooextreme values of λ, performance at equilibrium are very close to optimal ones. For instance, in several interesting scenarios, the prices of anarchy and stability are both equal to max{2λ, 1-λ}/ min{2λ, 1-λ} which never exceeds 2 for λ ϵ [1/5, 1/2]. Moreover, in many settings, we provide even better upper bounds on the prices of anarchy and stability, which are tight under mild assumptions.