Regularity of dynamic opinion games

We analyze a class of stochastic games where two lobbies compete by influencing the opinions in a society. We assume that the opinions evolve according to De Groot opinion formation and that the decisions of the lobbies change the structure of the network representing the society. We show that the regularity of discounted Nash equilibrium payoffs when players become patient is highly model-dependent. We provide two extreme cases. First, we present an example where the sequence of discounted Nash equilibrium payoffs does not converge. Hence, the solution is highly sensitive to the discount factor and a modeler needs to know the discount factor precisely in order to compute equilibria. Second, we focus on a subclass of problems where lobbies are restricted and show the existence of a uniform equilibrium. Hence, in this restricted framework, a modeler can predict some approximate equilibria with an imprecise knowledge of the discount factor.