Abstract : We introduce the mean-field games with explicit interactions. This model is a finite-state space mean-field game where the evolution and the cost function of the individual players depend not only on the actions taken, but also on the population distribution. We analyze these games in continuous and discrete time, over finite as well as infinite time horizons. We show the existence of a mean-field equilibrium in this type of games using an adapted version of Kakutani fixed point theorem. Besides, we also study the convergence of the equilibria of N-player games to mean-field equilibria. We define classes of strategies over which any equilibrium converges to a mean-field equilibrium when the number of players goes to infinity. We also exhibit equilibria outside this class that do not converge to mean-field equilibria. In discrete time the same non-convergence phenomenon implies that the Folk theorem does not scale to the mean field limit. Finally, we construct a mean-field game with explicit interaction to study vaccination strategies over an SIR infection propagation model and compute its mean field equilibrium is almost closed form. We also compare the Nash equilibrium with a centrally optimal strategy and show that, in all but degenerated cases, the equilibrium does not coincide with the optimal solution. We design a pricing mechanism that force the equilibrium to coincide with an optimal vaccination strategy.
