Mean Field Games with Substitutable Goods

We study continuous time Bertrand competition in which a large number of firms producing similar goods compete with one another by setting prices. Interactions are of mean field type in the sense that the demand faced by a producer is affected by the others through their average price. We consider the setting of dynamic game with uncertain market demand, where firms of different sizes compete on prices and produce to clear the market demand until they have depleted their lifetime capacity. We set up the nonzero-sum stochastic differential game of mean field type and its associated forward/backward system of partial differential equations in the case of linear demand functions. Asymptotic approximation enables us to deduce certain qualitative features of the game in the limit of small competition. The equilibrium of the game is further studied using numerical solutions. We find that, in accordance with the 2-player game, firms slow down production with a large degree of substitutability.