Macroscopic Market Making Games via Multidimensional Decoupling Field

Building on the macroscopic market making framework as a control problem, this paper investigates its extension to stochastic games. In the context of price competition, each agent is benchmarked against the best quote offered by the others. We begin with the linear case. While constructing the solution directly, the $\mathit{\text{ordering}}\mathit{\text{property}}$ and the dimension reduction in the equilibrium are revealed. For the non-linear case, we extend the decoupling approach by introducing a multidimensional $\mathit{\text{characteristic}}\mathit{\text{equation}}$ to analyse the well-posedness of the forward-backward stochastic differential equations. Properties of the coefficients in this characteristic equation are derived using tools from non-smooth analysis. Several new well-posedness results are presented.