The potential method for price-formation models

Abstract

We consider the mean-field game price formation model introduced by Gomes and Sa úde. In this MFG model, agents trade a commodity whose supply can be deterministic or stochastic. Agents maximize profit, taking into account current and future prices. The balance between supply and demand determines the price. We introduce a potential function that converts the MFG into a convex variational problem. This variational formulation is particularly suitable for machine learning approaches. Here, we use a recurrent neural network to solve this problem. In the last section of the paper, we compare our results with known analytical solutions.

A Fully-discrete Semi-Lagrangian scheme for a price formation MFG model

“…To evaluate the accuracy of our method, we use the semi-explicit solutions proposed in [11]. Then, we compare our scheme with other numerical methods: the variational method using a potential transformation given in [5] and a recurrent neural network method given in [4].…”

“…Under assumptions 1-5, from [11], we have existence and uniqueness of solution (u, m, ϖ), where m is bounded, u is Lipschitz, semiconcave, and differentiable in x. Moreover, from [3], ϖ is Lipschitz continuous. Finally, assumption 6 ensures compact support of the density function m(x, t) up to the terminal time T .…”

“…We also run our scheme with the same discretization parameters ρ = 0.05 and h = 0.05 and the same problem data as in [4], where authors used the variational method. Our scheme outperforms the variational method in accuracy for ϖ, u, and m, by approximately 10, 8, 1.5 times.…”

“…An ML training process was developed based on a min-max characterization of control and price variables in [12]. Using the potential transformation, [4] solved the price problem with common noise using a recurrent neural network. These methods are suitable for common noise problems where standard numerical methods fail.…”

“…Figures 2, 3, and 4 compare numerical solutions to analytical ones and illustrate absolute errors.Table1presents l ∞ error rates for various grid sizes, with all cases converging within 6 iterations for the tolerance parameter ε = 0.001.We also run our scheme with the same discretization parameters ρ = 0.05 and h = 0.05 and the same problem data as in[4], where authors used the variational method. Our scheme outperforms the variational method in accuracy for ϖ, u, and m, by approximately 10, 8, 1.5 times.We also compared our scheme to the recurrent neural network method with the same problem data as given in[5].…”