Linear Quadratic Mean Field Games with Quantile-Dependent Cost Coefficients

Abstract

This paper studies a class of linear quadratic mean field games where the coefficients of quadratic cost functions depend on both the mean and the variance of the population's state distribution through its quantile function. Such a formulation allows for modelling agents that are sensitive to not only the population average but also the population variance. The corresponding mean field game equilibrium is identified, which involves solving two coupled differential equations: one is a Riccati equation and the other the variance evolution equation. Furthermore, the conditions for the existence and uniqueness of the mean field equilibrium are established. Finally, numerical results are presented to illustrate the behavior of two coupled differential equations and the performance of the mean field game solution.

Figures (5)

• Figure 1: The probit function for the standard (scalar-valued) Gaussian distribution
• Figure 2: Trajectories of $V$ and $\Pi$. We see a clear reduction of the variance $V$ in (a) compared to that in (b), and $\Pi$ is larger in (a) than in (b). The parameters are $a = 0.5$, $\mu_0=0$, $b=r=1$, $\sigma =V_0=q=1$, $\alpha = 0.95$, and $T=1$. In this case, the inequality \ref{['eq:T-M relation']} is not satisfied, but the numerical solution to the two-point boundary value problem (specified by \ref{['eq:dyn-Pimv']} and \ref{['eq:dyn-Vmv']} with the coupling \ref{['eq:Q-constraint']}) still exists as the inequality in \ref{['eq:T-M relation']} is only a sufficient condition for the existence.
• Figure 3: The difference between the cost of the MFG tracking control and that of the optimal tracking control (assuming the data of all other agents are given a priori), as the number of agents increases, is illustrated in (a). The maximum difference (over time) between the limit population mean and the mean of the finite population, as the number of agents increases, is illustrated in (b). The parameters are the same as those in Figure \ref{['fig:simulation-trajectory-20000']}.
• Figure 4: The agent cost of the optimal tracking solution (assuming the state trajectories of other agents are given a priori) is illustrated in star points, and the agent cost using the MFG solution (assuming the state trajectories of other agents are given a priori) is illustrated in triangle points, and finally the agent cost (by computing empirical average of cost over simulations) of using the MFG solution is illustrated in circle points. All the simulations use the same parameters as those in Figure \ref{['fig:simulation-trajectory-20000']}.
• Figure 5: Simulation results with 2000 agents using MFG solutions. The parameters in the simulations are $a = -0.15$, $b = 0.75$, $r = 3.5$, $\sigma = 1$, $V_0 = 0.5$, $\alpha = 0.975$, $q = 0.45$, $T = 0.2$ and $\mu_0 = 1.0$. In this case, both inequalities \ref{['eq:T-M relation']} and \ref{['eq:contraction-condition']} are satisfied for $M = 3$.

Theorems & Definitions (14)

• lemma 1: MFG Solutions
• lemma 2: Simplified MFG Solutions
• remark 1
• remark 2
• proposition 1: Existence
• proof
• remark 3
• proposition 2: Uniqueness
• proof
• lemma 3