Deciding Bank Interest Rates -- A Major-Minor Impulse Control Mean-Field Game Perspective

TL;DR

This work addresses the problem of setting bank deposit rates in a competitive interbank market by formulating it as an impulsive major-minor mean-field game, where a distributional mean-field flow $\mu_t$ captures interactions among banks and a central bank rate $r_t^c$ acts as exogenous noise. A novel deep Q-network (DQN) approach with measure-projection and fictitious-play averaging is developed to compute Nash equilibria in continuous-state spaces, extending prior discrete-state methods. Key contributions include the first major-minor MFG model for interbank rate decisions, a scalable DQN algorithm that handles measure-valued inputs via a projection operator $\mathcal{A}$, and demonstrated convergence and robustness in simulated experiments. The framework enables analysis of major vs. minor banks’ strategies under Nash equilibrium and offers a practical tool for policy and risk assessment in the presence of impulse-driven rate adjustments.

Abstract

Deciding bank interest rates has been a long-standing challenge in finance. It is crucial to ensure that the selected rates balance market share and profitability. However, traditional approaches typically focus on the interest rate changes of individual banks, often neglecting the interactions with other banks in the market. This work proposes a novel framework that models the interest rate problem as a major-minor mean field game within the context of an interbank game. To incorporate the complex interactions between banks, we utilize mean-field theory and employ impulsive control to model the overhead in rate adjustments. Ultimately, we solve this optimal control problem using a new deep Q-network method, which iterates the parameterized action value functions for major and minor players and updates the networks in a Fictitious Play way. Our proposed algorithm converges, offering a solution that enables the analysis of strategies for major and minor players in the market under the Nash Equilibrium.

Figures (4)

• Figure 1: Diagram of constructing projection $\mathcal{A}$: $\mu_k\to \tilde{\mu}_k$
• Figure 2: Average losses of 10 individual trainings (shadowed) and its $200$ rolling average (bold).
• Figure 3: The evolution of major player's state and central bank rate.
• Figure 4: The evolution of mean field measure (minor players).

Theorems & Definitions (1)

• definition 1