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

Fan Chen; Francois Buet-Golfouse; Nicholas Martin; Po-Yu Chen; Xiaozhen Wang; Zhenjie Ren

REVIEW 5 major objections 5 minor 44 references

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

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper establishes that bank deposit-rate competition can be modeled as a major-minor mean-field game with impulsive rate changes, and that a deep Q-network algorithm with fictitious-play averaging converges to a Nash equilibrium of…

desk verdict Novel mix of major-minor MFG, impulse control, and DQN for bank rates, but the central convergence claim rests on self-referential loss and no Nash verification. read the letter →

arxiv 2411.14481 v2 pith:I7WYJWC3 submitted 2024-11-19 math.OC math.PR

Fan Chen , Nicholas Martin , Po-Yu Chen , Xiaozhen Wang , Zhenjie Ren , Francois Buet-Golfouse This is my paper · ORCID

classification math.OCmath.PR MSC 91A1649N2593E20

keywords BankInterestRateonDepositsCompetitioninProportionMajor-MinorPlayersMean-FieldGamesImpulsiveControlFictitiousPlayAlgorithmReinforcementLearningDeepQ-NetworkNashEquilibrium

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the long-standing problem of how a bank sets deposit rates while competitors respond can be framed as a game with one dominant bank and many small identical banks, and that the equilibrium strategies of that game can actually be computed. It builds a mean-field model in which the small banks are represented by a distribution, rate changes are costly and therefore impulsive, and the central bank rate provides common noise. It then argues that a deep Q-network algorithm, which averages past Q-functions in the manner of fictitious play, converges to a Nash equilibrium of this game in continuous state and action spaces. If true, this would give banks and regulators a quantitative way to see how deposit rates, market shares, and adjustment costs interact under competition.

What carries the argument

The central object is the parameterized action-value (Q) function of the major bank and of the representative minor bank, defined through the projected Bellman equations (8)--(9). A compactly supported mean-field measure is replaced by its projection onto a finite grid, so each Q-function takes a vector of grid masses as part of its input. The carrying identity is the representation of a one-hidden-layer network as an expectation under an empirical measure over parameters (equation (10)); this turns the fictitious-play moving average of neural networks into a convex combination of measures (equation (11)), which can be implemented by randomly merging neurons. The equilibrium notion is Definition 3.1: no unilateral deviation by the major bank or by an individual minor bank improves expected discounted profit, so the algorithm is searching for a fixed point of the best-response map.

What would settle it

Hold the trained major and minor strategies fixed, then compute the best response of one bank by high-accuracy dynamic programming over a fine grid of states, rates, and mean-field distributions, and compare its expected discounted profit with what the trained strategy earns; if either the major bank or any minor bank can improve by more than the algorithm's own loss tolerance, the computed profile is not a Nash equilibrium.

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Extended reading notes

Core claim

The central claim is that the inter-bank deposit-rate game has a Nash equilibrium in which each bank's rate policy is optimal given every other bank's policy, and that this equilibrium can be reached by a concrete iterative algorithm. The discovery is computational: a deep Q-network can parameterize action-value functions whose inputs include a projection of the mean-field measure, and the fictitious-play averaging step can be performed by averaging the parameter measures that represent the networks rather than by averaging the outputs. With this machinery the paper reports Bellman residuals below $1\times 10^{-7}$ after about half of training, stable across repeated runs, and it reads the resulting policies under the tested parameters---all banks keep rates low and market shares nearly flat---as an approximate Nash equilibrium.

Load-bearing premise

The algorithm's claim of reaching a Nash equilibrium rests on the premise that pushing the Q-networks' prediction error to a tiny value against targets produced by the same networks yields the true action-value functions of the game; the experiments only show that this training loss decreases, never whether any player could profitably deviate from the final strategies.

Editorial extensions

If this is right

The algorithm supplies an approximate Nash equilibrium for one major bank against many homogeneous minor banks under impulse control, in continuous state and action spaces, where tabular methods would fail.
Under the tested parameters, the equilibrium behavior is non-competitive: all banks set low deposit rates, and market shares remain roughly constant.
The measure-averaging trick makes fictitious-play averaging of neural networks implementable for major-minor mean-field games.
The framework extends to multiple major banks by increasing the number $M$ of major players in the dynamics (5)--(6), with the discrete-time experiment specialized to $M=1$.
Because the central bank rate is modeled as a jump process, the equilibrium rate policies can in principle trace how banks respond to policy-rate shocks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same machinery should carry over to other pass-through settings where one dominant firm prices against a competitive fringe with adjustment costs, such as mortgage rates, deposit sweeps, or retail lending.
A sharper test of the Nash claim would be to audit the final strategies for unilateral deviations on a coarse action grid; the paper does not report such a check, so the equilibrium conclusion is only as strong as the Bellman-residual fit.
One could extend the model to heterogeneous minor banks by replacing the single representative minor with several populations, at the cost of a higher-dimensional mean-field vector.
A formal convergence guarantee would likely need a contraction or monotonicity condition; without it, the observed decay of training loss is an empirical, not a proven, convergence result.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is worth a look for its modeling idea, but the headline claim is not backed by the evidence. It combines major-minor mean-field games, impulse control, and a deep Q-network with measure-averaged fictitious play, and applies it to bank deposit rate setting. That combination is new relative to the cited literature, and the measure-averaging trick for neural networks is a sensible workaround for the usual difficulty of averaging in fictitious play. The bank-rate application is also timely, and the model setup is explicit enough that someone could build on it.

The soft spots are real. The abstract says the algorithm converges and gives a Nash equilibrium, but there is no theorem and the numerical evidence is just a decreasing Bellman loss computed against targets generated by the same network being trained. That only shows self-consistency, not convergence to an equilibrium. The paper never checks whether either player could improve by a unilateral deviation, which is exactly what Definition 3.1 requires. The fictitious-play averaging rule in (11) also has no convergence justification in this continuous-state, major-minor setting. On top of that, the mean-field derivation in equations (3)-(5) has notational slips and ambiguities that made me re-read several times, and no code or baseline comparisons are provided.

The reader's take is fair. I agree that the convergence and Nash-equilibrium claims are unsupported as written. The stress-test note correctly identifies the missing unilateral-deviation test. Still, this is not a crank paper. The authors know the relevant literature and are attempting a real algorithmic contribution, but they overstate what they have shown. A revised version with a proper convergence theorem for the averaged Q-learning, a correction of the mean-field equations, and at least one baseline comparison (e.g., against the finite-state method of Cui et al.) could be a solid contribution.

I would send this to a serious referee, but with the expectation of major revision. The core idea deserves engagement; the current claims do not. I would not cite it in its present form, though I would keep an eye on a revised version.

Referee Report

5 major / 5 minor

Summary. The paper proposes a major-minor mean-field game model for interbank deposit-rate competition, incorporating impulse controls for rate adjustment costs. It formulates a discrete-time version with one major bank and a continuum of minor banks, defines a Nash equilibrium in Definition 3.1, and presents a deep Q-network algorithm with fictitious-play averaging of Q-functions by interpreting one-hidden-layer networks as measures. Numerical experiments report decreasing Bellman losses and simulated trajectories of the major bank's rate and the minor players' mean-field distribution.

Significance. If the algorithm really converged to a Nash equilibrium of a continuous-state major-minor impulse-control mean-field game, the paper would be a useful contribution: the application domain is novel, and the neuron-as-measure averaging trick in Eq. (10)-(11) is an elegant workaround for averaging neural-network Q-functions in fictitious play. The paper also correctly identifies that existing major-minor MFG algorithms are mostly restricted to finite state spaces. However, the current validation is not sufficient: the only convergence evidence is a self-referential Bellman loss, the Nash property is never tested against unilateral deviations, there is no baseline comparison, and no code is provided. The significance of the contribution is therefore currently a proof-of-concept rather than an established numerical method.

major comments (5)

[§4, Algorithm 1; §5.2, Fig. 2] The abstract's convergence-to-Nash claim is not established by the evidence. The loss minimized in Algorithm 1 (lines 6-7) is L(ω) = (1/B) Σ ||Q(θ_i;ω) - S_Q(θ_i;ω)||², where S_Q is constructed from the same network Q(·;ω) via Eqs. (8)-(9); there is no target network, no independent policy evaluation, and the sampling distribution for the batches in line 5 is not specified. A small residual only shows that the network is close to a fixed point of its own Bellman operator on those batches; it does not show that the greedy policies returned in line 9 satisfy the two inequalities in Definition 3.1. No unilateral-deviation test for the major player or for a representative minor player is reported, and no convergence theorem for the fictitious-play averaging (11) is given. To support the claim, the authors should evaluate the returned policies against best responses computed with frozen opponent policies and report the resulting P&L differences, not just the training loss.
[§3.2, Eqs. (3)-(6)] The mean-field derivation contains notational and mathematical ambiguities that prevent verification. In Eq. (3), the terms involving minor players mix r_t, δ^{(i)}, and p_t^{(i)} even when i indexes a major bank; in Eq. (4), the integral is written as ∫ κ(...)_+ ar p_t μ^N_t(dar p_t,dr_t) although μ^N_t is already an atomic measure on (p,r), so the role of the factor ar p_t is unclear; and Eq. (5) introduces μ_t as Law(ar p_t,r_t) only after Eq. (4) has used a similar expression. Eq. (6) then states the conservation law Σ_{i=1}^M ar p^{(i)}_t + ∫ ar p_t μ_t = 1 without specifying whether μ_t is the normalized empirical measure of the rescaled minor proportions or a probability law, and without indicating how this survives the N→∞ limit. These points should be clarified or corrected; as written, the model behind the algorithm is not well-defined.
[§3.3, Eq. (8)] There is an internal inconsistency in the Bellman target for the major player. In Eq. (7), the next mean field is T^u_t(x^0,u^0,r^c,μ), using the current central-bank rate r^c. In Eq. (8), the same position is written as T^u_t(x^0,u^0,\tilde r^c,A(μ)), using the next-period rate \tilde r^c, while Eq. (9) uses r^c. Since T^u_t is defined in Section 3.3 with the current r^c and the minor controls at time t depend on the current r^c, the version in Eq. (8) changes the information structure of the trained Q-function. Please fix Eq. (8) or explain why the mean-field transition should be evaluated at \tilde r^c.
[§5.1, central bank transition] The transition kernel for the central-bank rate is not a probability kernel. The paper defines P^c(r^c_{t+1}=r^{c,i'}|r^c_t=r^{c,i}) = λΔt 1_{i=i'} + (1-λΔt)1_{i≠i'} with λΔt=0.2 and three states; summing over the three states i' gives 0.2 + 2×0.8 = 1.8 instead of 1. The intended kernel is probably (1-λΔt) on the diagonal and λΔt/(|R|-1) off-diagonal, or an equivalent, but as written the numerical experiments simulate an ill-defined stochastic process. This directly affects Figure 2 and the reported trajectories, so the experiments need to be rerun with a corrected kernel.
[§3.4, Definition 3.1] The Nash equilibrium definition is incomplete. In the objective for a deviating minor player, J(u^0,u,\hat u) = E[Σ γ^t R_t(x^0_t,u^0_t,\hat x_t,\hat u_t,μ_t)], but the running reward R_t was defined earlier as a function of (x_t,u_t,r^c_t,μ_t); no dynamics for the deviating state \hat x_t are given, and the dependence on r^c_t is dropped. The major-player condition is only checked against fixed minor policies, which may be deliberate, but the paper should state the information structure and specify how \hat x_t evolves, otherwise the equilibrium conditions that Algorithm 1 is supposed to compute are not well-posed.

minor comments (5)

[§3.3] The text says the minor state evolves as x_{t+1}=P(...), but the operator P is never defined; it should presumably be K as defined in the same paragraph.
[§5.1] The grid formulas appear to have index errors: p^{(i)} = 20% + 4%·i with i∈[15] and r^{(j)} = 2.5% + 0.2%·j with j∈[5] give p starting at 24% and r starting at 2.7%, missing the stated lower endpoints 20% and 2.5%. Please check the intended indexing.
[§5.1, Eq. (12)] The bound (12) is |b|≤4.5%, but the text immediately says 'This implies that the proportion ... will change by at most 2%'; with Δt=1 the bound should be 4.5 percentage points. The two statements should be reconciled.
[§5.2, Fig. 2] The caption reports standard errors 'after 10000 iterations' while the horizontal axis extends to 20000 training loops; please specify which iterations are used and whether the plotted losses are computed on the same batches used for training.
[References] Several references are incompletely identified: [16], [21], [30], [36], [37], and [41] carry '[n. d.]' or lack venue and year, and [16] is cited without its publication details. This should be completed before resubmission.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Bellman residual concern is a correctness gap, not a circular reduction.

full rationale

The paper's derivation chain is self-contained. The Nash equilibrium is defined in Definition 3.1 as a pair of unilateral-deviation inequalities, and the Q-functions in equations (7)-(9) are defined by Bellman fixed-point equations. Algorithm 1 minimizes the residuals between the parameterized Q-functions and the right-hand sides of those equations, so a low residual is, at best, evidence of approximate Bellman consistency. That evidence is weak: the residual is computed with a target built from the same network being optimized, and the paper never checks the Definition 3.1 inequalities directly. But this is an omitted validation or convergence proof, not a circularity: the Bellman equation is an independent fixed-point characterization, and the paper does not define the Nash equilibrium as 'low training loss.' The self-citations to [14] and [23] support the representation of shallow neural networks as measures used in the fictitious-play averaging step; those results are not used to define the central equilibrium claim and are not load-bearing in the abstract's convergence statement. No quoted step exhibits a reduction in which a prediction is equal to its input by construction, so no circular step is found.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

All economic parameters are hand-picked for the simulation and none are calibrated to data. The mean-field limit and compact-support projection are standard machinery, but the Bellman-residual-implies-Nash assumption is an ad hoc step specific to this paper. The algorithm's convergence evidence is internal to the training objective.

free parameters (8)

Escape rate kappa0, kappa = 5, 5
Chosen in Section 5.1 without calibration to bank data; controls the speed of client switching and drives the state dynamics.
Viscosity threshold delta0, delta = 0.1%, 0.1%
Thresholds for interest rate differences in equation (1); set by hand.
Liquidity premium l0, l = 0 and 0.1%
Added to the running reward in Section 3.3; arbitrary values that directly affect the equilibrium rate level.
Adjustment cost function C = 0.1|Delta r| + 0.001 * 1_{Delta r != 0}
Chosen ad hoc in Section 5.1; this cost is the main force shaping impulse control and the resulting low-rate strategy.
Discount factor gamma = 0.9
Set in Section 5.1 with no economic justification and no sensitivity analysis.
Central bank transition rate lambda = 0.2 per period
Parameter of the three-state Markov chain for the central bank rate; arbitrary.
Projection grid X_F = 15 by 5 grid
The epsilon-net used for the mean-field measure projection in Section 5.1; its resolution is chosen without error estimates.
Network hyperparameters L, N, M, B, learning rate = 256, 100, 400, 240, 0.001
Chosen for the single reported experiment; no ablation or sensitivity study.

assumptions (6)

domain assumption Minor banks are homogeneous and infinitely many, so their aggregate state is a probability measure mu_t.
Section 3.2 adopts the mean-field limit; this is a standard modeling assumption but is not justified by bank market data.
domain assumption The proportion dynamics in equation (1) with escape rates kappa and viscosity thresholds delta describe client switching between banks.
Section 3.1 posits this dynamics without empirical calibration or comparison to actual deposit flow data.
ad hoc to paper The mean-field flow remains supported on a fixed compact set for all times, so the epsilon-net projection A is a valid approximation.
Section 4 relies on boundedness of b0 and b to justify projecting mu_t onto X_F; no quantitative error bound is given.
ad hoc to paper A neural network with near-zero Bellman residual computed against its own target approximates the true Q-functions and hence a Nash equilibrium.
Section 5.2 uses decreasing Bellman loss as evidence of convergence, but no theorem or unilateral deviation check connects this residual to a Nash equilibrium.
ad hoc to paper Impulse controls in continuous time can be treated as regular controls in discrete time with a fixed cost on rate changes.
Section 3.3 states that impulse controls are 'treated similarly to regular controls'; this changes the timing problem into a per-period rate selection problem.
standard math Standard dynamic programming and measure-theoretic background for Bellman equations and Poisson random measures.
Used for equations (2), (7), (8), (9); not proved in the paper.

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Pith. "Pith review of Deciding Bank Interest Rates -- A Major-Minor Impulse Control Mean-Field Game Perspective." pith.science (2026). https://pith.science/paper/I7WYJWC3

@misc{pith2026241114481,
  author       = {Pith},
  title        = {Pith review of: Deciding Bank Interest Rates -- A Major-Minor Impulse Control Mean-Field Game Perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7WYJWC3}},
  note         = {Machine review of arXiv:2411.14481}
}

read the original 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

Figures reproduced from arXiv: 2411.14481 by the authors.

**Figure 2.** Average losses of 10 individual trainings (shadowed) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 4.** The evolution of mean field measure (minor players). [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Yves Achdou, Jiequn Han, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll. 2022. Income and wealth distribution in macroeconomics: A continuous- time approach. The review of economic studies 89, 1 (2022), 45–86

work page 2022

[2] [2]

René Aïd, Matteo Basei, Giorgia Callegaro, Luciano Campi, and Tiziano Vargi- olu. 2020. Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications. Mathematics of Operations Research 45, 1 (2020), 205–232

work page 2020

[3] [3]

LHR Alvarez. 2001. Stochastic forest growth and Faustmann’s formula. Turku School of Economics and Business Administration, mimeo (2001). 1Disclaimer. The authors’ views are their own. This paper was prepared for infor- mation purposes and is not a product of any Research Department. The authors’ employers past and present make no representation and warra...

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