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On Convergence Rates of General $N$-Player Stackelberg Games to their Mean Field Limits

Alain Bensoussan, Ziyu Huang, Sheng Wang, Sheung Chi Phillip Yam

TL;DR

This work analyzes convergence rates from a general N-player Stackelberg game to its mean-field limit, incorporating time delays, empirical distribution interactions, and diffusion coefficients that depend on the distribution. It introduces a Wasserstein- and regular-conditional-distribution framework to manage delays and mixture empirical measures, and derives explicit rates that depend on the follower state dimension $n_1$ and moment order $q>4$, with faster rates in several subcases. The main result shows that the limiting mean-field Stackelberg solution yields an approximate Stackelberg Nash equilibrium for the finite-N game, with rates of the form $ ext{O}ig((f(N-1))^{ rac{q-2}{3q-4}}ig)$ and notable improvements in diffusion-independent or discretized-delay scenarios, including $ ext{O}(1/ oot 2N)$ in certain linear settings. By extending mean-field Stackelberg theory to nonstandard features such as delays, distribution-based interactions, and control-dependent diffusion, the paper provides rigorous, practically relevant convergence guarantees for large-scale hierarchical decision problems in economics, engineering, and finance.

Abstract

In this article, we establish precise convergence rates of a general class of $N$-Player Stackelberg games to their mean field limits, which allows the response time delay of information, empirical distribution based interactions, and the control-dependent diffusion coefficients. All these features makes our problem nonstandard, barely been touched in the literature, and they complicate the analysis and therefore reduce the convergence rate. We first justify the same convergence rate for both the followers and the leader. Specifically, for the most general case, the convergence rate is shown to be $\mathcal{O}\left(N^{-\frac{2(q-2)}{n_1(3q-4)}}\right)$ when $n_1>4$ where $n_1$ is the dimension of the follower's state, and $q$ is the order of the integration of the initial; and this rate has yet been shown in the literature, to the best of our knowledge. Moreover, by classifying cases according to the state dimension $n_1$, the nature of the delay, and the assumptions of the coefficients, we provide several subcases where faster convergence rates can be obtained; for instance the $\mathcal{O}\left(N^{-\frac{2}{3n_1}}\right)$-convergence when the diffusion coefficients are independent of control variable. Our result extends the standard $o(1)$-convergence result for the mean field Stackelberg games in the literature, together with the $\mathcal{O}(N^{-\frac{1}{n_1+4}})$-convergence for the mean field games with major and minor players. We also discuss the special case where our coefficients are linear in distribution argument while nonlinear in state and control arguments, and we establish an $\mathcal{O}(1/\sqrt{N})$ convergence rate, which extends the linear quadratic cases in the literature.

On Convergence Rates of General $N$-Player Stackelberg Games to their Mean Field Limits

TL;DR

This work analyzes convergence rates from a general N-player Stackelberg game to its mean-field limit, incorporating time delays, empirical distribution interactions, and diffusion coefficients that depend on the distribution. It introduces a Wasserstein- and regular-conditional-distribution framework to manage delays and mixture empirical measures, and derives explicit rates that depend on the follower state dimension and moment order , with faster rates in several subcases. The main result shows that the limiting mean-field Stackelberg solution yields an approximate Stackelberg Nash equilibrium for the finite-N game, with rates of the form and notable improvements in diffusion-independent or discretized-delay scenarios, including in certain linear settings. By extending mean-field Stackelberg theory to nonstandard features such as delays, distribution-based interactions, and control-dependent diffusion, the paper provides rigorous, practically relevant convergence guarantees for large-scale hierarchical decision problems in economics, engineering, and finance.

Abstract

In this article, we establish precise convergence rates of a general class of -Player Stackelberg games to their mean field limits, which allows the response time delay of information, empirical distribution based interactions, and the control-dependent diffusion coefficients. All these features makes our problem nonstandard, barely been touched in the literature, and they complicate the analysis and therefore reduce the convergence rate. We first justify the same convergence rate for both the followers and the leader. Specifically, for the most general case, the convergence rate is shown to be when where is the dimension of the follower's state, and is the order of the integration of the initial; and this rate has yet been shown in the literature, to the best of our knowledge. Moreover, by classifying cases according to the state dimension , the nature of the delay, and the assumptions of the coefficients, we provide several subcases where faster convergence rates can be obtained; for instance the -convergence when the diffusion coefficients are independent of control variable. Our result extends the standard -convergence result for the mean field Stackelberg games in the literature, together with the -convergence for the mean field games with major and minor players. We also discuss the special case where our coefficients are linear in distribution argument while nonlinear in state and control arguments, and we establish an convergence rate, which extends the linear quadratic cases in the literature.

Paper Structure

This paper contains 26 sections, 15 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Wasserstein space and some properties
  4. Problem formulation
  5. $N$-player Stackelberg game
  6. Limiting mean field Stackelberg game problem
  7. Assumptions and Estimates for Controlled SDEs in Section \ref{['sec:formulation']}
  8. Convergence of an $N$-player system
  9. Convergence for state process under optimal control
  10. $\Delta\equiv a\equiv b$
  11. Discrete random variable $\Delta$
  12. General $\Delta$
  13. Convergence for state process under arbitrary control
  14. A class of sub-cases with standard convergence rate
  15. Approximate Stackelberg Equilibrium
  16. ...and 11 more sections

Key Result

Lemma 2.1

Let $\left(X_k\right)_{k \geqslant 1}$ be a sequence of i.i.d. random variables in $\mathbb{R}^{n_1}$ with a common distribution $\mu \in \mathcal{P}_q\left(\mathbb{R}^{n_1}\right)$ for some $q>4$ , then, for each dimension $n_1 \geqslant 1$, there exists an universal constant $C=C\left(n_1, q\right where $\delta_x$ is the Dirac measure with a unit mass at $x$, and

Theorems & Definitions (33)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 1
  • Remark 2.6
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • Remark 3.4
  • ...and 23 more