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Zero-Sum Games between Large-Population Teams: Reachability-based Analysis under Mean-Field Sharing

Yue Guan, Mohammad Afshari, Panagiotis Tsiotras

TL;DR

The paper tackles zero-sum mean-field team games with two large competing teams whose intra-team dynamics are cooperative. It introduces a mean-field approximation and a common-information-based coordinator game to compute decentralized strategies that are ε-optimal in the original finite-population game. The core achievements include an explicit suboptimality bound of order O(1/√N), Lipschitz continuity of the coordinator values, and a reachability-based analysis that links finite-population behavior to the infinite-population limit. Numerical experiments validate the theoretical guarantees and illustrate the coordinator game’s value behavior under different scenarios. The work provides a scalable framework for analyzing and solving large-scale competitive multi-agent systems with mean-field interactions and shared information structures.

Abstract

This work studies the behaviors of two large-population teams competing in a discrete environment. The team-level interactions are modeled as a zero-sum game while the agent dynamics within each team is formulated as a collaborative mean-field team problem. Drawing inspiration from the mean-field literature, we first approximate the large-population team game with its infinite-population limit. Subsequently, we construct a fictitious centralized system and transform the infinite-population game to an equivalent zero-sum game between two coordinators. We study the optimal coordination strategies for each team via a novel reachability analysis and later translate them back to decentralized strategies that the original agents deploy. We prove that the strategies are $ε$-optimal for the original finite-population team game, and we further show that the suboptimality diminishes when team size approaches infinity. The theoretical guarantees are verified by numerical examples.

Zero-Sum Games between Large-Population Teams: Reachability-based Analysis under Mean-Field Sharing

TL;DR

The paper tackles zero-sum mean-field team games with two large competing teams whose intra-team dynamics are cooperative. It introduces a mean-field approximation and a common-information-based coordinator game to compute decentralized strategies that are ε-optimal in the original finite-population game. The core achievements include an explicit suboptimality bound of order O(1/√N), Lipschitz continuity of the coordinator values, and a reachability-based analysis that links finite-population behavior to the infinite-population limit. Numerical experiments validate the theoretical guarantees and illustrate the coordinator game’s value behavior under different scenarios. The work provides a scalable framework for analyzing and solving large-scale competitive multi-agent systems with mean-field interactions and shared information structures.

Abstract

This work studies the behaviors of two large-population teams competing in a discrete environment. The team-level interactions are modeled as a zero-sum game while the agent dynamics within each team is formulated as a collaborative mean-field team problem. Drawing inspiration from the mean-field literature, we first approximate the large-population team game with its infinite-population limit. Subsequently, we construct a fictitious centralized system and transform the infinite-population game to an equivalent zero-sum game between two coordinators. We study the optimal coordination strategies for each team via a novel reachability analysis and later translate them back to decentralized strategies that the original agents deploy. We prove that the strategies are -optimal for the original finite-population team game, and we further show that the suboptimality diminishes when team size approaches infinity. The theoretical guarantees are verified by numerical examples.
Paper Structure (50 sections, 28 theorems, 175 equations, 10 figures, 1 table)

This paper contains 50 sections, 28 theorems, 175 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Work
  4. Mean-field games.
  5. Mean-field teams.
  6. Markov team games and information structure.
  7. Common-information approach.
  8. Road Map
  9. Notations
  10. Problem Formulation
  11. Dynamics
  12. Reward Structure
  13. Information Structure.
  14. Optimization Problem.
  15. Examples
  16. ...and 35 more sections

Key Result

Lemma 1

Let $\mathbf{X}^{N_1}_t$, $\mathbf{Y}^{N_2}_t$, $\mathcal{M}^{N_1}_t$ and $\mathcal{N}^{N_2}_t$ be the joint states and the corresponding EDs of a finite-population game. Denote the next EDs induced by an identical policy pair $(\phi_t, \psi_t) \in \Phi_t \times \Psi_t$ as $(\mathcal{M}_{t+1}^{N_1}, where $\mathcal{M}_{t+1} = \mathcal{M}^{N_1}_{t} F_t^{\rho}(\mathcal{M}_{t}^{N_1}, \mathcal{N}_{t}^

Figures (10)

  • Figure 1: The road map of the proposed approach.
  • Figure 2: An example of ZS-MFTG over a two-node graph, where $N_1=2$, $N_2=2$ and $\rho=0.5$.
  • Figure 3: A schematic of the proposed common-information approach for the mean-field zero-sum team games.
  • Figure 4: An illustration of Lemma \ref{['lmm:mf-apprx-team-policy']}.
  • Figure 5: Subplots (a)-(c) present the game value computed via discretization. The x- and y-axes correspond to $\mu^\rho_t(x^1)$ and $\nu^\rho_t(y^1)$, respectively. Subplot (d) illustrates the reachable sets starting from $\mu_0=[0.96, 0.04]$ and $\nu_0 = [0.04, 0.96]$.
  • ...and 5 more figures

Theorems & Definitions (76)

  • Definition 1: Empirical Distribution
  • Definition 2
  • Remark 1
  • Definition 3
  • Definition 4: Identical Blue Team Strategy
  • Definition 5
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • ...and 66 more