Cooperation, Correlation and Competition in Ergodic N-player Games and Mean-field Games of Singular Controls: A Case Study

Paper: arXiv 2404.15079

Authors: Federico Cannerozzi, Giorgio Ferrari

Abstract

We consider a class of $N$-player games and mean-field games of singular controls with ergodic performance criterion, providing a benchmark case for irreversible investment games featuring mean-field interaction and strategic complementarities. The state of each player follows a geometric Brownian motion, controlled additively through a nondecreasing process, while agents seek to maximize a long-term average reward functional with a power-type instantaneous profit, under strategic complementarity. We explore three different notions of optimality, which, in the mean-field limit, correspond to the mean-field control solution, mean-field coarse correlated equilibria, and mean-field Nash equilibria. We explicitly compute equilibria in the three cases and compare them numerically, in terms of yielded payoffs and existence conditions. Finally, we show that the mean-field control and mean-field equilibria can approximate the cooperative and competitive equilibria, respectively, in the corresponding $N$-player game when $N$ is sufficiently large. Our analysis of the mean-field control problem features a novel Lagrange multiplier approach, which proves crucial in establishing the approximation result, while the treatment of mean-field coarse correlated equilibria necessitates a new, specifically tailored definition for the stationary setting.

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents an exceptionally rigorous mathematical framework for N-player and mean-field games of singular control with ergodic criteria, establishing explicit equilibria and new analytical tools like a Lagrange multiplier approach. However, it is purely theoretical and numerical, lacking empirical market data or real-world backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze Ergodic N-Player & Mean-Field Games of Singular Control] --> B[Model Inputs: Geometric Brownian Motion & Power-Type Profit with Complementarity]
    B --> C[Methodology: Apply Novel Lagrange Multiplier & Tailored Stationary CCE Definitions]
    C --> D[Solve 3 Optimality Concepts: Mean-Field Control, Correlated Equilibria, Nash Equilibria]
    D --> E[Compute & Compare Explicit Equilibria, Payoffs, and Existence Conditions]
    E --> F[Key Outcome: Prove Mean-Field Solutions Approximate N-Player Cooperative & Competitive Equilibria]

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1 of 51 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.