Paper: arXiv 2610.30891

Authors: Matteo Onger, Luigi Foscari

Abstract

We propose a market-making game played among $N \ge 2$ strategic market makers and a single non-strategic representative trader. In a continuous quote space, standard finite-game arguments do not apply, while non-convex and discontinuous payoffs rule out classical continuous existence results and complicate learning. Nonetheless, we characterize pure-strategy Nash equilibria (PSNEs) and show that at least one PSNE always exists under minimal assumptions on the underlying asset value distribution, and that the equilibrium bid-ask spread widens strictly with asset value volatility. To obtain a tractable learning framework, we discretize prices and, under an additional regularity condition, show that the time-averaged play of no-internal-regret learners approaches the continuous Nash price envelope as the tick size vanishes. Finally, we empirically validate our theoretical results through numerical simulations.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly mathematical game-theoretic model for competitive market making, characterizing Nash equilibria in a continuous quote space. While primarily theoretical, it includes numerical simulations for empirical validation, demonstrating a strong blend of advanced mathematical derivation and practical relevance. The novelty lies in bridging Bertrand-style oligopoly and classic market-making models to analyze competition and learning dynamics in two-sided inelastic markets.

Research Flowchart

  flowchart TD
    A[Research Goal: Characterize Equilibria and Learning in Competitive Market Making] --> B{Methodology: Game Theory & Strategic Interaction};
    B --> C[Model Formulation: N Strategic Market Makers, 1 Non-Strategic Trader, Continuous Quote Space];
    C --> D{Analysis: Characterize Pure-Strategy Nash Equilibria (PSNEs)};
    D --> E[Computational Processes: Discretize Prices for Learning & Numerical Simulations];
    E --> F{Key Findings: PSNE Existence, Spread Widens with Volatility, No-Regret Learning Converges};