Equilibrium and Learning in Competitive Market Making

This paper proposes a market-making game model involving N≥2 strategic market makers and one non-strategic representative trader. It proves that under minimal assumptions on the underlying asset value distribution, at least one pure-strategy Nash equilibrium always exists, and the equilibrium bid-ask spread widens strictly with the volatility of the asset value. By discretizing prices, a tractable learning framework is constructed, proving that the time-averaged game of no-internal-regret learners approximates the continuous Nash price envelope as the tick size approaches zero. Numerical simulations validate the theoretical results.