This paper studies liquidity provision in a market with exogenous competition, focusing on a reference market maker who monitors both her own inventory and the aggregated inventory of competing market makers. The competing market makers follow a simple linear rule-of-thumb for their posted depths, depending on their inventory, while the reference market maker optimizes her depths to maximize a linear-quadratic goal functional that includes terminal wealth, inventory penalty, and a running inventory cost. The fill probability for the reference market maker's limit orders depends exponentially on the difference between her depths and the competition's depths, capturing the idea that orders are more likely to be filled if she posts more aggressively than competitors.

The authors derive an approximate closed-form solution for the optimal depths by solving a matrix Riccati ODE, which is computationally efficient and provides insights into the trade-offs faced by the market maker. The solution shows that optimal depths are linear functions of both the reference market maker's inventory and the competitor's inventory, with coefficients depending on model parameters. The paper also compares the approximate solution with numerical solutions obtained via an Euler scheme and a reinforcement learning agent using proximal policy optimization (PPO), finding good agreement for realistic parameter choices.

The model highlights a key trade-off: the reference market maker must balance managing her own inventory risk against the impact of missed trades on future market conditions. When she fails to fill a market order, the competitor fills it, altering his inventory and thus his future depths, which in turn affects her future fill probabilities. This feedback loop makes the problem dynamic and non-trivial. The results show that competition leads to narrower spreads and more aggressive quoting, especially when the competitor's inventory is large. The paper contributes to the market making literature by providing a tractable framework for analyzing competition without resorting to complex stochastic differential games, and it offers practical insights for algorithmic traders operating in competitive electronic markets.