Equilibria and Learning in Competitive Market Making
Onger and Foscari analyze competitive market making as a two-sided pricing game where strategic liquidity providers face a non-strategic informed trader with inelastic demand. Because bid and ask quotes jointly set the market mid-price that determines trade direction, standard equilibrium existence theorems fail and payoffs are non-convex and discontinuous. The authors prove that pure-strategy Nash equilibria exist via a novel fixed-point argument on a midpoint operator, derive a structural linear scaling of spreads with asset volatility, and show that no-internal-regret learning on fine price grids recovers the competitive benchmark as tick size vanishes.
The paper delivers an elegant theoretical bridge between Bertrand competition and market microstructure, proving PSNE existence through a novel fixed-point argument on the midpoint operator $H(m)$ and deriving a sharp linear volatility-spread scaling law. However, the model assumes a non-strategic trader and perfectly inelastic demand, while the learning guarantee relies on an unmotivated regularity condition $a'(m)+b'(m)\leq 2$; additionally, the simulation details are not present in the provided text, leaving empirical claims unverified.
The strongest element is the equilibrium existence proof in continuous quote spaces where standard theorems fail. The authors define the midpoint operator $H(m)\coloneq(\mathbb{E}[V\mid V>m]+\mathbb{E}[V\mid V
The model assumes a single representative trader who is fully informed and non-strategic, eliminating the classic adverse-selection and inventory-risk channels that dominate market-microstructure theory. The authors acknowledge that 'quotes determine the direction of trade rather than its occurrence,' but the zero-profit benchmark may be fragile once traders react strategically or demand becomes elastic. Additionally, the learning guarantee in Section 4 hinges on the regularity condition $a'(m)+b'(m)\leq 2$, which bounds the midpoint operator's slope; while sufficient for concentration of grid correlated equilibria, it is merely assumed rather than derived from primitive distributional properties, and the paper does not discuss whether it holds for common families such as uniforms or truncated normals.
The theoretical claims are internally consistent and the proofs are rigorous given the model's assumptions. The volatility-spread scaling law is derived cleanly from location-scale families and contrasted with information-based explanations. However, the paper positions itself against algorithmic collusion tests by Hartline et al., arguing volatility-driven widening is a 'benign, competitive source' of correlated price movement, yet this discussion is speculative without an empirical counterpart. The learning comparison to Stoltz and Lugosi is fair, but the convergence result is asymptotic in tick size $\Delta$ and does not characterize finite-grid rates, leaving a gap relative to practical market tick sizes.
Reproduction relies entirely on the theoretical framework: the model is fully specified in Section 2, the payoff functions are explicit, and the equilibrium characterization reduces to finding fixed points of $H(m)$. However, the paper does not mention code availability, and the simulation section is truncated in the provided text, so hyperparameters, learner algorithms, and grid sizes remain unknown. Independent reproduction of the learning experiments would require the missing experimental details from Section 5, without which the empirical validation claim cannot be assessed.
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.
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