Doubling fundamental-value volatility doubles the competitive spread in Onger and Foscari's market-making game, within any location-scale family. Nobody colludes. Write V = μ + σX with E[X] = 0 and Var(X) = 1, with support inside [0,1]. The midpoint operator maps a proposed mid to the mid implied by its zero-profit quotes. Its fixed points contain every equilibrium mid-price and shift as μ + σ times the unit-volatility fixed points. At any equilibrium μ + σz, the spread equals σ·w₁(z), where w₁ is the standardized variable's spread. The relationship is exactly linear.
This is a comparative static within a model. It supplies the slope for a purely competitive dealer market. Distinguishing volatility from collusion in observed spreads would require market data, which the paper does not use.
Dealers compete by undercutting each other Bertrand-style and price informed flow in the manner of Glosten and Milgrom. The authors target inelastic markets and cite commodity markets: total volume stays unchanged as prices move, while quotes determine trade direction. Each maker earns zero expected profit on each side in equilibrium. Trading between makers and the trader is zero-sum, leaving the trader with no average extraction either.
The evidence is entirely synthetic. Two makers learn quotes for a Gaussian value with mean 0.5, on grids with tick sizes between 0.1 and 0.5.
A trader who knows the value
N ≥ 2 makers quote bids and asks on [0,1]. A single trader observes the realized value v and must trade. She buys at the best ask when v exceeds the mid of the best quotes, and sells at the best bid when v falls below it. Inelastic demand means quotes determine which way she trades, never whether she trades. Makers tied at the best price divide the order equally.
Every fill faces someone who knows the answer. The competitive ask must therefore be E[V | V > m], the average value among the buyers it attracts. The bid must be E[V | V < m]. Each side separately earns zero expected profit; an equilibrium also requires at least two makers at the best ask and two at the best bid. The difference between those conditional means is entirely adverse selection. Inventory is absent from the model and appears among the authors' suggestions for future work. Treating the spread as inventory compensation would add a mechanism they left out.
The mid must also agree with the quotes it generates. The authors call the average of the conditional means H(m), their midpoint operator, and seek its fixed points. Because H is nondecreasing, the Knaster-Tarski theorem establishes that at least one exists. Checking the ends of the support rules out boundary fixed points. Equilibrium requires more, however. A maker could undercut the ask and withdraw its bid, lowering the mid to attract more flow at a thinner margin. The authors show that whenever a profitable deviation exists, a profitable one-sided move of this sort, or its mirror image, exists as well.
They use this result to construct extremal equilibria m₋ and m₊, which bracket all equilibrium mid-prices. Existence requires only a continuous value with bounded support. The appendix gives the full existence proof. The two necessary conditions, zero profit on each side and two makers at each best quote, receive informal arguments as observations.
What does learning converge to?
The learning theorem needs more: a continuous positive density and a′(m) + b′(m) ≤ 2. Here a and b give the zero-profit quotes as functions of the mid. With this condition, the fixed points form the interval [m₋, m₊], and each point in that interval is an equilibrium. The Nash price envelope is the rectangle [b₋, b₊] × [a₋, a₊].
For a grid with tick Δ, every correlated equilibrium keeps market prices within ε_Δ of the rectangle, with ε_Δ → 0. Learners with no internal regret then achieve a time-averaged expected distance of at most ε_Δ. The limits have a prescribed order. T goes to infinity with Δ fixed; Δ shrinks afterward. The paper gives a rate for neither limit.
The authors are explicit: "The envelope is only an outer bound". Their results "do not exclude that learners occasionally play price pairs away from the Nash curve, such as (b₋, a₊), but they cannot settle on one". That last restriction matters. A learner converging to a single profile must end up near the curve. Yet their example combines the lowest equilibrium bid with the highest equilibrium ask. It produces the rectangle's widest spread, exceeding every equilibrium pair whenever m₋ < m₊. The guarantee reduces to a point only with a unique fixed point.
Two learners on a ten-tick grid
The theory allows N ≥ 2 makers. The learning section treats two, with the extension claimed in a footnote. For simulations, the authors put Exp3 inside the Blum-Mansour reduction to internal regret and set η = 0.05 for every expert. They use a Gaussian value with mean 0.5, clamped to [0,1].
Implementation raises questions immediately. Clamping creates atoms at 0 and 1 despite the model's assumption of a continuous V. The authors describe this mass as negligible. At σ = 0.20, however, the boundaries are only 2.5 standard deviations away. We did not find a check that this clamped Gaussian meets a′ + b′ ≤ 2, the condition needed for convergence. Tie allocation also matters. The authors defend equal splitting as having the same expectation as a uniform random pick, with less variance; many venues instead use price-time priority.
In the tick sweep, σ = 0.1 stays fixed while Δ takes values 0.50, 0.33, 0.25, 0.20 and 0.14. Each setting gets 100,000 rounds and 100 runs. Even the finest grid has only seven ticks across the support. Quotes approach the envelope, though non-monotonically.
For the volatility sweep, Δ = 0.1 gives ten ticks. The authors vary σ from 0.05 to 0.20 and use 1,500,000 rounds over 15 runs. Learned spreads increase almost linearly. According to the authors' own figure caption, they also remain consistently above theory. The paper offers neither an explanation nor a size for that gap. We did not find its size reported numerically anywhere.
The paper's grid bounds suggest a candidate explanation. Under Lemma 21, the market ask in any grid correlated equilibrium can fall one tick below the zero-profit response to its extremal bids, or rise as much as C₀ ticks above it. The bid can fall up to C₀ ticks below its own response. Both allowances widen the spread, matching the direction of the simulation bias.
A run at Δ = 0.01 with a table of the gap would resolve more than another figure at Δ = 0.1.
Volatility and the pattern a collusion test flags
The introduction says spread widenings "can arise purely from competitive risk adjustments, rather than algorithmic collusion". By the conclusion, this becomes advice: collusion tests "should control for volatility". The authors themselves acknowledge the difficulty when volatility has to be estimated.
The case remains conceptual. We did not find a price-based detector of the sort Hartline and coauthors propose tested on these simulated markets. Such tests are designed to flag simultaneous widening across makers. A competitive mechanism producing that pattern gives them reason to control for volatility. It establishes neither that a detector fails nor the size of any failure.
The envelope's corner makes the concern sharper. The authors leave open occasional learner visits to (b₋, a₊). Whenever m₋ < m₊, that pair would produce a spread wider than every equilibrium pair without collusion. Our earlier note on a Hawkes-structured simulator criticized a setup without adverse selection. In this model, adverse selection supplies the entire spread.
We could not test the mechanism on our data. It requires competing quotes and tie allocation, plus counterfactual fills as quotes change and the realized fundamental value the trader compares with the mid. One-minute OHLCV bars provide none of the three. A synthetic rerun would merely test the model against itself.
The paper leaves a clean benchmark: competitive spreads within location-scale families equal σ·w₁(z), with the result proved for any N ≥ 2. Learning carries a narrower guarantee. It covers two makers subject to a regularity condition, applies only as the tick shrinks, and reaches only a rectangle.