Arbitrage depth tells a market maker how much half-spread it needs to avoid being picked off. Cotton defines it as the worst sure gain an arbitrageur can extract per unit traded from the cost schedule. That is the paper's most useful quantity. The remaining results largely establish the floor and work out what the maker gives up to meet it.
What does the chord test require?
A cost-function market maker maintains a potential C over its inventory q of outstanding shares. It charges C(q+s) minus C(q) for a fill of s shares; the slopes of C give the quotes. Standard theory takes C to be convex. Cotton asks how much of that assumption no-arbitrage needs, defining arbitrage here as a fill that makes strictly positive profit in every outcome.
The answer is a condition on chords. Every fill's average price, [C(q+s) minus C(q)]/s, must fall within the convex hull of the security's possible payoffs. For a scalar contract settling in [−1, 1], C need only be 1-Lipschitz. Convexity has a separate source: Cotton derives it from monotone information incorporation, an axiom requiring prices to rise when buyers arrive.
The paper stays entirely within theory, using propositions, lemmas and analytic counterexamples. There are no data or sample period. It directs readers to an external repository for reference implementations and numerical tests. We did not find those results in the paper text.
Sure-profit protection has limits
A non-convex schedule can pass the test for sure-profit trades. Cotton shows how narrow that protection is with C(q) = −α|q|, where α is below 1. The schedule satisfies the chord condition. Yet at q = 0, a trader whose belief is μ = 0 has an infinite expected-profit supremum. Coherence limits the certain extraction from a fill. It leaves a speculator's expected gain uncapped.
An affine minorant changes the picture, though −α|q| has none. When C has one, rational flow trades its convex envelope, the convex schedule beneath C. Optimal fills finish only at points where C meets that envelope. A trader starting elsewhere takes the gap between C and the envelope as well as the envelope profit. Flow therefore jumps across concave stretches of inventory.
Cotton's symmetric double well makes the consequence concrete: moving all the way from −c to c is free, while the half trade from −c to 0 costs αc. The block cannot be divided. With the arctan cost, the envelope is the constant −π/2 and never touches the schedule. At μ = 0, inventory must tend to −∞ to approach the supremum π/2.
Rational flow's use of the envelope explains the abstract's claim that non-convexity costs expressiveness rather than soundness. Cotton is right on soundness. The paper makes no argument that the maker earns more from a non-convex schedule. Its purpose is to make coherent markets out of non-convex predictors, such as a trained model's loss. Lost expressiveness is the price.
Paying the arbitrage depth
A proportional fee f ≥ ε restores no-arbitrage when chord slopes stray outside the hull by at most ε per unit. Quotes become m − f on the bid and m + f on the ask, giving a spread of 2f. Lemma 5 identifies the beliefs for which trading is unprofitable under any cost and any fee f ≥ 0. The lower boundary is the highest average sell price minus f; the upper boundary is the lowest average buy price plus f. Intersect that interval with the payoff hull to obtain the admissible no-trade beliefs. For chord-coherent C, this intersection is non-empty exactly when 2f spans the gap between the chord extremes.
The resulting region need not match ±f around the marginal quote. At a convex kink, its boundaries are the one-sided slopes. The kink thus contributes a spread of its own even when f = 0. Take C(q) = a·sin q at q = 0: a trader with belief μ = 0 stops exactly when f ≥ a. Without a fee, every belief profits at that state. A sufficiently wide spread allows the maker to stay where frictionless rational flow would jump away.
Intersecting with the hull matters.
Without fees, C(q) = 100q gives a no-trade interval consisting only of {100}, well outside [−1, 1]. Every admissible belief wants to trade. The maker can be arbitraged at every state.
Cotton is explicit about what the fee buys:
> "A venue quoting a wide spread to cover a deeply incoherent cost is arbitrage-free but uninformative in proportion."
In his account, the market prices the model's incoherence as uncertainty. He describes the headline equivalence as "close to definitional": a coherent implementation exists if and only if arbitrage depth does not exceed the permitted friction. He bases the result's value on the finiteness of that depth for costs excluded by every axiom set in the literature. Propositions 1 to 3 then establish what that finiteness costs.
Pooling needs an internal clearing trade first
Cotton credits Bhaskara et al. (2023) with the result that pooling parallel makers gives an infimal convolution. He adds fees to the costs and develops the resulting execution rules. This section assumes differentiable, strictly convex costs for each pooled maker; supply curves otherwise become set-valued. It also assumes the order has a clearing price.
Before quoting the pool, the books must be netted against each other. Consider two quadratic makers with zero fees and different marginal quotes. At zero demand, their merged cost is −(m1 − m2)²/(2(1/λ1 + 1/λ2)), a negative amount. It represents a free internal cross-trade. The pool must clear it first.
Once netted, each maker supplies according to its inverse marginal cost, shifted by the fee. Supply stays at zero within the maker's band. Adding these curves leaves one monotone equation to solve for the clearing price, regardless of how many makers participate. A maker trades exactly zero if its band strictly contains that price.
Small orders execute solely at the best quote unless that quote is tied. As orders grow, execution reaches other makers in quote-price order. This follows fee order only when marginal prices coincide. The combined market reads like a consolidated limit order book.
For local depth, S′(p) = Σ 1/C″_i over the active makers. This expression requires C² costs with C″ > 0 and a price p away from band edges. The quartic example shows why the conditions matter: Cotton's maker at q = −1 supplies (p/4)^{1/3} + 1, which is not differentiable at p = 0. Fees remain given throughout this analysis. He expressly leaves unsolved the game in which makers choose them.
More liquidity leaves a gap
Adding a quadratic co-quoter with liquidity λ makes the merged cost a Moreau envelope, smoothing the cost through a quadratic penalty. The chord bounds hold at every λ. For the double well, however, the midpoint price jump remains 2α whenever λ is below c/α. Beyond that depth, the jump becomes 2c/λ.
The midpoint cost is αc − λα²/2 in the first region and c²/(2λ) thereafter. It stays positive at every finite depth. Up to c/α, the gap falls linearly with λ; beyond it, the decline follows 1/λ. No finite depth closes the excluded range.
An LP backing a non-convex engine with a deep quadratic pool therefore buys no reduction in the jump until the co-quoter's liquidity λ exceeds c/α. The midpoint gap still declines as λ increases.
The cash result comes from inconsistent cross-venue quotes
We could not backtest these results. Testing requires each maker's cost schedule and live inventory. Historical price bars cannot reconstruct an executable schedule from closing quotes. Synthetic maker simulations could verify the algebra, though they would supply no trading record.
The cross-venue result gets closer to cash. Quotes on products x_i·x_j can be arranged as a unit-diagonal matrix. If that matrix is not positive semidefinite, the eigenvector bundle earns a sure |λ_min|. This profit requires fixed linear prices that can be executed in the traded size. For a nonlinear maker, the paper gives a local profit whose rate is limited.
Cotton cites Saguillo et al. (2025), who measure $39.6M captured by Polymarket arbitrageurs enforcing logical coherence. That inconsistency differs from the PSD case. He identifies the missing link himself: there is still no map from lumpy endpoint jumps to something observable in a quoted book. Testing venues with non-convex liquidity would require that map first.