Competition and Incentives in a Shared Order Book

Mathieu Rosenbaum; Philippe Bergault; Ren\'e A\"id

REVIEW 3 major objections 4 minor 28 references

Competition and Incentives in a Shared Order Book

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Two exchanges sharing one order book can end up with neither offering market-making incentives.

desk verdict The math is real but the headline result is contradicted by the paper's own Figure 2(d): offering incentives is a dominant strategy in their numbers, so the claimed free-rider equilibrium with no incentives does not exist. read the letter →

arxiv 2509.10094 v1 pith:XPZEZQNZ submitted 2025-09-12 q-fin.TR

René Aïd , Philippe Bergault , Mathieu Rosenbaum This is my paper · ORCID

classification q-fin.TR MSC 91A8091G8060H3093E20

keywords make-takefeesmarketmakingsharedorderbookfree-riderproblemprincipal-agentstochasticcontrolintradayelectricitymarketsfinancialregulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that when two exchanges share a single limit order book, incentive payments from one exchange to its own market maker are a public good: they tighten both market makers' quotes, and the rival exchange benefits without paying. Because providing incentives is costly, this spillover produces a free-rider problem, and the paper's main equilibrium result is that neither exchange offers any incentive in the Nash equilibrium. If right, this means shared-order-book regulation, motivated by European intraday electricity markets, can reduce rather than increase market-making competition. The supporting analysis is a continuous-time principal-agent model with exponential (CARA) utility for all players, proving existence and uniqueness of the Nash equilibrium and characterizing optimal contracts through coupled PDEs.

What carries the argument

The load-bearing object is the map $\Delta(z,q)$, the unique optimal fixed point of the two market makers' Hamiltonians, which converts contract payment rates $z$ and inventories $q$ into equilibrium bid and ask quotes. Around this sits a partial-execution rule: a market maker whose quote is not the best can still be filled, but the trade pays a penalty factor $\beta\in(0,1)$, so one market maker's tighter quote pulls order flow away from the other and forces a competitive response. The paper represents every admissible contract as a terminal value of a controlled process indexed by order arrivals and the asset price; this representation converts each exchange's optimisation into a coupled Hamilton-Jacobi-Bellman PDE system whose unique solution yields the optimal contracts and the exchanges' certainty equivalents.

What would settle it

Recompute the exchanges' Nash equilibrium under strict price-time priority, where a non-best quote never executes at all; if the unique equilibrium still has zero incentives from both exchanges, the free-rider conclusion is robust to the execution rule, and if it has positive incentives, the paper's main result is an artefact of its partial-execution assumption.

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Extended reading notes

Core claim

The paper's central claim is that incentive provision by an exchange to its dedicated market maker is a public good in a shared order book. An optimal contract paid by $E_0$ makes $M_0$ quote tighter spreads; the extra order flow then increases competitive pressure on $M_1$, who tightens his own quotes even though $E_1$ pays nothing. The paper shows the resulting game between exchanges has a free-rider structure and that the unique Nash equilibrium is the absence of any incentive contract by either platform. On the mathematical side, it establishes existence and uniqueness of the equilibrium, derives the optimal contracts from the unique solution of a system of coupled PDEs for the exchanges' certainty equivalents, and numerically illustrates that almost all the liquidity gain comes from the first incentive contract.

Load-bearing premise

The argument depends on the book's execution rule in which a non-best quote can still capture trades with pay scaled down by $\beta$, a rule the authors themselves note never occurs in a real limit order book; it also assumes each exchange's market maker is permanently dedicated to that exchange.

Editorial extensions

If this is right

A unilateral incentive contract by one exchange raises the value of the passive exchange without any cost to it, because both market makers end up quoting tighter spreads.
In the unique Nash equilibrium of the two-exchange game, neither platform offers incentives, so the shared order book can suppress liquidity provision instead of encouraging it.
The numerical results show that almost all of the liquidity improvement comes from the first incentive contract; a second contract adds only marginal value.
Policy evaluation of shared-order-book regulation should treat market-making incentives as a public good and expect free-riding between exchanges.
The equilibrium contracts and value functions are computable from a coupled PDE system, so the effects of parameter changes such as volatility, risk aversion, and connection efficiency can be quantified.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If real shared books enforce strict price priority so that only the best quote executes, the spillover channel weakens or disappears; the free-rider equilibrium is contingent on the paper's partial-execution rule.
The same public-good logic likely applies to other venues where one platform's liquidity is visible to another's users, such as decentralised-exchange aggregators; the authors mention the analogy but do not model it.
Allowing market makers to switch exchanges or to trade on both venues would break the pure public-good structure, because an exchange could then capture some benefit of its own incentives through the market maker's behaviour on the other venue.
A regulation that requires both exchanges to contribute to a joint liquidity-rebate fund would be a natural remedy suggested by the public-good diagnosis, though the paper does not analyse such mechanisms.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The honest take: the model is real work and the mathematics is mostly sound, but the headline result is contradicted by the paper's own numbers. The free-rider conclusion, as stated, does not hold.

What is new: a two-exchange shared-order-book extension of the El Euch et al. principal-agent make-take fee framework, with two dedicated market makers and a connection parameter beta. That extension is not in the prior literature. The verification theorems and PDE uniqueness results are nontrivial, and the authors are transparent about the modeling simplifications. If you need a tractable model of two venues sharing a book, this is a useful starting point.

The soft spot is load-bearing. Section 3.1 asserts that the Nash equilibrium results in no incentive provision by any exchange. But the binary contract-choice game is never solved. The paper computes three separate regimes—no contract, one contract, two contracts—and compares utilities. Figure 2(d) shows no-contract utility around -30, one-contract utility around -13 for both exchanges, and two-contract utility around -12. So offering is strictly dominant: against a non-offering rival you prefer to offer (-13 > -30), and against an offering rival you also prefer to offer (-12 > -13). The unique pure-strategy Nash equilibrium is (offer, offer). Their own numbers overturn the paper's central claim. The abstract's claim of existence and uniqueness of the Nash equilibrium between exchanges is also not established; Theorems 2.2 and 2.3 are conditional verification theorems, and Lemma 2.4 is uniqueness of a PDE solution.

The partial-execution rule is a second concern. Allowing the non-best quote to capture trades with a beta penalty is a simplification the authors admit never happens in a real limit order book. The spillover result may depend on it. That is worth exploring, but it is secondary to the equilibrium flaw.

What is good: the spillover effect itself—a passive exchange benefits from the competitor's contract—is supported by the numerics. The mathematical machinery is carefully developed. The paper is honest about its assumptions.

Who gets value: anyone building on this class of models or designing shared-order-book incentives. But the policy conclusion should be ignored until the authors actually solve the contract-choice game and find parameter regions where no-incentive is an equilibrium.

Recommendation: this deserves a serious referee only conditionally—the framework is substantial enough that peer review could salvage it with a corrected equilibrium analysis. As submitted, I would not accept.

Referee Report

3 major / 4 minor

Summary. The paper develops a continuous-time principal-agent model of two exchanges sharing a limit order book. Each exchange is the principal of a dedicated market maker, and the exchanges choose whether to offer incentive contracts while market makers choose bid and ask quotes. The authors prove a representation theorem for admissible contracts, characterize the market-maker Nash equilibrium as a fixed point, provide PDE-based verification theorems for optimal contracts in the unilateral and bilateral cases, and report numerical solutions. The advertised economic conclusion is that incentive provision is a public good with a competitiveness spillover and a free-rider problem, so that the Nash equilibrium between exchanges may involve no incentive provision by either platform.

Significance. If the headline result held, it would be a policy-relevant finding: shared-order-book regulation could reduce, rather than increase, market-making incentives. The mathematical apparatus is substantial and partly self-contained: Lemma 2.1 gives an explicit characterization of the market-maker fixed point, Theorems 2.2 and 2.3 provide verification results for the principal problems, and Lemma 2.4 gives uniqueness of the relevant PDE solutions. The contract representation builds on El Euch et al. (2021) and is not circular. However, the central economic claim is not supported by the paper's own numerical results. The utilities reported in Figure 2(d) imply that offering a contract is strictly dominant for each exchange, so the asserted no-incentive Nash equilibrium in Section 3.1 is contradicted by the paper's own numbers. The mathematical core may be salvageable, but the paper's main message, abstract, and policy discussion are currently inconsistent with the reported findings.

major comments (3)

[Section 3.1, Figure 2(d)] The no-incentive Nash equilibrium assertion is contradicted by the paper's own numerical results. Figure 2(d) reports utilities of about -30 in the no-contract regime, about -13 for both the active and the passive exchange in the one-contract regime, and about -12 in the two-contract regime over the displayed range of risk-aversion parameters. By symmetry, each exchange compares: offering yields -13 if the rival does not offer and -12 if the rival offers, while not offering yields -30 if the rival does not offer and -13 if the rival offers. Thus 'offer' is strictly dominant and (no, no) is not a Nash equilibrium; the unique pure-strategy equilibrium is (offer, offer). Section 3.1 states flatly that 'the Nash equilibrium results in no incentive provision by any exchange,' and the abstract claims that the strategic interaction 'may lead to an equilibrium in which neither platform offers incentives.' Both statements are inconsistent with the figure the paper itself presents.
[Section 3.1; Theorems 2.2, 2.3; Lemma 2.4] The exchange-level game is never actually solved. The paper asserts in Section 3.1 that the Nash equilibrium results in no incentive provision, but no proposition computes the payoff matrix or the equilibrium of the discrete contract-choice game between the two exchanges. Theorems 2.2 and 2.3 are verification theorems for the optimal contract conditional on the rival's contract, and Lemma 2.4 establishes uniqueness of the PDE solution, not uniqueness of the exchange-level equilibrium. The claimed existence and uniqueness of the Nash equilibrium between exchanges in the abstract is therefore not established. The authors need to formulate the binary participation game, compare the four payoff outcomes, and derive the equilibrium; the current text conflates 'best response conditional on the rival's contract' with 'equilibrium of the game.'
[Section 2.1, beta-execution rule] The beta-penalized partial execution rule is load-bearing for the spillover mechanism, yet the authors acknowledge that it allows a market maker with the largest quote to capture trades at the expense of a more competitive market maker, something that never occurs in an actual limit order book. Because this rule is what generates the shared-execution benefit that drives the public-good and free-rider conclusions, the robustness of those conclusions to strict price priority is not established. A numerical experiment with beta=1 or with an explicit price-priority matching rule should be reported before the policy conclusions can be drawn.

minor comments (4)

[Section 2.1, notation before Lemma 2.1] The notation for the minimum of two quotes is unclear: the text writes 'for d=(di,dj) in R^2, we defined d = min(di,dj)' but then uses the same symbol d for the vector and for the minimum; please introduce separate symbols.
[Appendix A.2, proof of Lemma A.1] The proof contains an unresolved citation placeholder '(?, Proposition VI.I.I)' that should be replaced with a proper reference to a martingale convergence theorem.
[Table 1] The entry for A0,A1 is printed as '10010^-5'; please clarify the intended value and its units, as this is ambiguous.
[Figures 2 and 3 captions] The word 'vignette' is used throughout; please replace it with 'panel' for clarity, and state explicitly in the Figure 2(d) caption that the no-contract curve uses the right axis and the contract curves use the left axis.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation is self-contained; the unproved no-incentive Nash claim is a correctness gap, not a circular step.

full rationale

The paper's mathematical derivation is self-contained. The contract representation (Theorem 2.1), the fixed-point characterization of the market-maker equilibrium (Lemma 2.1), and the PDE verification results (Theorems 2.2 and 2.3, with Lemmas 2.2-2.4) are proved in the text or appendix using standard stochastic-control and Cauchy-Lipschitz arguments. Prior work such as El Euch et al. (2021) and Baldacci et al. (2021) is used as modeling inspiration and technique templates, not as the source of the existence or uniqueness claims. The spillover effect is a direct consequence of the shared-execution intensity structure in Section 2.1 and is not fitted or definitionally imposed. The one serious issue is Section 3.1's assertion 'as the Nash equilibrium results in no incentive provision by any exchange': no exchange-level game with the outside option of offering no contract is actually solved, and the paper's own Figure 2(d) appears to contradict the assertion, since unilateral contracting improves the active exchange's utility from about -30 to about -13 and two contracts improve both exchanges from about -13 to about -12. This is a missing-derivation and correctness gap, not a circular reduction: the claim is not equivalent by construction to any model input, fitted parameter, or self-cited theorem. Therefore the circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The model relies on standard market-making assumptions plus several ad hoc devices, notably the beta partial-execution rule and the equilibrium selection in Assumption 1. The central economic conclusion is demonstrated only on a hand-picked parameter set with no sensitivity analysis.

free parameters (4)

Connection parameter beta = reported as 5 in Table 1, must lie in (0,1)
Introduced ad hoc to model partial execution of non-best quotes; the magnitude of the spillover and the free-rider conclusion depend on it, and it is not calibrated to data.
Order arrival intensity parameters kappa, A_j, c_j = kappa=8, A_j and c_j from Table 1
Shape the exponential order arrival intensity; chosen by hand and not fitted to market data; they determine the quantitative equilibrium outcome.
Risk aversion parameters eta_i, gamma_i = eta=0.01, gamma=0.6 in Table 1
CARA risk aversions of exchanges and market makers; chosen by hand; the utility comparisons in Figure 2(d) depend directly on these values.
Inventory bound q_bar and spread cap delta_infinity = values unclear from Table 1 in the text
Boundary parameters for the inventory constraint and the admissible quote range; chosen by hand for the numerical illustrations.

assumptions (5)

domain assumption Trades arrive as Poisson processes with exponential intensities Lambda_j(d)=A_j exp(-kappa(d+c_j)/sigma) and the fundamental price is an arithmetic Brownian motion.
Standard market-making assumptions in the Avellaneda-Stoikov tradition; the entire equilibrium and contract analysis is carried out under these dynamics (Section 2.1).
domain assumption Each exchange has a dedicated market maker who never switches platform, and liquidity takers interact with a specific exchange.
Stated as a core assumption in Section 1; if market makers could switch venues, the strategic interaction between exchanges would change.
ad hoc to paper A market maker whose quote is not the best can still capture trades, with the payment penalized by factor beta; the authors note this may happen in the model but never occurs in an actual limit order book.
Section 2.1; this partial-execution device is introduced for tractability and is load-bearing for the spillover effect.
ad hoc to paper Assumption 1: market makers always select the optimal fixed point policy Delta(Z_t,Q_t) from Lemma 2.1.
Section 2.1 and 2.2; this equilibrium selection is imposed rather than derived, and it is needed to compute the exchanges' value functions.
standard math Standard stochastic calculus tools (dynamic programming, martingale representation, verification theorems) apply in this setting.
Used throughout the appendices for the contract representation and the PDE verification arguments.

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Cite this review

Pith. "Pith review of Competition and Incentives in a Shared Order Book." pith.science (2026). https://pith.science/paper/XPZEZQNZ

@misc{pith2026250910094,
  author       = {Pith},
  title        = {Pith review of: Competition and Incentives in a Shared Order Book},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XPZEZQNZ}},
  note         = {Machine review of arXiv:2509.10094}
}

read the original abstract

Recent regulation on intraday electricity markets has led to the development of shared order books with the intention to foster competition and increase market liquidity. In this paper, we address the question of the efficiency of such regulations by analysing the situation of two exchanges sharing a single limit order book, i.e. a quote by a market maker can be hit by a trade arriving on the other exchange. We develop a Principal-Agent model where each exchange acts as the Principal of her own market maker acting as her Agent. Exchanges and market makers have all CARA utility functions with potentially different risk-aversion parameters. In terms of mathematical result, we show existence and uniqueness of the resulting Nash equilibrium between exchanges, give the optimal incentive contracts and provide numerical solution to the PDE satisfied by the certainty equivalent of the exchanges. From an economic standpoint, our model demonstrates that incentive provision constitutes a public good. More precisely, it highlights the presence of a competitiveness spillover effect: when one exchange optimally incentivizes its market maker, the competing exchange also reaps indirect benefits. This interdependence gives rise to a free-rider problem. Given that providing incentives entails a cost, the strategic interaction between exchanges may lead to an equilibrium in which neither platform offers incentives -- ultimately resulting in diminished overall competition.

Figures

Figures reproduced from arXiv: 2509.10094 by the authors.

**Figure 2.** Value functions of the exchanges. (a) Value function of Exchange 0 as a function of her inventory q 0 when the inventory q 1 of Exhange 1 is zero (b) Value function of Exchange 1 as a function of her inventory q 1 when the inventory q 0 of Exhange 0 is zero (c) Sum of the value functions of exchanges as a function of q 0 when q 1 = 0; (d) Value function of the exchanges as a function of the risk-aversion parameters … view at source ↗

**Figure 3.** Bid quote of the market makers as a function of [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Optimal bid quote of the market makers as a function of the inventory of market maker [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Aqsha, P

A. Aqsha, P. Bergault, and L. S \'a nchez-Betancourt. Equilibrium Reward for Liquidity Providers in Automated Market Makers. arXiv preprint arXiv:2503.22502, 2025

arXiv 2025

[2] [2]

Avellaneda and S

M. Avellaneda and S. Stoikov. High-frequency trading in a limit order book. Quantitative Finance, 8(3):217–224, 2008

work page 2008

[3] [3]

Baldacci, D

B. Baldacci, D. Possamaï, and M. Rosenbaum. Optimal make-take fees in a multi market-maker environment. SIAM Journal on Financial Mathematics, 12(1):446–486, 2021

work page 2021

[4] [4]

Baldacci and P

B. Baldacci and P. Bergault. Optimal incentives in a limit order book: a SPDE control approach. Market Microstructure and Liquidity, 2024

work page 2024

[5] [5]

Baldacci, I

B. Baldacci, I. Manziuk, T. Mastrolia, and M. Rosenbaum. Market making and incentives design in the presence of a dark pool: a Stackelberg actor--critic approach. Operations Research, 71(2):727–749, 2023

work page 2023

[6] [6]

Baldacci, P

B. Baldacci, P. Bergault, J. Derchu, and M. Rosenbaum. On bid and ask side-specific tick sizes. SIAM Journal on Financial Mathematics, 14(4):1215–1248, 2023

work page 2023

[7] [7]

Baldacci, P

B. Baldacci, P. Jusselin, and M. Rosenbaum. How to design a derivatives market? Peter Carr Gedenkschrift: Research Advances in Mathematical Finance, 657–699, 2024

work page 2024

[8] [8]

Simulating and analyzing a sparse order book: an application to intraday electricity markets

P. Bergault and E. Cognéville. Simulating and analyzing a sparse order book: an application to intraday electricity markets. arXiv preprint arXiv:2410.06839, 2024

work page Pith review arXiv 2024