{"id":"7c7062dd-f38a-4daa-8198-964aa9202512","arxiv_id":"2509.10094","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"In a two-exchange shared order book model, incentive provision by one exchange improves liquidity for both, creating a free-rider problem that the paper claims can eliminate incentives entirely.","lead":"Two exchanges sharing one order book can each pay their market maker to tighten quotes, but the benefits spill over to the rival exchange. The paper argues this makes incentive provision a public good that may collapse to zero in equilibrium, which would undermine shared order book regulation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The no-incentive Nash equilibrium is not derived and is contradicted by the paper's own Figure 2(d), where each exchange does better by offering a contract against either rival action.","rationale":"I read the paper as attempting to show that shared order books create a public-good/free-rider problem so that exchanges may end up offering no market-making incentives. The mathematical engine (exponential intensities, CARA principal-agent contracts, PDE verification) is substantial and appears coherent: Lemmas 2.3 and 2.4 establish well-posedness of the PDE systems, and Theorems 2.2 and 2.3 provide admissible optimal contracts for the unilateral and bilateral cases. Credit is due for those. However, the economic headline requires an equilibrium analysis of the two exchanges' binary contract-choice game, and none is given. The assertion in Section 3.1 is an informal extrapolation from the spillover observation, not a solved game. Worse, the paper's own Figure 2(d) supplies payoffs under which offering a contract is a strictly dominant strategy, so the no-incentive equilibrium cannot be an equilibrium of the model as parameterized. This is the load-bearing issue: even a perfect execution-rule model would not rescue the conclusion unless the payoff matrix is changed. The reader's concern about the partial-execution parameter beta is legitimate for external validity, but the internal contradiction is more direct. I therefore agree with the REJECT verdict, though via a different emphasis.","tokens_in":34135,"tokens_out":5479,"duration_ms":47775,"concrete_test":"Re-run the numerical solver for the three regimes with Table 1 parameters and produce the 2x2 payoff matrix for each exchange at q0=q1=0, gamma0=gamma1=gamma (e.g., gamma=0.3): entries (No,No), (Yes,No), (No,Yes), and (Yes,Yes). Then compute best responses. If, as Figure 2(d) indicates, each exchange's payoff is higher under 'offer' for both possible actions of the rival, the claimed no-incentive equilibrium does not exist. A minimal version: check whether (No,No) is a mutual best response; if not, Section 3.1's central claim is refuted by the model's own output.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires solving the discrete contract-choice game in which each exchange decides whether to offer an incentive contract, given the other's decision. The paper never computes this game; Section 3.1 simply asserts the free-rider conclusion and states that 'the Nash equilibrium results in no incentive provision by any exchange.' The numerical results it presents contradict that assertion. In Figure 2(d), with a common risk aversion of the market makers, the no-contract regime gives each exchange a utility around -30, the unilateral-contract regime gives the active exchange about -13 and the passive exchange about -13, and the two-contract regime gives each about -12. Thus, against a rival that does not offer, offering is strictly better (-13 > -30); against a rival that offers, offering is also strictly better (-12 > -13). 'Offer' is a strictly dominant strategy, so (no, no) is not a Nash equilibrium; the unique pure-strategy Nash equilibrium is (offer, offer). This is not a subtle calibration issue: the paper's own numbers overturn the headline result. Moreover, the abstract's claim of existence and uniqueness of the Nash equilibrium between exchanges is not established anywhere; Theorems 2.2 and 2.3 are verification theorems for optimal contracts conditional on the rival's contract, and Lemma 2.4 gives uniqueness of the PDE solution, not of the exchange-level equilibrium.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":34460,"tokens_out":8211,"duration_ms":66866,"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":[{"comment":"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":"Section 3.1, Figure 2(d)"},{"comment":"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":"Section 3.1; Theorems 2.2, 2.3; Lemma 2.4"},{"comment":"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.","section":"Section 2.1, beta-execution rule"}],"minor_comments":[{"comment":"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.","section":"Section 2.1, notation before Lemma 2.1"},{"comment":"The proof contains an unresolved citation placeholder '(?, Proposition VI.I.I)' that should be replaced with a proper reference to a martingale convergence theorem.","section":"Appendix A.2, proof of Lemma A.1"},{"comment":"The entry for A0,A1 is printed as '10010^-5'; please clarify the intended value and its units, as this is ambiguous.","section":"Table 1"},{"comment":"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.","section":"Figures 2 and 3 captions"}],"recommendation":"reject","confidential_remarks":"The mathematical framework (Lemma 2.1, Theorems 2.2-2.3, Lemma 2.4) appears technically serious and could support a future revised paper, but the central economic claim as stated is falsified by the paper's own numerical results: the reported utilities make incentive provision a strictly dominant strategy. A revision that merely corrects the conclusion would change the paper's main contribution, and the current version's abstract, Section 3.1, and policy discussion are not consistent with the reported findings. I therefore recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe 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.\n\nWhat 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.\n\nThe 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.\n\nThe 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.\n\nWhat 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.\n\nWho 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":34948,"tokens_out":3767,"would_cite":false,"duration_ms":31650,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A80","91G80","60H30","93E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two exchanges sharing one order book can end up with neither offering market-making incentives.","keywords":["make-take fees","market making","shared order book","free-rider problem","principal-agent problem","stochastic control","intraday electricity markets","financial regulation"],"falsifier":"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.","tokens_in":33896,"feed_emoji":"📉","tokens_out":9663,"duration_ms":85791,"temperature":0.7,"pith_summary":"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.","feed_headline":"Shared order books can push both exchanges to offer no incentives","feed_subtitle":"One exchange's market-making bonus helps the rival too, so in equilibrium nobody pays for liquidity.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the exponential order-arrival intensity that links quoted spread to fill rates.","marker":"Avellaneda and Stoikov (2008)"},{"why":"is the single-market-maker principal-agent benchmark whose optimal make-take fee contract is extended here to two competing exchanges.","marker":"El Euch et al. (2021)"},{"why":"provides the multi-market-maker contract framework and the PDE structure used for the exchanges' optimisation problems.","marker":"Baldacci et al. (2021)"},{"why":"frames competition between exchanges as the research agenda the paper contributes to.","marker":"Cantillon and Yin (2011)"},{"why":"is the baseline for how competition between trading venues shapes fees and liquidity, and is contrasted with the shared-order-book setting.","marker":"Colliard and Foucault (2012)"},{"why":"provides the continuous-time principal-agent foundations behind the contract representation.","marker":"Sannikov (2008)"},{"why":"supplies the interacting-agents contracting theory used for the Nash equilibrium between the two market makers.","marker":"Elie and Possamaï (2019)"}],"fun_headline_variants":["Shared order books turn incentives into a public good","Free-riding exchanges starve shared order books of incentives","Why shared order books end with zero incentives","In shared order books, nobody pays for liquidity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Shared order books turn incentives into a public good","Free-riding exchanges starve shared order books of incentives","Why shared order books end with zero incentives","In shared order books, nobody pays for liquidity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"total_tokens":1389,"prompt_tokens":916,"completion_tokens":473,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":413}},"tokens_in":532,"tokens_out":473,"duration_ms":4340,"temperature":1.0,"reasoning_tokens":413,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:58:04.336023+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}