{"id":"d762214f-b092-4197-85c2-d754b520f508","arxiv_id":"2411.15727","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"New constant-factor approximation algorithms for revenue-maximizing menus in two-sided matching markets with arbitrary pairwise rewards under MNL choice.","lead":"This paper gives algorithms that earn a guaranteed constant fraction of the best possible revenue in two-sided matching markets where every customer-supplier pair has its own reward and both sides choose according to standard customer-choice models. The result is the first constant-factor approximation for such pairwise rewards, in both a personalized-display model and an all-customers-displayed model.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.5's proof of Property 2 has an algebraic error: the per-customer denominator alpha_{ell j} cannot be factored out of the sum, so the constructed point need not satisfy the required denominator bound; the inclusive-model approximation rests on this step.","rationale":"The reader and I agree on the core problem: the inclusive-model result is not established as written, and the key defect is in Theorem 5.5, whose proof is invalid. I examined the algebraic manipulation and reproduced a concrete instance where the proposed hat{x} violates Property 2. This directly breaks Observation 5.6, the relaxation (RDL_-), and ultimately Theorem 1.2. The reader's stated weakest_assumption was the unstated nonnegativity of rewards; that is a legitimate secondary modeling concern, but the proof error is more immediate and load-bearing, since it invalidates the derivation even under nonnegative rewards. I therefore mark agreement as partial. I also note that the customized-model result (Theorem 1.1) has a coherent 1/3 analysis and appears far more secure; the rejection should focus on the inclusive-model proof gap rather than the whole paper. The proposed concrete test checks both the algebraic error and, via a small instance, whether the structure theorem itself is salvageable or false.","tokens_in":1166,"tokens_out":1035,"duration_ms":75878,"concrete_test":"Independently recompute the proof of Theorem 5.5 Property 2 using the actual definition hat{x}_{ell j} = x*_{ell j} / alpha_{ell j}, keeping alpha_{ell j} inside the summation. Then verify the explicit low-weight instance: one supplier, customers 1-4, w_{ij}=1, x*=(0.1,0.5,0.5,0.4) in P_C. The constructed hat{x} violates Property 2 for customer 1. To determine whether the theorem itself (not just its proof) is false, solve the nonlinear program OPT(RD_-) for a reward vector such as r=(10,6,4,1), and check whether any x in P_C achieves sum r_i x_i >= OPT(RD_-) while satisfying sum_{ell != i} x_{ell j} <= 1 for all i. If no such x exists, Theorem 5.5 is false, and the inclusive-model guarantee is unsupported.","verdict_should_be":"REJECT","load_bearing_attack":"The inclusive-model result Theorem 1.2 depends on Theorem 5.5 through Observation 5.6, which asserts that the linear program (RDL_-) is a relaxation of (RD_-). The proof of Theorem 5.5 is invalid. It defines alpha_{ij} = max{sum_{ell in C_{-i}} w_{ell j} x*_{ell j}, 1} and sets hat{x}_{ij} = x*_{ij} / alpha_{ij}. In proving Property 2, the paper claims sum_{ell in C_{-i}} w_{ell j} hat{x}_{ell j} = (1 / alpha_{ell j}) times sum_{ell} w_{ell j} x*_{ell j} <= 1. This is incorrect: the denominator inside the sum is alpha_{ell j}, which varies with ell, so it cannot be factored out. Consequently the inequality does not follow from alpha_{ell j} >= 1 or from the definition of alpha_{ij}. A concrete instance exposes the failure: take one supplier j, customers 1-4, all w_{ij}=1, and x* = (0.1, 0.5, 0.5, 0.4). Since each coordinate is at most 0.5, x* is in P_C. Then alpha_{1j}=1.4, alpha_{2j}=alpha_{3j}=1, alpha_{4j}=1.1, so hat{x} = (0.0714, 0.5, 0.5, 0.3636). For customer 1, sum_{ell != 1} hat{x}_{ell j} = 0.5 + 0.5 + 0.3636 = 1.3636 > 1, violating Property 2. This is a proof gap in a load-bearing theorem: without a valid structure theorem, the chain Lemma 5.3, Observation 5.6, Theorem 4.2, Lemma 4.5, and Theorem 1.2 collapses. The issue is independent of reward sign; it persists even with all rewards positive.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies revenue maximization in two-sided sequential matching markets under Multinomial Logit choice. The platform offers each customer a menu of suppliers; customers choose a supplier, and each supplier then chooses among the customers who selected her. The paper distinguishes a customized model, where the platform can filter the set of customers shown to each supplier, and an inclusive model, where each supplier sees all selecting customers. The main claims are a polynomial-time 1/3-approximation for the customized model (Theorem 1.1) and a polynomial-time (10/539 - epsilon)-approximation for the inclusive model (Theorem 1.2). The proofs proceed by reformulating menus as points in an MNL-choice polyhedron, then developing linear relaxations; the inclusive model is split into low-weight and high-weight regimes, with the low-weight regime relying on convex stochastic orders and a structure theorem (Theorem 5.5).","tokens_in":33511,"tokens_out":14883,"duration_ms":147535,"significance":"If the results were correct, the paper would resolve a natural open direction by giving the first constant-factor approximation algorithms for pairwise rewards in sequential two-sided matching, where the objective is neither submodular nor subadditive. The customized-model analysis in Section 3 is explicit and appears internally coherent; it is a meaningful contribution in its own right. The inclusive-model analysis, however, rests on a structure theorem whose proof contains an invalid algebraic step and whose proposed construction fails on a concrete instance. Since Theorem 1.2 is one of the two headline results and depends directly on that theorem, the inclusive-model contribution is not established by the manuscript as written.","major_comments":[{"comment":"The proof of Property 2 in Theorem 5.5 contains an invalid algebraic step. After defining alpha_ij = max{sum_{ell in C_{-i}} w_{ell j} x*_{ell j}, 1} and hat{x}_ij = x*_ij / alpha_ij, the proof claims that sum_{ell in C_{-i}} w_{ell j} hat{x}_{ell j} = (1 / alpha_{ell j}) times sum_{ell in C_{-i}} w_{ell j} x*_{ell j} <= 1. The denominator alpha_{ell j} varies with the summation index ell, so it cannot be factored out of the sum, and the inequality does not follow from alpha_{ell j} >= 1. This is not a cosmetic issue: for a single supplier j, four customers with u_ij = w_ij = 1, and x* = (0.1, 0.5, 0.5, 0.4), each coordinate is at most 1/2, so x* lies in P_C. Then alpha_{1j}=1.4, alpha_{2j}=alpha_{3j}=1, alpha_{4j}=1.1, so hat{x} = (0.0714, 0.5, 0.5, 0.3636). For i=1, sum_{ell != 1} hat{x}_{ell j} = 1.3636 > 1, violating Property 2. Because Observation 5.6, Theorem 4.2, Lemma 4.5, and ultimately Theorem 1.2 all rely on Theorem 5.5 as the structure theorem for the low-weight inclusive regime, the inclusive-model approximation is currently unsupported.","section":"5.3 (Theorem 5.5)"},{"comment":"The paper advertises 'arbitrarily-valued' pairwise rewards but never states a nonnegativity assumption on r_ij. Several load-bearing arguments assume nonnegative rewards. In particular, Lemma 6.3 sorts rewards descending and bounds the contribution of a realization with Q ∩ [k_j] = empty by r_{sigma_j(k_j),j}; this bound uses that the factor (sum w)/(1+sum w) <= 1 multiplies a nonnegative quantity. If r_{sigma_j(k_j),j} < 0, the inequality can fail. Similar monotonicity assumptions appear elsewhere, for example in the reward-ordering step in Section 6.3 and in parts of the customized-model lower bound. If the intended model is nonnegative rewards, the statements should say so explicitly; if negative rewards are allowed, the proofs need an argument that all negative-reward edges are set to zero in an optimal solution, which is not provided. I do not treat this as the primary reason for rejection, but it is a modeling-precision issue that must be fixed if the theorems are restated.","section":"1.1 and 6.3"}],"minor_comments":[{"comment":"The phrase 'the suppliers’ polyhedron P^C is downward-closed' appears where the set being decremented is x in the customers’ polyhedron P^C defined in (6); the wording should be corrected to avoid confusion between P^C and P^S.","section":"3.2"},{"comment":"Lemma 4.5 states a bound of (10/539 - 2 epsilon) OPT(R), while Theorem 1.2 states 10/539 - epsilon. This is likely harmless after renaming epsilon, but the constants should be reconciled in the final statement.","section":"4.3"},{"comment":"In equation (42), the denominator for x*-light suppliers is written over ell in C_j^+, while for x*-heavy suppliers it is written over ell in C; since x* and hat{x} are zero outside E+, the expressions coincide, but the notation should be made uniform to avoid confusion.","section":"6.3"}],"recommendation":"reject","confidential_remarks":"The rejection is driven entirely by the invalid proof of Theorem 5.5 in the inclusive-model part. The customized-model section (Section 3) appears coherent and seems to be a genuine contribution, but the inclusive model is one of the two headline claims and is not established. If the authors can supply a valid structure theorem for the low-weight regime, or otherwise repair the gap, I would be willing to reconsider. The sign-of-rewards issue should also be clarified in any revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know before reading this one. The customized-model result (Theorem 1.1, 1/3 approximation) is real: the LP relaxation and analysis in Section 3 hold together. The inclusive-model result (Theorem 1.2, 10/539) is not established as written. Theorem 5.5's proof has an algebraic error that I can't wave away.\n\nThe problem is in Property 2. They define hat{x}_ij = x*_ij / alpha_ij, where alpha_ij = max{sum_{l != i} w_lj x*_lj, 1}. Then they claim sum_{l != i} w_lj hat{x}_lj = (1/alpha_{lj}) * sum_l w_lj x*_lj <= 1. That factors a per-l denominator out of a sum over l. It's simply wrong. The concrete instance from the stress-test note confirms the failure: one supplier, four customers, all w=1, x* = (0.1, 0.5, 0.5, 0.4). The constructed hat-x gives sum_{l != 1} hat{x}_l = 1.3636 > 1. Property 2 fails. Since Observation 5.6, Lemma 5.3, and Lemma 4.5 all lean on this structure theorem, the entire low-weight-regime chain lacks a valid base.\n\nAlso worth flagging: the paper never states that rewards are nonnegative, but several proofs sort rewards and discard below the k-th largest. If negative rewards are allowed, those bounds break. I suspect the intended model assumes nonnegativity, but it should be explicit.\n\nThe paper does real work. It's the first to handle arbitrary pairwise rewards in sequential two-sided matching, and the customized model is a clean contribution. The LP-relaxation framework with MNL polyhedra is well presented, and the use of convex stochastic orders is inventive. The writing is clear and the related-work discussion is honest.\n\nWho is this for? Researchers in algorithmic game theory and revenue management. The customized model deserves attention; the inclusive model needs a corrected proof or a revised theorem. As submitted, I would not accept the inclusive claim, but I would send it to peer review. A good referee can separate the solid half from the fixable gap. I'd engage with it—but don't quote the 10/539 guarantee until the algebra is fixed.","headline":"Customized-model result is solid, but the inclusive-model proof has a genuine algebraic gap that pulls the rug from under the second headline guarantee.","tokens_in":34129,"tokens_out":3315,"would_cite":true,"duration_ms":28449,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C27","91B68"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes constant-factor polynomial-time approximation guarantees for revenue maximization in two-sided matching markets with arbitrary pairwise rewards under Multinomial Logit choice.","keywords":["two-sided matching markets","revenue maximization","assortment optimization","Multinomial Logit choice","pairwise rewards","constant-factor approximation","MNL-choice polyhedron","convex stochastic orders"],"falsifier":"Run the Section 3 LP and brute-force enumeration on all small instances with nonnegative integer rewards and unit MNL weights; a single instance where the LP menu's expected reward is below $1/3$ of the optimal menu's reward would refute Theorem 1.1. Separately, a two-customer, two-supplier instance with $r_{11}=-1$ and all other rewards $1$ checks the missing sign assumption: if the LP violates the claimed bound there, the theorem as written without a nonnegativity restriction is false.","tokens_in":32870,"feed_emoji":"🤝","tokens_out":11428,"duration_ms":88400,"temperature":0.7,"pith_summary":"The paper asks how a platform should present suppliers to customers, and customers to suppliers, when every customer-supplier pair carries its own reward and both sides choose according to multinomial logit preferences. It claims that, despite the objective being neither submodular nor subadditive, near-optimal personalized menus can be computed in polynomial time. For the customized model, where the platform can filter which customers each supplier sees, the guarantee is a $1/3$ fraction of the optimal expected reward. For the inclusive model, where suppliers see all customers who chose them, the guarantee is $10/539-\\epsilon$ for any $\\epsilon>0$. These are the first constant-factor bounds for general pairwise rewards in sequential two-sided matching.","feed_headline":"Pairwise-reward two-sided matching gets constant-factor guarantees","feed_subtitle":"Customized menus reach 1/3 of optimal revenue; inclusive menus reach 10/539, first constant bounds for pairwise rewards.","key_machinery":"The central object is the MNL-choice polyhedron $P^C$, the set of customer choice probability vectors satisfying $x_{ij}/u_{ij} \\le 1-\\sum_k x_{ik}$. Any point in it can be realized in expectation by a distribution over prefix assortments, so optimizing over $P^C$ is equivalent to optimizing over menus. The customized model adds a linked supplier-side polyhedron and caps customer weights at $\\hat{w}_{ij}=\\min\\{w_{ij},1\\}$, which supports a Jensen-inequality argument yielding $1/3$. The inclusive model decomposes the problem by supplier weights: low-weight edges ($w_{ij}\\le1$) are handled by a deterministic relaxation justified by convex stochastic orders (Poissonization), and high-weight edges ($w_{ij}>1$) by a structure theorem that caps the expected number of customers selecting each supplier at $3/5$; an approximate dynamic program estimates the expected reward of each candidate menu so the better one can be selected.","core_discovery":"The paper establishes that revenue maximization in sequential two-sided matching markets with general pairwise rewards admits constant-factor polynomial-time approximation algorithms. In the customized model, the platform chooses personalized menus for customers and may also tailor the set of selecting customers shown to each supplier; Theorem 1.1 gives a $1/3$-approximation via a linear relaxation over the MNL-choice polyhedron. In the inclusive model, each supplier must see every customer who selected her; Theorem 1.2 gives a $(10/539-\\epsilon)$-approximation by splitting edges into low-weight and high-weight regimes, approximating each regime separately, and using a $(1\\pm\\epsilon)$ estimator to pick the better menu. The proof works around the absence of submodularity and subadditivity by working with choice probabilities rather than menus.","pith_inferences":["The model definition does not explicitly say rewards are nonnegative, but every proof sorts by reward and bounds contributions by the $k$-th largest reward; if negative rewards are allowed, the constant-factor guarantees as stated may fail.","The inclusive-model constant comes from balancing a $10/39$ low-weight approximation against a $1/50$ high-weight approximation, so a tighter analysis of either regime would improve the final factor substantially.","The exact equivalence between menus and the MNL-choice polyhedron suggests the same continuous relaxations could be reused for other two-sided objectives, such as maximizing the number of matches or balancing fairness, whenever the supplier-side reward can be bounded by capped weights.","The $(1\\pm\\epsilon)$ evaluation procedure for expected rewards is a standalone tool: any candidate menu or randomized menu whose choice probabilities lie in $P^C$ can be compared approximately, which may be useful outside this paper."],"forward_implications":["If the guarantees hold, they resolve the open question of constant-factor approximability for pairwise rewards in sequential two-sided matching.","A platform in the customized model can compute a random menu whose expected reward is at least $1/3$ of optimal in polynomial time.","A platform in the inclusive model can compute a random menu whose expected reward is at least $10/539-\\epsilon$ of optimal, with running time polynomial in the input size and $1/\\epsilon$.","The LP-based construction works even though the expected reward is neither submodular nor subadditive, which had blocked earlier techniques."],"supporting_citations":[{"why":"Introduced the sequential two-sided matching model with assortment optimization and gave the first constant-factor approximation for uniform rewards, which this paper extends to pairwise rewards.","marker":"Ashlagi et al. (2022)"},{"why":"Extends the model to supplier-specific rewards under monotone submodular choice; its submodularity assumption is what the current paper removes.","marker":"Torrico et al. (2023)"},{"why":"First to study pairwise customer-supplier rewards, providing the reward objective and showing that simple greedy heuristics can be arbitrarily bad.","marker":"Ahmed et al. (2022)"},{"why":"Proved the MNL-choice polyhedron characterization that the paper uses to move from menus to choice probabilities.","marker":"Topaloglu (2013)"},{"why":"Supplies the sales-based linear program and general attraction model used for the MNL-choice polyhedron in the relaxations.","marker":"Gallego et al. (2015)"},{"why":"Provides the LP formulation for MNL assortment optimization used to express the customized supplier reward function.","marker":"Gallego and Topaloglu (2019)"},{"why":"Source of the convex stochastic order and Poissonization tools used in the low-weight regime of the inclusive model.","marker":"Shaked and Shanthikumar (2007)"}],"fun_headline_variants":["Constant-factor approximations for revenue in matching markets","Pairwise-reward matching: first constant guarantee for revenue","Customized matching achieves 1/3 revenue; inclusive gets 10/539","Two-sided matching: revenue maximization gets constant factors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proofs assume every pairwise reward $r_{ij}$ is nonnegative; with negative rewards, sorting by reward and bounding contributions by the $k$-th largest reward can fail, so the constant-factor guarantees may not hold.","fun_headline_variants_meta":{"raw":{"variants":["Constant-factor approximations for revenue in matching markets","Pairwise-reward matching: first constant guarantee for revenue","Customized matching achieves 1/3 revenue; inclusive gets 10/539","Two-sided matching: revenue maximization gets constant factors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000439,"raw_usage":{"total_tokens":2206,"prompt_tokens":903,"completion_tokens":1303,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":1235}},"tokens_in":519,"tokens_out":1303,"duration_ms":11659,"temperature":1.0,"reasoning_tokens":1235,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:01:26.123029+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the Section 3 LP and brute-force enumeration on all small instances with nonnegative integer rewards and unit MNL weights; a single instance where the LP menu's expected reward is below $1/3$ of the optimal menu's reward would refute Theorem 1.1. Separately, a two-customer, two-supplier instance with $r_{11}=-1$ and all other rewards $1$ checks the missing sign assumption: if the LP violates the claimed bound there, the theorem as written without a nonnegativity restriction is false.","supporting_citations":[{"cited_title":"Krishnaswamy, Rahul Makhijani, Daniela Saban, and Kirankumar Shiragur","cited_arxiv_id":null,"evidence_quote":"Introduced the sequential two-sided matching model with assortment optimization and gave the first constant-factor approximation for uniform rewards, which this paper extends to pairwise rewards."},{"cited_title":"Sohoni, and Chaithanya Bandi","cited_arxiv_id":null,"evidence_quote":"First to study pairwise customer-supplier rewards, providing the reward objective and showing that simple greedy heuristics can be arbitrarily bad."},{"cited_title":"A general attraction model and sales-based linear program for network revenue management under customer choice","cited_arxiv_id":null,"evidence_quote":"Supplies the sales-based linear program and general attraction model used for the MNL-choice polyhedron in the relaxations."},{"cited_title":"Revenue Management and Pricing Analytics","cited_arxiv_id":null,"evidence_quote":"Provides the LP formulation for MNL assortment optimization used to express the customized supplier reward function."},{"cited_title":"Stochastic Orders","cited_arxiv_id":null,"evidence_quote":"Source of the convex stochastic order and Poissonization tools used in the low-weight regime of the inclusive model."}],"review_version":1}