Revenue Maximization in Choice-Based Matching Markets

Alfredo Torrico; Dan Nissim; Danny Segev

REVIEW 2 major objections 3 minor 19 references

Revenue Maximization in Choice-Based Matching Markets

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.15727 v1 pith:4Q2UIMRT submitted 2024-11-24 cs.GT cs.DSmath.OC

Dan Nissim , Danny Segev , Alfredo Torrico This is my paper · ORCID

classification cs.GTcs.DSmath.OC MSC 90C2791B68

keywords two-sidedmatchingmarketsrevenuemaximizationassortmentoptimizationMultinomialLogitchoicepairwiserewardsconstant-factorapproximationMNL-choicepolyhedronconvexstochasticorders

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

Also 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.

The 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.

Who 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.

Referee Report

2 major / 3 minor

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).

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 (2)

[5.3 (Theorem 5.5)] 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.
[1.1 and 6.3] 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.

minor comments (3)

[3.2] 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.
[4.3] 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.
[6.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the approximation chain is analytically self-contained; the Theorem 5.5 proof gap is a correctness issue, not circular reasoning.

full rationale

The paper's derivation chain is self-contained in the sense relevant to circularity. Theorems 1.1 and 1.2 rest on LP relaxations (LP), (RDL_-), and (RDL_+) whose constraints come from the MNL-choice polyhedron and from structural bounds; the approximation constants (1/3, 10/539, etc.) are produced by analytic inequalities (Jensen, Markov, Poissonization, dynamic programming), not by calibrating parameters to the target objective. The realization lemmas (Lemmas 2.2 and 2.4) are exact equivalences whose inputs are the feasibility conditions of P_C and P_S, not the theorems being derived. The citations to prior work are external mathematical tools (Shaked-Shanthikumar stochastic-order closure, Topaloglu/Gallego MNL polyhedron) or background context (Ashlagi et al.; Torrico et al.), and they are not invoked to justify the claimed constant-factor approximation. The one serious concern flagged by the skeptic is in the proof of Theorem 5.5, Property 2: the paper writes a sum with the per-customer denominator alpha_{ell j} inside the sum and then appears to factor it out, which is an algebraic error; this threatens Observation 5.6 and the chain through Lemma 5.3 to Theorem 4.2/1.2. However, that is a correctness gap, not circularity: the claimed bound does not reduce to an input by definition, and no fitted parameter is renamed as a prediction. Thus the paper merits a circularity score of 0.

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

No fitted parameters and no invented entities. The model rests on MNL choice assumptions, independence of customers, nonnegative rewards (implicit), and standard results on MNL polyhedra and stochastic orders. The unstated nonnegativity of rewards is the main ledger item a reader must supply.

assumptions (5)

domain assumption Customers and suppliers follow Multinomial Logit choice with fixed, known preference weights.
Section 1.1 defines the sequential matching process with MNL choice; every approximation guarantee is relative to this model.
domain assumption Customer choices are independent across customers.
Used throughout to represent C_j as independent Bernoulli variables, e.g., in Lemma 2.3 and Section 5.2.
domain assumption Pairwise rewards r_ij are nonnegative.
Proofs in Sections 4 through 6 use monotonicity of sorted rewards and discard low-reward options; negative rewards would invalidate bounds such as Lemma 6.3. Section 1.1 does not state a sign constraint.
standard math The MNL-choice polyhedron characterization and randomized-assortment realization hold as stated.
Used in Section 2 to reformulate menus as choice probabilities, citing Topaloglu 2013 and Gallego et al. 2015.
standard math Convex stochastic order facts hold as cited.
Used in Lemma 5.1 and Corollary 5.2 to bound expectations by Poissonization, citing Shaked and Shanthikumar 2007.

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Cite this review

Pith. "Pith review of Revenue Maximization in Choice-Based Matching Markets." pith.science (2026). https://pith.science/paper/4Q2UIMRT

@misc{pith2026241115727,
  author       = {Pith},
  title        = {Pith review of: Revenue Maximization in Choice-Based Matching Markets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4Q2UIMRT}},
  note         = {Machine review of arXiv:2411.15727}
}

read the original abstract

The primary contribution of this paper resides in devising constant-factor approximation guarantees for revenue maximization in two-sided matching markets, under general pairwise rewards. A major distinction between our work and state-of-the-art results in this context (Ashlagi et al., 2022; Torrico et al., 2023) is that, for the first time, we are able to address reward maximization, reflected by assigning each customer-supplier pair an arbitrarily-valued reward. The specific type of performance guarantees we attain depends on whether one considers the customized model or the inclusive model. The fundamental difference between these settings lies in whether the platform should display to each supplier all selecting customers, as in the inclusive model, or whether the platform can further personalize this set, as in the customized model. Technically speaking, our algorithmic approach and its analysis revolve around presenting novel linear relaxations, leveraging convex stochastic orders, employing approximate dynamic programming, and developing tailor-made analytical ideas. In both models considered, these ingredients allow us to overcome the lack of submodularity and subadditivity that stems from pairwise rewards, plaguing the applicability of existing methods.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Sohoni, and Chaithanya Bandi

Asrar Ahmed, Milind G. Sohoni, and Chaithanya Bandi. Parameterized approximations for the two-sided assortment optimization. Operation Research Letters, 50 0 (4): 0 399--406, 2022

work page 2022

[2] [2]

Competition in two-sided markets

Mark Armstrong. Competition in two-sided markets. The RAND Journal of Economics, 37 0 (3): 0 668--691, 2006

work page 2006

[3] [3]

Krishnaswamy, Rahul Makhijani, Daniela Saban, and Kirankumar Shiragur

Itai Ashlagi, Anilesh K. Krishnaswamy, Rahul Makhijani, Daniela Saban, and Kirankumar Shiragur. Assortment planning for two-sided sequential matching markets. Operations Research, 70 0 (5): 0 2784--2803, 2022

work page 2022

[4] [4]

Matching impatient and heterogeneous demand and supply

Angelos Aveklouris, Levi DeValve, Maximiliano Stock, and Amy Ward. Matching impatient and heterogeneous demand and supply. Operations Research, 2024. (Forthcoming)

work page 2024

[5] [5]

Information provision in two-sided platforms: Optimizing for supply

Kostas Bimpikis, Yiangos Papanastasiou, and Wenchang Zhang. Information provision in two-sided platforms: Optimizing for supply. Management Science, 70 0 (7): 0 4533--4547, 2023

work page 2023

[6] [6]

Submodular functions maximization problems

Niv Buchbinder and Moran Feldman. Submodular functions maximization problems. In Teofilo F. Gonzalez, editor, Handbook of Approximation Algorithms and Metaheuristics, pages 753--788. Chapman and Hall/CRC, 2018

work page 2018

[7] [7]

Chicken & egg: Competition among intermediation service providers

Bernard Caillaud and Bruno Jullien. Chicken & egg: Competition among intermediation service providers. The RAND Journal of Economics, 34 0 (2): 0 309--328, 2003

work page 2003

[8] [8]

Maximizing a monotone submodular function subject to a matroid constraint

Gruia C a linescu, Chandra Chekuri, Martin P \'a l, and Jan Vondr \'a k. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40 0 (6): 0 1740--1766, 2011

work page 2011