GenAI vs. Human Creators: Procurement Mechanism Design in Two-/Three-Layer Markets
Authors: Rui Ai
David Simchi-Levi
Haifeng Xu
Venue: Accepted by TMLR ()

Abstract: With the rapid advancement of generative AI (GenAI), mechanism design adapted to its unique characteristics poses new theoretical and practical challenges. Unlike traditional goods, content from one domain can enhance the training and performance of GenAI models in other domains. For example, OpenAI’s video generation model Sora (Liu et al., 2024b) relies heavily on image data to improve video generation quality. In this work, we study nonlinear procurement mechanism design under data transferability, where online platforms employ both human creators and GenAI to satisfy cross-domain content demand. We propose optimal mechanisms that maximize either platform revenue or social welfare and identify the specific properties of GenAI that make such high-dimensional design problems tractable. Our analysis further reveals which domains face stronger competitive pressure and which tend to experience overproduction. Moreover, the growing role of data intermediaries, including labeling companies such as Scale AI and creator organizations such as The Wall Street Journal, introduces a third layer into the traditional platform–creator structure. We show that this three-layer market can result in a lose-lose outcome, reducing both platform revenue and social welfare, as large pre-signed contracts distort creators’ incentives and lead to inefficiencies in the data market. These findings suggest a need for government regulation of the GenAI data ecosystem, and our theoretical insights are further supported by numerical simulations.

Published in Transactions on Machine Learning Research (02/2026)
GenAl vs. Human Creators: Procurement Mechanism De-
sign in Two-/Three-La Markets
Rui Ai
ruiai@mit edu
Massachusetts Institute of Technology
David Simchi-Levi
dslevi@mit. edu
Massachuselts Institute of Technology
Haifeng Xu The Universily of Chicago
wwwpi/iggnwuuccho....oo edu
Reviewed on OpenReview: https: opeereeveee.. net/ forum? id= Bukf4TBHS7
Abstract
With the rapid advancement of generative AI (GenAl), mechanism design adapted to its unique characteristics poses new theoretical and practical challenges. Unlike traditional goods, content from one domain can enhance the training and performance of GenAl mod- els in other domains. For example, OpenAl's video generation model Sora (Liu et al., 2024b) relies heavily on image data to improve video generation quality. In this work, we study nonlinear procurement mechanism design under data transferability, where online platforms employ both human creators and GenAl lo satisfy cross-domain content demand. We propose optimal mechanisms that maximize either platform revenue or social welfare and identify the specific properties GenAI that make such high-dimensional design problems tractable. Our analysis further reveals which domains face stronger competitive pressure and which tend to experience overproduction. Moreover, the growing role of data interme- diaries, including labeling companies such as Scale Al and creator organizations such The Wall Street Journal, introduces third layer into the traditional platform -rretoor structure. We show that this three-layer market can result in a lose-lose oulcome, reducing both plat- form revenue and social welfare, as large pre-signed contracts distort creators' incentives and lead to inefficiencies in the data market. These findings suggest a need for goverument regulation of the GenAl data ecosystem, and our theoretical insights are further supported by numerical simulations.
1 Introduction
Large models have entered the era of multimodality (Yin et al. 2023). Modern commercial systems such as GPT-4 AAchiam et al. 2023), Gemini (Team et al. 2023). and Clande (Anthropic] 2024) are trained on diverse types of data, including lext, images, audio, and video. These modalities are not isolated; transferability among them is well established both theoretically and empirically through transfer learning (Howard & Ruder 2018). Just as film directors may draw inspiration from Shakespeare's works, image datasets can enhance the performance of video generation models. This interdependence links the valuation of data across domains into a unilied economic problem and naturally raises the question:
How do we price content across different domains?
The answer depends on the degree transferability, a feature absent from traditional goods. Bricks, for instance, can only build houses but not chips, while data can improve the performance of models across tasks. Data with high transferability should therefore command higher price, whereas specialized data that benefits only single task should be valued lower. In this paper, we incorporate parameters capturing data transferability into formal production model to analyze how this property shapes market outcomes.

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The transferability of data also raises important ethical concerns. Recent debates have centered on whether GenAT will cause certain professions to vanish, alongside growing discussions on the implications of artificial general intelligence (George et al. 2023 We want to ask:
From on economic perspective, will any domains disappear?
In Section 2.2 show that although GenAI may roduce employment in some domains, it cannot completely replace them. This result follows from the principles of diminishing marginal returns and increasing marginal costs. Moreover, in domains with high data transferability, we observe tendency toward overproduction, which distorts the allocation of creative effort. Such imbalances highlight the need for regulatory oversight and further research on how GenAI reshapes data generation markets.
The growing demand for data to train large models has driven many AT companies to purchase high-quality datasets. For example, OpenAl reportedly spent about $250 million acquiring data from The Wall Street Journal (WSJ 2024). Industry forecasts further predict that the global generative AI market for content creation will expand from USD 15.2 billion in 2024 to USD 175.3 billion by 2033 (Market.us 2024). This surge in data demand can be traced back to the creation of ImageNet (Deng et al. 2009), which catalyzed the modern data economy. The emergence of specialized data annotation companies such as Scale AI, Lionbridge, Aurora AI, and Amazon Mechanical Turk has since transformed the landscape, pushing the market from a two-layer structure toward a three-layer structure that includes professional data brokers. This evolution raiscs key question:
How does the three-lager market impact revenuue and social welfare?
In Scction 3 we examine two types of platforms-those that maximize revenue and those that maximize social welfare and show that in the three-layer market, both platform revenue and social welfare decline relative to the two-layer setting, resulting in a clear lose-lose outcome. The presence of data brokers distorts creators' incentives and reduces overall efficiency, highlighting the need for regulatory oversight to address inefficiencies in the data market.
1.1 Our Contributions
We summarize our contributions in three main aspects and claborate on each below.
Mechanism design with competition between humans and generative AI. Recent studies have begun examining market equilibria that arise from competition between AT content generators and human creators (Yao et al.) 2024). Our work is the first to approach this problem from the buyer's perspective. We study online sharing platforms that rely on both human creators and GenAI for content production, situating the analysis within the broader framework of procurement mechanism design. Going beyond the classical single-dimensional setting (Myerson] 1981), we derive optimal mechanisms for five of six multi- dimensional enviromments and establish light upper and lower bounds for revenue and social welfare in the remaining case. Our results show that although GenAl reduces the overall demand for human-created content, human creators remain indispensable from an economic standpoint. Their outputs not only satisfy subscriber preferences but also provide essential data for GenAl training. Furthermore, in domains with high data transferability, we find that overproduction more likely to occur, reflecting the complex interactions between human and algorithmic production incentives.
Distinguishing between two/threc-layer markets regarding data brokers. Finding an efficient market structure has been a long-standing goal in digital economics (Liang et al.) 2018 Agarwal et al. 2019). For online platforms, a central question is whether to buy content directly from human creators as crowdsourcing, or through data companics. Our analysis shows that the latter, a threo-layer market whore the platform purchases data from a broker who sources it from humans, leads to a lose-lose outcome: both platform revenue and social welfare decline. The inefficiency stems from asymmetric information. Large, transparent contracts reveal the platform's 's and broker's valuations to creators, making price discrimination impossible and weakening incentives for efficient production. As result, both welfare and revenue fall, underscoring the need for policy intervention.
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Numerical experiments validate the conclusions about market dynamics. Finally, we validate our theoretical results using simulated data. In the two-layer market, the experiments reveal content overproduc- tion in specific domains. We also compare platform revenue and social welfare across the two- and threc-layer settings, illustrating the "lose-lose" outcome in the latter. These numerical experiments offer valuable inspi- ration and insights for future real-world mechanism design and implementation using real data.
1.2
Related Works
Our work is closely related to the literature on procurement mechanism design (Myerson 1981 McAfee &
McMillan 1987 Laffont & Tirole, 1993), data pricing (Jia et al., 2019 Ghorbani & Zou 2019 Schoch ct al.
2022 Ai et al. 2024], human vs. GenAl compelilion (Esmaeili et al. 2024 Yao el al. 2024 Fish el al. 2024
Duetting et al. 2024) and multi-layer markets (Fallah al. 2024] Xu et al. 2020). We generalize physical
goods procurement to transferable digital content and analyze the ensuing human GenAl competition across
multi-layer markets. A thorough comparison and additional related work are provided in Appendix A.1
2
Content Procurement in (Two-Layer) Platform-Createn Markets
We begin with the two-layer market, in which online platforms procure content directly from human creators. For example, platforms such as YouTube invite crcators to upload content in exchange for monctary rewards, providing a representative case of this market structure.
2.1 A Model of Content Procurement in the GenAl Era
An online platform features K domains (e.g., sports, pop music, knowledge sharing), denoted by [K] = 1.2...., K. To serve its user base, the platform requires dy E R units of content in each domain E [K], produced by human creators and/or GenAIs. We assume that each human creator specializes in a single domair]] For instance, @dream has attracted over 30 million YouTube subscribers through Minecraft challenge videos. For analytical convenience, cach domain & is represented by single creator who incurs cost caar to produce Ig units of content, following standard models of the creator economy (Hu et al. 2023 Jagadeesan et al.) 2024 Yao et al.) 2024). We interpret æ& as calibrated quantity, meaning that production is weighted by quality metrics such as click-through rate or content accuracy (Radlinski et al. 2008). The parameter сı captures the creator's production efficiency, which is private information, while ps reflects the population-level growth speed of the production cost and assumed to be publicly known, as it can be inferred from market data. This representative-crator assumption without loss of generality: when multiple creators exist in the same domain with heterogencous cost paramcters Ck. they can be aggregated into single "meta" creator with effective cost coefficient C₁ (see Appendix C.4 for details). Consistent with prior work, we assume A 2 1 to reflect increasing marginal costs (Marshall 2009). This captures the empirical observation that creative ideas become progressively harder to generate over given time, resulting in a convex production cost function.
To meet its demand requirements asks human creators report their private cost parameters (cI,..., To account for possible misreporting, let == (a...., denote the reported costs. The mechanism then proceeds according to the standard timeline in mechanism design (Hart & Tirole, 1988 Salant 1989):
dx), the platform (e.g., a video-sharing site such as YouTube)
1. The platform commits to a mechanism x(-)) and p(-);
2. The creators' costs é are then elicited;
3. The platform implements "allocation outcomes" x(²) = (a (x))) by procuring x (²) amount of domain-k content from crcator is for each k;;
4. The payments p(č)-
(c) is then executed by paying (c to creator k.
While procurement auctions have been extensively studied in prior work (Laffont & Tirole 1993 our setting differs by incorporating the platform's 's ability to use GenAI to augment content creation at negligible cost
Although some creators may produce content across domains, most major creators focus one area in practice; see Figure for examples.
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Marques Browniee
Markiplirr
Figure 1: @mkbhd focuses on tech reviews, and @markiplier is dedicated to gaming content, especially horror-themar Let's Plays.
(We discuss the non-negligible case in Appendix C.1 Specifically, we assume that the platform can employ GenAT, together with human-created content, to partially meet demand with minimal additional expense relative to human creation. Motivated by the scaling laws of GenAl's creative capacity (Kaplan et al. 2020 Yao et al. 2024). we model this process as follows: I; units of content in domain enable GenAI to generate 2 1& units of essentially new content in domain k; where jaa captures the transferability from domain i to domain k, and "isk represents the capability of employed GenAl tool. Yao et al.) (2024) adopts a similar assumption to analyze the symbiosis or conflict between GenAl and hurnan creators. This formulation highlights three key distinctions between GenAI, human creators, and traditional machines. First, GenAT exhibits transferability, allowing knowledge from one domain to benefit others with varying efficiency. Second, its marginal cost of production is negligible compared with human effort. Third, GenAl's productivity depends on human creation, unlike traditional machines. The power-law relationship between model capability and training data is well documented in the GenAl literature (Kaplan et al. (0220), and we assume that these capability parameters are known, as they can be empirically estimated @Albddulohssnn et al. 2022 Goyal el al. 2024 Lin el al. 2024). Finally, as is slandard, we restrict 7 E [0, I] to reflect the diminishing efficiency of data. Accordingly, given reported costs ē, the platform's design must satisfy the following demand constraints:
Demand: (c) + Σ (C) 2 dg for any k E [K].
The revelation principle (Roughgarden 2010 implies that, without loss of generality, we can restrict attention to incentive-compatible (IC) mechanisms in which truthful reporting constitutes a Nash equilibrium (Myer-
son 1986). In Appendix AA.2] we provide illustrations about the economic background for these and other relevant terms. Accordingly, the incentive-omppatibliyy condition given by
IC: [_R[-cu-f (c) (c)) E_ (Cx, c-R) Pa (C,, c=]]] for any k E [K],
where C_k is e except the k-th entry. The expectation is taken over domains i + k. The nonlinearity of the cost function hinders the application of existing methods (Myerson 1981). We first focus on the Nash equilibrium (Nash Jr. 1950), assuming no collusion, so it suffices to consider one-shot deviations. We will later extend the analysis to more complex settings involving creator union and data brokers. An interesting future direction is extend to correlated equilibrium and coarse correlated equilibrium (Aumann) 1987) with limited signals.
Moreover, since human creators can always reject any allocation payment pair offered by the platform and quit the market, the mechanism must also satisfy the individual rationality (IR) condition:
IR: E (c) + (x(c) 2 0 for any k E [K].
We consider the following class of allocation rules, which are natural when items are substitutable, as in content production. In a valid mechanism, creator's production 2y decreases with her own cost and increases weakly as the costs of other creators rise.
Definition 2.1 (Valid Mechanisms). We say mechanism is valid if its allocation rule satisfies
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TA(Ck C-t) 2 Iz Ckk c-k) for any cN 2 ck and (2 C-N) Ix(C,, C-R) if à 2 a for any + k.
In practice, platforms typically pursue one of two objectives. Some aim to maximize revenue, or equivalently, minimize total cost. Others, such as large technology firms like Google (Google 2024 and Meta (Meta) 2024), emphasize social welfare, defined as the aggregate utility of all market participants, or equivalently, the minimization of creators' total costs, since monetary transfers do not affect overall welfare. These considerations motivate the two optimization objectives:
Type 1: minz,p Ee Σ Pc(c)]; Type 2: minzzp E. D k= Cxa (c)..
For the second objective, when multiple mechanisms yield the same level of social welfare, we select the one with the minimum total payment, breaking tics in favor of the platform.
For simplicity, we impose the following assumption on the distribution of cost parameter c.
Assumption 2.1 (Independent cost). We assume that the cost components 01 -1---CC CK independent with p.d.f. fi(c)),.... fe((K) and corresponding c.d.f. Fi(ci),... Fx (ck)),, respectively. Additionally, we define f(c) = -I fi((s)) F(c) II Fa(cx), f-((c-1) = IT fa(C)) and F-i(c-s) II Fe(c)).
To avoid lengthy discussions of corner cases, we impose standard regularity assumptions on the above distributions, as commonly adopted in both the economics and machine learning literature (Myerson 1981
Wang et al. 2019 Ai et al. 2022
Assumption 2.2 (Bounded cost and density). We assume that for any l E [K], Cj bounded, say cj E bx), and (c() is lower bounded from 0.
Assumption 2.3 (Monotone reverse hazard rate). We assume that for all is [K], holds that Fl(()) is log-concave, in other words, the ratio fx(). is monotone decreasing.
Fx(.]
Assumption 2.2 can always be ensured through distribution truncation, while Assumption 2.3 is standard
and widely popular assumption in economics (Kleiber 2003 Bagnoli & Bergstrom, 2006 Wang et al. 2024
These assumptions are mild and hold for many commonly used distributions, such as the truncated Gaussian and the uniform distribution (Golrezaei et al. 2019).
2.2
The Optimal Mechanism Mi for
Procurement
We first consider the case in which the platform aims to maximize its own revenue, Accordingly, the platform faces the following revenue-maxinizalin (or cost-mininization) problem:
Rev = maxx,p
B.1-Ea Σ
k= px(c)]
s.t.
I& (c) + Σ ==1 (c) 2 dy and zx(c) 2 0 for any k E [K]
(1)
E + c (c) + (c) 2 E-xl-caff (Ck, c- + Pk (Cx, c))
E_ E - (c) + (c) 2 0,
where the expectation is laken over c ~ F(c). In addition, the corresponding social welfare is SW, =
Ekm Ek cu-T (c)]. We now present the optimal mechanism Mi under information asymmetry between human creators and the platform in Algorithm
Theorem 2.1. Mechanism M1 is valid mechanism salisfying IC and IR condilionn and achieves the highest expected revenue/lowess cost for the platform.
While Theorem 2.1 is formulated in proof arises from the nonlinearity of the designer's allocation constraint, namely y in Algorithm This constraint, induced by the nonlinear cost function and data transferability, prevents the direct application of the standard analysis by Myerson 1981 We address this difficulty in two steps. First,
mechanism design setting, the main challenge in its
1/px + E Payy qw/pt 2 de
"Since Algorithm is valid mechanism, actually satisfies ex-post 1C and IR conditions.
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Algorithm Mechanism M₁ for revenue revenue-maximizing platforms without union in the two-layer market.
Input: Report č
Calculate virtual cost: (C) Cx
for any & E [K].
fa (c4)
Call some oracle to solve the auxiliary optimization problem
K
argmin M
DCk (V)
s.b. yk
1/p0
2 ds and ye 2 0 for any k [K].
Calculate the allocations: (x(() = 10k
1/px for any k E [K]-
Calculate the payments: (C) Ckckk + Eor yt (C Ck-1 tx. Ck+1 ..., CK dtt for any E [K].
Pk Output: Allocation-payment pair (x()).p(()) = (Z), ..... TK Px (e)).
by applying a nonlinear transformation, we replace the search for the optimal allocation x(-) with that for v(+), monotonic transformation of z<-), which lincarizes the cost term. Second, we show that the nonlincar constraints introduced by GenAI still preserve convexily. Together, these steps reformulate Problem into convex optimization problem, for which we establish the existence of an efficient solution algorithm.
Proposition 2.1. Algorithm
be implemented to output c-optimal mechanism in (()) time. Furthermore, if max; 24k for any k E [K], the computational complerity reduces to (
Randomized payments can further reduce the computational complexity to O(L/²²) and (1/e)) as detailed in
Proposition C.1 of Appendix C.2 These results demonstrate that online platforms can efficiently maximize revenue even when incorporating GenAI into content creation. This extends the scope of procurement mechanism design and provides theoretical foundation for pricing in the rapidly evolving GenAT economy.
Furthermore, we derive the following corollary from Theorem 2.1 which shows that original human-croated content remains essential in every domain and that complete substitution does not occur. However, unlike markets without GenAT, overproduction may emerge as a new phenomenon.
Corollary 2.2. Assuming data has transferubility, iee,, all Pik and "is are not zero, then no domain will disappear, i.e., all zz(c) will be positive, no matter the value of e. However, some demand constraints will be non-binding, i.e., some [x(c) + Eiz (c)x,Cc)is > dy. It shows that with the development of GenAl, some domains will be overproduced to augment knowledge transfer.
2.3 The Optimal Mechanism M₂ for Welfare- Maximizini Procurement
To maximize social welfare, an intuitive way is to allocate I corresponding to argmin, Σn Cx2f(c) for each c. This yields the following optimization formulation and the corresponding revenue for such kind of platforms 1 (c)].
is Rev2 = Ee[- Ek= Pk
SW max s.1.
Ed - Σ EK_ aath
Ph (c)]
nx (c) + Σ i (e)
K
IVV IV 2. da and xx(c) for any k E [K]
(2)
E E_ E_Il-ca2 c2
(c) + (a(e] E E=Il-cx0"" (Ōx, c..) + (et c-+)]
E E=Il-cu (c) pk (e)
(
0.
However, two key challenges arise in deriving the optimal design: (a) incentivizing each creator lo report their cost truthfully, and (b) identifying a payment rule p(-) that satisfies both IR and validity constraints. We resolve both challenges the affirmative and derive the optimal mechanism M2 for this setting by substituting vc(?) with é Details are provided in Algorithm of Appendix B
"It means compared the optimal mechanism, the extra loss most

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We conclude this section with our main result, which establishes the optimality of Algorithm A i.c., mecha- nism M2. The availability of the optimal mechanism also explains the rationale for the presence companies in the market that pursuc social welfare maximization.
Theorem 2.2.
Mechanism M₂ is valid mechanism satisfying IC and IR conditions and achieves the highest social welfare with polynomial time complerit
2.4 Creator Union and Optimal Procurement Mechanism Design
Recently, OpenAl and The Wall Street Journal reached data agreement valued at up to $250 million (WSJ 2024). At the same time, data-selling companies such Scale AI have experienced rapidly rising valua- tions, driven by the growing demand for high-quality data to fine-tune large language models (The Scale Team 2023). Economically, these intermediaries can be viewed as aggregating data from human creators and repackaging it for sale, similar to a creator union. We defer the analysis of the case in which such intermediaries secure contracts before sourcing data from creators to Section 3 This raises a key question: when a creator union appears, represented by a single agent producing content across all K domains, how will the revenue of data-obsessing platforms and social welfare be affected?
We retain the setting from Section 2.1 except that the representative creator now has a cost vector (cI....,ck' .K). We also let (01, (o,,..., ax) and (bI...., bk). Unlike before, one-shot deviations are insufficient to capture strategic behavior, as the creator may misreport multiple entries simultancously. This possibility of collusion complicates the mechanism design problem by introducing correlations, analogous to the distinction between Nash and correlated equílibria (Gilboa & Zemel 1989..
2.4.1
Revenue-Maximizn Procurement Mechanism M3 with a Creator Union
Pessimistically, general high-dimensiona mechanism design problems remain open and agnostic (Briest et al, 2010 Hart et al. 2013 Daskalakis 2015 Hart & Nisan 2017). Therefore, we formulate the corresponding optimization problem and derive bounds on platform revenue and social welfare, lcaving explicit analytical solutions for future research. The revenue-maxximiznng platform now faces the following optimization problem (Mechanism M3), where the payment rule p(-) is a scalar function and the corresponding social welfare is
SW, = Eel-> Ek=n CxaR (c)]:
Rev3 max
pcl-p(c)] xx(c) + u=
K (c) 2 ds and xx(c) 2 0 for any E [K]
s.t.
(3)
p(c)- Σ k cH (c) > p(() Σk carl (?)
p(c) Σ Ca (c) 0.
The main challenge in this problem is that the union may misreport multiple components of the cost vector c simultaneously. The resulting correlations make the optimization problem effectively contain infinitely many constraints and require the use of path integrals in defining the payment rule. Finding the optimal solution remains difficult open problem in economics. To be specific, recall in the one-dimensional case, the optimal payment in Algorithm has the form pk = ck + S yk (c1,..., Cx-1, tky Ck+l. cx )dt, with truthful reports.
However, in the high-dimensional case, the corresponding payment would become p = c.y+ S v()). When high-dimensional vector, it raises an extra question that the integration results might be different for different paths from to b, which contradicts the fixed contract. Hence, simply applying methods in the one-dimensional case won't yield a valid mechanism, and it hinders the optimal high-dimensional mechanism design in relevant scenarios. In contrast, as shown in the next section, platform that maximizes social welfare can efficiently obtain the optimal solution, highlighting a notable advantage.
"In application, it is also known Fully Polynomial-Timee Approximation Scheme (FPTAS), which means that we find e-optimal solution within polynomial time with respect to 1/r. The symbol represents the dot product.

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2.4.2 Welfare-Maximizze Procurement Mechanism M4 under a Creator Union
The main obstacle to obtaining explicit solutions high-dimensional mechanism design lies in the IC con- dition. To ensure truthful reporting by the creator union, the payment rule must involve high-dimensional integral, and it is generally difficult to construct one that is independent of the integration path.
In this section, we turn to studying the properties of social welfare maximizers who aim to minimize total social cost Eee 1 (c)]. Similarly, we use Rers = Be(-p(c)] to denote corresponding platform revenue.
SW4 = max
Ee[- Σx ca2k (c)
s.t.
(c) + > K
xi=l (c) 2 dx and (c) 2 0 for any k E [K]
p(c) Σ + C (c) 2 p(o) -E c (c)
K
p(c) Ek=1 Ch (c) N O.
Nonetheless, the analysis shows that the optimal mechanism can be derived in an explicit form and computed in polynomial time. This may explain why some companies choose to maximize social welfare, as the corresponding mechanism is easier for customers to understand and involves only a small loss in revenue. We now present the optimal mechanism Mi under this setting in Algorithm 2
Algorithm 2 Mechanism M₁ for Type 2 platforms with union in the two-layer market.
Input: Report č Replace virtual cost: vc(2) by ट.
Call Algorithm I for y.
Calculate the allocations: 2n (T) y) 1/px for any e [K]-
Calculate the payment: p(c) = é fe u(t) dt.
Output: Allocation-payment pair (x()).p(() =
(()))()))
Note that calculating the payment involves an integration path. A natural question whether p(.) well- defined, that is, independent of the chosen path. We answer the question in the following Lemma 2.3
Lemma 2.3. The payment p(-) in Algoritiom 2 is path independent and so well-defined.
We now state the main theorem establishing the optimality of our Algorithm 2
Theorem 2.3. Mechanism Ma is volid mechanism satisfying IC and IR conditions and achieves the highest social welfare with polymomial time complerity.
This theorem reveals surprising result: despite the high-dimensional setting, the welfare-maxinizing mech- anism can be computed efficiently. The proof is constructive, identifying an optimal payment rule p(-) that is independent of the integration path. The distinctive propertics of GenAI are crucial this result. Since GenAI introduces only convex demand constraints, the allocation, whether a vertex or along smooth segment, remains orthogonal to cost after transformation. We construct potential function (c), detailed in Appendix C.6 and establish the relationship between its gradient and the allocation rule. This approach provides a new method for addressing high-dimensional mechanism design problems, distinct from cycle monotonicity (Lavi & Swamy 2007), and advances understanding of this open question. Building on the above results, the next section presents a partial comparison of REV1-4 and SW144
2.5 Comparison of Revenue and Social Welfare for Mechanisms M,-4
Intuitively, a revenue-maxinee mexemme-m platform (Type 1) should generate lower social welfare than a welfare- maximizing platform (Type 2) under the same setting, while Type platform typically earns less revenue. Regarding the creator union, since it aggregates information from all human creators, it is expected to possess greater market power, potentially leading to higher total utility for creators. The following theorem formalizes these comparison results.
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Theorem 2.4. Let Rev, and SW; denote the revenue and welfare of volid mechanism M, For the reveruue objective, we have Rev, 2 Revz and Rev 2 Renz 2 Rev-- For the social welfore objective, we have SW2 = SWa 2 5Wx)
The most technically interesting comparison lies between Revy and Rees- Since the payment rule of Ma involves a path integral, we select a specific integration path and, using the validity of the mechanism, show that Rev2 2 REv4. The complete proof is provided in Appendix C.7
As shown in Theorem 2.4 we establish the comparative rankings of revenue and social welfare among Al- gorithms 1 2 and 4 and provide upper bounds for M3 under mild assumptions, since it lacks explicit solution. We find that the presence of a creator union reduces platform revenue by weakening its pricing power. In contrast, for a social welfare-maxinn maxiimiiiiiiiiin platform, the union does not affect the outcome. Intu- itively, social welfare depends solely on the creators' total production cost, so while the union may alter the payment rule, it has no incentive to distort the allocation. As social welfare is allocation-dependent only, its optimality remains preserved.
3 Content Procurement in Haaform--rookr-Creen Markets
In this section, we extend our analysis beyond the union case and consider three-layer market Fallah et al. (2024). In this setting, the platform first offers a contract (2,t) to a data broker (Liu et al. 2021 who decides whether to accept it. The broker then engages with human creators and faces a mechanism design problem similar that Section 2.T. Unlike the union, which maximizes the total utility of creators, the broker focuses solely on profit, defined as the difference between and Σ Σk=1 Pk- In practice, companies such SchoolDigger and Datarade serve examples of data brokers (Zhang et al. 2024a). To build intuition, Figure in Appendix D.1 illustrates the timeline of this mechanism design setting.
3.1 Gevenue-Maximizze Procurement Mechanism MMs with a Data Broker
We first consider the scenario in which the platform seeks to maximize its revenue. To identify the optimal mechanism, we begin by formulating the broker's 's optimization problem given (z, Since both the allocation and payment rules depend on (z,t), we denote them by 2(=;2, t) and 2,t,,,,, respectively. Because the broker signs the contract with the platform in advance, the credibility constraint (Akbarpour & Li 2020 imposes additional condition that, for any report ē
(c) 2 for any k E [K].
Consoquently, the broker needs to solve the following optimization problem:
maxz,p s.t.
Be[l E (c + pa(c; 2, t)] E- cLk, (p c,);
(4)
E- E-1l-ch.o - (c (c,,,) + pa(c; 2, 2,t)] 2
T (c. 2 for any & E [K],
where t is fixed constant and z is a fixed K-dimensional nonnegative vector. Let x² (12, and p² (:: (1,,t) denote the optimal allocation and payment rules of Problem g The platform's objective is then given by
Revs max;,1 E-l-t] s.t.
Ee 2 Thh 2 d, and 2 2 for any [K] Ex(t - Em (c 22))2 20
2 Σ K
(5)
The second constraint ensures the participation of the data broker in the market, serving as the broker's IR
condition. Moreover, the corresponding social welfare given by SW, = Ey[ Σ e cx 2, t))*).
A natural question is whether optimal mechanisms for both (z,t) and (x, p) can be derived in polynomial time. The following Algorithm and Theorem 3.1 establish their optimality and computational complexity.

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Published in Transactions on Machine Learning Research (02/2026)
Algorithm 3 Mechanism Ms for Type 1 platforms with data broker in the three-layer market.
# First-stage mechanism design.
Replace virtual cost: ve(e) by b.
Call Algorithm ! for y.
Calculate the allocations: 2 Vı 1/PN for any & [K]. Output: Allocation-paymentt pair (z, = ((z1,..., =x); b, v). # Second-stage mechanism design.
Input: Allocation 2, payment and report č
Calculate the payment: (c( = b
Output: Allocation-paymmen pair (z()).p(²)) = (b1 ...., bx bx=k))
Mechanism Ms is valid optimal solution to Problems 4 and and achieves the highest expected snvenee/lowest cost for the platform with polynomial time complerity.
Theorem 3.1.
3.2 Welfare-Maximizing Procurement Mechanism MM with Data Broker
We now turn to platforms that aim to maximize social welfare. The first-stage optimization problem is formulated follows, while the second-stage problem remains the same as Problem 4 The corresponding platform revenue is Rees = Be(-t].
SWa max,,t s.t.
Ee! K
(x [
2k + 2 Tik 2 ds and 2 2 for any E [K]
(6)
E. - k==1 (c 2 0.
Since the platform must determine prior to the revelation of c, the optimal solution expected to depend on E[c]. Formally, we substitute in Algorithm 3 with E(c] and derive the mechanism MMg presented in Algorithm 5 of Appendix B and Theorem 3.2
Theorem 3.2. Mechanism M6 valid optimal solution Problems 4 and and achieves the highest social welfare with polynomial time complerity.
3.3 The Lose-Lose Effect in Three-Layer Markets
Lastly, we compare the platform's revenue and social welfare in the three-layer market with those in the two- layer setting. We show that both revenue and social welfare decline relative to M2 and M4, indicating that the presence of data brokers distorts allocations and reduces market efficiency. Public contracting further hinders the effective incentivization of downstream creators and prevents the platform from optimizing social welfare based on realized lypes. This finding highlights the growing need for government oversight and regulation of data markets (Brooks 2024).
Theorem 3.3. Compared with M2 and Ma, both Ms and Ms face a lose-lose situation that Reve 2 Revs = Rev; 2 Rev; and SW2 = SW4 2 SW. 2 SW,
An interesting result Theorem 3.3 is that Revs equals Revs Recall that M4 corresponds to a platform maximizing social welfare, while Ms represents a revenue- maximizin platform. The loss of market power due the creator union and the welfare-oriented objective is economically equivalent the reduction in price discrimination ability resulting from pro-signed contracts. This phenomenon, observed for the first time, offers new insights into procurement. mechanism design in the era of GenAT. From Theorem 3.3 also observe that the introduction of a data broker reduces both platform revenue and social welfare. More surprisingly, the revenuc of a Type platform in the threo-layer market falls below that of Type 2 platform in the two-layer market. Moreover, under mechanisms Ms and Ma. the data broker's profit is exactly zero, indicating no benefit even to himself. This outcome arises because, in the three-layer market, the flatform must make decisions before observing creators' cost reports, leading to conservative and distorted allocations to satisfy hard demand constraints. These findings underscore the need for stronger government regulation of third-party data platforms to improve overall social welfare.
10

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Published in Transactions on Machine Learning Research (02/2026)
Ostinal for variuus
Ostiiaal for varieus
213
37.
15.0
15.3
20.0
10
15
5.0
2.5
25-
23
200
(a) Optimal under different di and d2.
(b) Optimal under different di and d2.
Figure 2: The overproduction phenomenon with respect to demand d.
Finally, we present the following byproduct lower bound on the revenue of M3, based on Ms and obtained by constructing feasible solution to Problem 3
Corollary 3.1. It holds that Recs 2 Rees, providing lower boundd of Reys.
We end this section by noting that although the proof of the lose-losc effect relics on the specific modcl, wo conjecture that it is pervasive phenomenon three-layer markets. The fundamental reason for the lose-lose effect as follows. When a platform and broker sign a public contract for content procurement, such the agrecment between OpenAT and The Wall Street Journal, such largo-scale deals are typically difficult to keep private due to regulatory requirements. As a result, creators in the relevant domain anticipate higher demand for their content and raise their payment requirements. This weakens the platform's 's ability to price discriminate among human creators and naturally reduces the platform's revenue. For social welfare, once human creators become aware the demand specified the broker's contract, the broker must not only compensate them for their production costs but also incentivize them to generate at least a larget level of content. This requires the broker to provide additional informational rents ensure participation and effort. Since social welfare depends solely on the true production costs, these additional incentives introduce distortions in the allocation. As result, the maximization of social welfare is distorted, leading to lower social welfare in the three-layer market compared to the two-layer market.
4 Empirical Studies
Impact of costs overproduction. We conduct synthetic experiments to examine the phenomenon of overproduction under the setting of Section 2.2 For visualization, consider K = domains. The hyperparameters and are drawn from Unif(0, 1], while follows Unif[1, All hyperparameters are vector-valued, and subscripts are omitted for clarity. We vary di and d2 between 1 and 20 and plot their relationship with the optimal allocations Σ1 and I2 derived from Algorithm [ As shown in Figure ((a) when d2 high (e.g., between and 20), 21 remains nearly constant regardless of di, indicating that the constraint Ø1 + Ei=1 7" 2 d1 non-binding and overproduction occurs in the first domain. The realized costs in this case are 0.035 and 0.992, respectively. Similarly, Figure 2(b) shows overproduction the socond domain when d, 2 12.5, where 22 changes little as d2 increascs, implying that 222 + + P=20?"² 2 dg becomes loose. The corresponding realized costs are 0.989 and 0.055. When relative production costs are low, overproduction more likely to occur. The economic intuition lies the transferability and substitution effects across domains: producing low-cost content in one domain can help satisfy demand in others through data transfer. Although this leads to excessive production and additional expenditure in the low-cost domain, the resulting savings in high-cost domains can more than offset. it. Hence, overproduction in some domains
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(a) Optimal 21 under differ- (b) Optimal 22 under differ- (c) Optimal #1 under differ- (d) Optimal 22 under differ- ent di and d2 when 412 = 1. ent d1 and d2 when f12 = 1. ent di and d2 when 712 = 1. cut di and d2 when 7/12 = 1.
Figure 3: The influence of content transferability on the optimal production.
can ultimately increase platform revenue. Moreover, these experiments empirically support Corollary 2.2 confirming that the optimal allocation zy(c) remains positive for all cost realizations
Impact of transferability parameters on overproduction. We next examine the specific effects of the transferability parameters j and y. To isolate their influence, we fix e₁ = 02 = 0.5, di d2 = 10, and Pi = P2 4. We first study the role of μ by setting Y11 = 712 = 721 = 722 0.5. Consider an extreme case where 112 = 1 and μn1 = P21 = 122 0, meaning that only content in domain 1 can facilitate production in domain 2, but not vice versa. As shown in Figure 3(a] when d2 is high and dj is low, overproduction occurs in domain 1 indicated by the uniform color in the upper-left region. This happens because domain content has higher transferability, enabling the platform to train GenAl models that generate outputs in both domains. For instance, when d2 = 20 and d1 S 10, the optimal #1 is around 10, well above the demand di; so the constraint #1 + Ei= 1110," 2h1 2 d, is non-binding. As dj increases, this constraint becomes binding and 21 rises accordingly. In contrast, Figure 3(b] shows no such pattern, since the constraint related d2 remains binding across all parameter combinations. Interestingly, as di increases, the optimal 22 decreases because larger 21 transfers more knowledge to domain 2, reducing the need for direct content production there. These results highlight the asymmetric effect of μ. We then turn to the influence of y by fixing P11 112 121 = 122 0.5 and setting 712 1, while 711 = 721 722 0. In this case, domain 1 content can continuously support the production of domain content, while the reverse effect remains constant. We again observe overproduction in domain 1. Moreover, when d1 is large but d2 is small, human-created content in domain provides sufficient transferable knowledge, keeping I2 nearly constant and close zero (Figure 3(d] Overall, these experiments confirm that higher transferability, whether through i or Y increases the likelihood of overproduction across domains.
Experimental evaluation of revenue and welfare among mechanisms. We next conducted a se- ries of experiments to compare the performance of Algorithms to and 5 The results corroborate Theorems 2.4 and 3.3. showing that the empirical rankings of revenue and social welfare align with the theoretical predictions. As illustrated in Figure 4 mechanism M1 exhibits more outliers and extreme values compared with M2. This suggests that enviromment, where both social welfare and revenue fluctuate less, offering a possible explanation for why some companies choose to prioritize social welfare. Detailed results are provided in Appendix E
platforms may operate in a more stable market
With and Without GenAl. Finally, we simulate the difference between scenarios with and without GenAI, that is, with μ = 0. Using the same setup as the preceding evaluation and mechanism M1 illustrative example, we report detailed results in Table 2 of Appendix E] In our framework, GenAT leverages numan-created content to generate additional material, allowing platforms to meet total demand with less direct human crcation. Consequently, both revenuc and social welfare increase in the presence of GenAL. Specifically, the sample-averaged revenue rises from -30.90 to -16.52, and sample average social welfare improves from -5.66 to -4.47, representing gains of 46.54% and 21.02%, respectively. These results indicate that GenAl enhances production efficiency while increasing both platform revenue and social welfare. However, as the primary beneficiary of GenAl, the platform enjoys a substantially larger increase in revenue than the corresponding improvement in social welfare. Although the magnitude of improvement depends
12

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-10
-10-
50031 wettrr
-15
18000
-20
-50
-25
-10
200
-55
Mecharsmm
Mechenems
Macranam
Mechersmm
Mecharam
Mecharam
Mecheram
Mecranar Hecheram Mechanam
Pectoncess
(a) Box plot of revenue.
(b) Box plot of social welfare.
Figure 4: Concrete revenue and social welfare of different mechanisms (red: mean; orange: median).
on parameter initialization, the asymmetric growth between revenue and welfare reflects a broader pattern observed in practice (Hosseini & Khanna 2025), highlighting reflections regarding the distributional impact of GenAI adoption.
5 Conclusion and Discussion
In this paper, we study non-linear procurement. mechanism design in markets where human creators coerist with GenAL. We develop optimal mechanisms for platforms sccking to maximize either revenue social welfare across three market structures: two-layer markets (with without creator union) and three-layer markets involving data brokers. Our analysis shows that although the rapid advancement of GenAl substan- tially affects creators in certain domains, cannot fully replace them in competitive environments (Chiang
& Lee 2023 Finally, we uncover striking result: the three-layer market structure leads to a lose-lose
outcome, echoing growing societal concerns over AT regulation and governance (Smuha 2021 de Almeida et al. 2021 Minssen et al., 2023).
This work leaves open many interesting future directions. For general high-dimensional mechanism design problems, such as Mechanism M3, can the optimal solution be computed exactly in polynomial time? Given that our optimization problem retains partial linearity, could approximate solvers be effective when parameters P and are unknown? Since the three-layer market leads to simultancons declines both revenue and social welfare, how should government intervention be structured, for instance, what constitutes an appropriate tax rate for data brokers? Furthermore, does competition among human creators yield excess profits for data intermediaries? Finally, our static model leads predictions and insights about the ultimate equilibrium outcomes. How will these insights change in dynamic settings when GenAI and human creators dynamically interact? In light of the growing demand for data in the AI era and the distinctive nature of data as an economic good, these questions open several promising directions for future exploration.
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A Omitted Details in Section 1
A.1 Related Works
We summarize the following four lines of existing literature pertinent to our work.
Procurement Mechanism Design. The topic of optimal mechanism design was introduced by the semi- nal paper Myerson 198T). Over decades of academic attention, researchers have not only focused on optimal mechanism design from the seller's perspective but have also begun to consider it from the buyer's perspec- tive, especially when buyers are either significantly large or even monopolistic, leading to studies on optimal procurement mechanism design (McAfee & McMillan 1987 Laffont & Tirole, 1993). In the economics liter- ature, Burguet et al. (2012) examines mechanism design in procurement under limited liability and explores ways to reduce the sponsor's is cost, while Gerding et al. (2010)
addresses the service procurement problem in
the presence of uncertain duration. Garg & Narahari 20088 adopts the framework of Stackelberg game
to study procurement auction design. In management science and supply chain literature, Zhang (2010)
investigates procurement with prico-scnsitive demand, while Prasad & Rao (2013); Ketankumar et al. (2015)
focus on the procurement of cloud computing resources. Jin & Wu (2002); Huang et al. (2011) attempt lo design optimal procurement auctions to enhance suppliers' profit margins. Our paper goes beyond tradi-
tional procurement mechanism design for physical goods by focusing on the digital content market, which features transferability. This extends the boundaries of procurement mechanism design in this domain.
GenAI Content Creation and Data Pricing. The development of generative AI has brought AI- generated content into the economic market, sparking research on the nature of AI products and traditional goods. Liu et al. (2024a) discusses the impact of large language models (LLM) on traditional search engines, while Zhang ct al. (2024b) explores advertising business in the context of LLMs. [yer et al. (2024) conducts
case study on the role of LLMs in e-commerce. [Chkirbene et al. (2024) provides comprehensive survey of GenAl's applications, challenges, and trends from several aspects, including content creation and marketing. Mcanwhile, due to the enormous data demand for GenAI training, public datasets are becoming insufficient, leading companies to shift toward private datasets, which has raised concerns about data pricing. Data Shapley and its variants are the most commonly used methods for data pricing (Jia et al.] 2019 Ghorbani & Zou, 2019 Schoch et al., 2022), while recent Ai et al. (2024) proposes a novel data pricing method based instrumental value. Our paper studies how much payment is needed to incentivize human creators to generate sufficient content within an agent-based framework, broadening the perspective on data pricing related to GenAI.
Human vs. Generative AI Competition. With the rapid development of generative AI, competition between human creators and GenAI has gradually emerged (Esmacili et al. 2024). In contrast to traditional aulomation (Acemoglu & Restrepo 2019 GenAl relies on human-crealed data for training, thereby simul- taneously complementing and competing with human labor. Yao et al. (2024) employs the framework of a Tullock contest to examine the symbiosis and conflict between humans and generative AI. [mmorlica et al. (2024h examines the augmentation of users by AI agents and the corresponding shifts in equilibrium. Gao et al. 2025), within game-theoretic framework, investigates the impact of AT tools on market outcomes and consumer welfare. It highlights that in certain scenarios, both platforms and creators may be reluctant to adopt GenAl. Moreover, some papers discuss the unique economic characteristics of LLMs, or broadly GenAl; please refer to Fish et al. (0244]; Duetting et al. (2024) for further details. Meanwhile, some papers explore the collaboration between humans and GenAT in various tasks (Fui-Hoon Nah et al. 2023 Singh et all. 2023 Han et al. 2024). We adopt a mechanism design perspective to study the collaboration and competition between human creators and GenAI. GenAI relies on human-created created content for training, while the content generated by Al partially substitutes human-created content, providing a novel approach to studying the competition between humans and AI.
Three-Layer Markets. The three-layer market offers a more comprehensive framework for characterizing data markets compared to the two-layer market. For further details, Zhang et al. (224a)) provides survey data markets. Fallah et al. (2024) models the three-layer data market multistage game and focuses
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on the subgame Nash equilibrium, while Xu et al. (2220)
considers a three-layer Stackelberg game in the car-sharing market. Besides, Reinders et al. 2018); Song & He (2019) consider three-layer structures in the domain of supply chain. Wang et al. (2023) investigates privacy protection issucs in a blockchain- based P2P three-layer market. [allseiro et al. (2021): ); Zeithammer & Choi (2024) investigate the impact of intermediaries
on the payofis of buyers and sellers in auctions, as well as the phenomenon of double-shading in the market. We compare the impacts of two-layer and threo-layer markets, within the framework of competition between humans and GenAl, both platform revenue and social welfare. We theoretically demonstrate that the three-layer structure would lead to a lose-lose outcome, offering novel perspective for future AT data regulation.
A.2 Relative Terms
We use this subsection to illustrate some terminology used in this paper for readers with less economic background. These terms are definitional, but we also provide the economic intuition behind them to assist the reader's understanding.
Virtual costs: A variant of the original cost, formally defined as c f. Because the platform must incentivize creators to report their true costs, the effective cost the platform faces is the human creator's production cost c plus the incentive-related virtual cost f.
Reverse hazard rates: Originating from survival analysis (Kabbftcisch & Prentice 2002). and defined in this paper as 4. It is used to characterize the log-concavity of F.
Incentive compalibility: The terminology for desirable mechanism property that for any creator with private cost truthfully reporting to the platform always maximizes the creator's 's expected utility (payment minus cost). Hence incentive incentive-compatible mechanisms can help avoid dishonest agent behaviors during information elicitation.
Individual rationality: The terminology of another desirable mechanism property that human cre- ators always have non-negative expected utility (so they prefer to stay in the market than to exit).
Path independence: When payments are defined through a path integral (e.g., f y(t) dt in Mech- anism M4), we require that the integral yields the same value for any path from č to b.
Myerson's analysis: This refers classic and well-known analysis framework of Myerson (9881) that converts payments into integrals of the allocation functions, which reduces mechanism design to allocation rule design.
Convex procurement: We hope the feasible set induced by demand constraints to be convex, so that the optimal mechanism can be computed efficiently in practice.
Valid mechanisms: A creator's higher cost. should lead the platform to procure less from them, while other creators' higher costs should lead the platform to procure more from this creator. This reflects substitutability in production.
B Omitted Optimal Mechanisms and Algorithms
We now detail the concrcte implementation of mechanisms M2 and Mg.
C Omitted Details in Section 2
C.1 Including GenAl Costs
In practice, the generation cost of GenAl is typically negligible compared to that of human creators. For example, once large model such Sora 2 is already trained, producing new video takes only a few
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Algorithm 4 Mechanism M2 for welfare- platforms without union in the two-layer market.
Input: Report č
Replace virtual cost: vc(2) by č.
Call Algorithm | for y-
1/pk for any E [
Calculate the allocations: 2& (c) 4)
Calculate the payments: P& (c] = CkUkk + t yk (e....... Ck-1, Lh Ck+1, 3x df for any k E [K].
Output: Allocation-paymernt pair (z()).p(c)) = ((z1(0), LK (o)), (p))))(p(()))))))) (p.
Algorithm Mechanism MM for Type 2 platforms with data broker in the three-layer market.
# First-stage mechanism design.
Replace virtual cost: ve(ū) by Ejc].
Call Algorithm | for y.
Calculate the allocations: 2 = yk l/x for any & [K]. Output: Allocation-payment pair (z, = ((21,.., b). # Second-stage mechanism design.
Input: Allocation 2, payment and report ?
Calculate the payment: Pz(2) bx=f*
Output: Allocation-payment pair (z()).((O) = (b, ..... bx xx
minutes of computation, whercas professional human creators require substantial compensation and effort to produce comparable content.
However, model naturally extend to scttings with small but non-zero GenAl training costs. We assume the training cost of GenAl is proportional to human production cost, and the training cost parameter of GenAl is G in domain k. In other words, the platform needs to pay & xf to obtain a GenAl agent for domain k. Consider Mechanism MM as an example: our transformation is derived entirely from IC and IR constraints, independent of the objective function. Therefore, incorporating GenAl costs is equivalent to adding Σk & yn to the platform's 's objective. The optimization problem in Algorithm becomes as follows:
y argmin
Σ k=1
UCC (c) + ))
K
1/pm
mon /ul 2 du and y/ 2 for any kt E [K].
s.t. yk
As result, we only need to adjust the virtual cost from vcp vck + ch recover the optimal mechanism under non-zero GenAI costs.
The same modification applies to other mechanisms as well. For instance, in Mechanism M2, replacing 2 with ê + & leaves the structure and insights unchanged. For other kinds of GenAl costs, we can similarly include the cost in the objective without modifying the demand constraints, with or without access to a closed-form solution, depending on the concrete cost form.
C.2 Further Complexity Reduction
Note that we adopt a deterministic payment rule in Algorithm 1 However, we can move to a random manner as follows to reduce the computational complexity in Proposition 2.1 Here, (o, Ck-1, CR2 1k indicates that the input report is CG....,k-1 Ck. Ōx) without ambiguity.
Sample E ~ Unif(Cr, by) and (C( (C) + (bu- - Ge. C [x)
(7)
Proposition C.1. Replocing the payment rule in Algorithm 1 by Equation using the subgradient
method the oracle, the computational complexity of getting a e-optimal mechanism is O Specially, if max; Tik] < for any & [K], the computational complerity reduces to O
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C.3 Omitted Proof in Section 2.2
C.3.1 Proof of Theorem 2.1
Recall Problem 1 and we use (x(c) 2 (c) to simplity the notations. Besides, we use Qx(y, oi) and (x(U:p,ck) to denote J [=(c)f-<(c==tdd- and J ['(x(cc) (=)=(c)]f=((cxx)))) respectively. Therefore, Q represents the units human creators will produce and u represents their expected utility.
Then, we have the following propositions regarding (.(-.) and u. (.,.))).
Proposition C.2. We have that
for any cı E (ak, by], it holds that (V.P,, Cx) us (V.PP,bbb) bk) + S Qu(U,ta)d, for any & [K], it holds that us p. bk) 2 0.
for any to 2 cb. it holds that Quly, 1x) S Qu(y, cx).
Proof. From the individual rationality condition, we know that the expected utility for human creators in domain k at every time is no less than zero. Then, it holds especially when the cost corresponds to by- It immediately leads to (y p, b&) 2
Assuming the real cost parameter is tx but the human report ck- Assume ck I ty without loss of generality. From the incentive compatibility condition, it holds that
uu tx) 2
[Px(CH)) -xu
f - cow(())f- (cx (cx-tx) f vnlo)f-a(t-xddtt
u. ck) (c+- - +x) ck).
Then, we know that un un p, (x) un (y P. ck) 2 (cr (cn-tih -xx)Qu ck).
Similarly, we have that uu(u, (2) uulu. (x) I (c - (x) (x) by changing the order of la and ck- Thus, it holds that
(c+ (a--t)]ucu1 S uu(y,,,te) uu(v.p,c)) S (ce (x))
Since C Lx, we know that Qu (u) s Qu(y,c)).
In addition, dividing e* tx, it holds that dus Qu (y, ck). Here use the fact that Qu -) is continuous dee which will be shown soon. Otherwise, we can still get the following from the definition of Riemann integral. We then obtain that un (v.pp, + fth Qu(y, dd un mx(y,p,4e) (x). Therefore, with simple algebra, we have that
un (x(..p.c)) uu(.ppbb)) 1/m Qulu,ta)de
which ends the proof.
Now, let's turn the problem of how to maximize content creation platffrrms' expected revenue (minimize expected cost). We use uo(v,)) = Ekn -pppp(p -px((c)(ccdd to denote the expected revenue for shorthand. Recall that the demand constraints are now
1/p yk
(c) +
(c))(c) ds
i=1
and
(c) 2 0
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for any & E [K].
We first decompose uo p) as follows, namely,
uo uo(v.))
(-Pe(c) + (a
(c)
Then, it holds that
91 = Σ
uu D, (e)
Σ
/ /lux(V. [ (u(dutt)dttallacca)dd
1 SrM
bN
Σ
uk p. bk) - - pNx
Σ
Fu(ta)Qn(y,taddt tx)
Σ uk (u. (.b.P.bb)) - FullH) / (oni)/u(1)00 (n(())sx((
-u (.....Pbb)) - Fa((t)
Σ
yn (t))((t)d
fu(tw)
The third equation holds due to the property of p.d.f. while the fourth equation holds because change the order of integration. Combining 92, it holds that
uo(..p) uplypp / |ca + F(((*) (c)
yR
fx(cx)
By defining the virtual cost (c) = Ck + Ee(ce) (e(ex) then we have
uu(y.p)= Σ (y..ppbb)) vcalav)ux()))(c)d --uaaoaa---mtcaaatrtoo
// minimize Σ vcs(c*) subject to 1/px (c) + IK (c) 2 d+. and Vi((e) 2 0 for any & E [K], so we only need to prove that the payment rule indeed results in us (v by) = 0. Here, since we consider an IC mechanism, we have naturally e C.
Note that in Algorithm
Recall that from Proposition
C.2 it holds that
u (v.pp bx) ux (ul.pppc.)) f/h Qu(y, tx)
f (enva(()
f be (
/ c-ccua()) + pp(c) - -1 (u(tts---)dttllf-A(C ddc
bo
Here, we use (1x, cx)) to represent (cI, Ck- txs Co+1s CK) with a little abuse of notation. Since we set the payment rule (c) = (cUa(c) + S (ll,,, ddd in Algorithm we know that us (v. p. by) 0 exactly.
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Together with Proposition C.2 we know that the first term in uo p) is maximized as well. Therefore, since two parts in (o(v: p) are maximized simultaneously, "o(y, p) takes its maximum using Algorithm shows the optimality of our mechanism.
which
Additionally, since the alloca ue(..-) 2 O, we know that
1h) -cyyk(c) + P&(C)= = 1. (R(CI,..,,C-II11 Ck+l CK ck)dtx 2
for every C. Thus, actually obtain much stronger individual rationality condition, say -c k (c) + (c(c) 2 0. In other words, the utility of human creators is non-negative for any h E [K] and cost veclor c E (a1,b) bi] x .... x lak, ba]. So, of course, the individual rationality holds in the expectation manner, and human creators won't exit the market due to a negative utility.
As for the incentive compatibility condition, we first show that ves is increasing with respect to eg- Since we have Assumption 2.3 we know that log(Fi(-)) is concave, holds that P is decreasing. Therefore, it holds that ucz (c) ck + Fe(c))
Jalen) increasing in C for all [K]. Together with the implementation of Algorithm | we know the objective function is linear in y, and from the following Lemma C.3 we will see the feasible region is convex, then we can actually derive similarly a stronger version of the TC condition, namely, -cyaf (c) + (c) 2 -cxaf (Gx, c-x) + Pk (Ck, c-a). Specifically, when the true cost is ck but the human creator reports Ck2 it holds that the creator will have utility loss
/ ce
nck
-
Ck-1, Ck+l, CK) +
(c1, Ck-1, 1.,,, Ck+l, CK )dtk.
For overreport Ō₂ > Ck. holds that (c ... Ck-1 Ck Ck+1, CR) S (,. 1 tk Ck+1: CK) for every la E [cE, [k] in our convex optimization problem. Meanwhile, for underreport &h Ck. know that (e, Ck-1: Cr Ck+1, ck) (e., the agent incurs a non-negative loss. Thercfore, truthful reporting is dominant strategy, which implies incentive compatibility. Similarly, we get that Qn (v,-) is continuous almost everywhere, and Mi is valid for the same reason.
Cet, for every to E [x]. In both cases,
C.3.2 Proof of Proposition
We first prove the convexity of the area and then give the computational complexity.
Lemma C.3. yk 1/px + +EK pa,,, /u/ee > de and S yk de for any k [K] construct conver sel. Specially, when max;
(ik (2th Pi 1, it becomes strongly convex set.
Proof. Assuming fx(y) = Vic 1/pl EK Maay? Ta/pp it holds that
fu) = diag, (Hin Yik Yikk 1)v,"
74k 2
--- 1)vE*
1fi [1}
Pi Pi
Pk Pk
Since have Pk 1 and Fik < 1, it holds that (²fe(y) is diagonal matrix and all components are non-positive due to 0 yk S dft Hence, we know that fu (y) is concave function, showing that fa(y) ds is a convex set.
When max; ( <1, it holds that all components of 2 (v)) are negative. Therefore, we have that fu(x) is a strongly concave function, and similarly, fu(u) 2 de yields strongly convex set.
Note that the intersection of (strongly) convex sets is also a (strongly) convex set. We know that constraints Vi 1/pu + EK Mu33 /upp dę and < yk S de for any k E [K] yield a convex set. In the meanwhile, we know that if maxx (i } < 1, the set will become strongly convex set, which ends the proof.
7
P.
With Lemma C.3 in hands, know that the computational complexity of Algorithm ! equalling finding the minimum of (strongly) convex function with primal-dual methods. Therefore, using the subgradient
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method Nesterov 2009 2014 and constraining 0 S yk I de lead to an O(2) complexity to calculate
one allocation y immediately.
When max,1 Tut < 1, the strongly convexity leads to O complexity
instantly Nesterov
2005
For the price voctor p(c), we use the Monte Carlo method to estimate the second term, i.c., Sex yo C Since from the implementation of Algorithm we know that is bounded by de Then, Rubinstein & Kroesc 2016) tells us that we need O ((22) times of simulation to estimate with tolerance c. Therefore, the
total computational complexity ((+) for general cases and O when max; Ie < respectively.
C.3.3 Proof of Proposition C.1
Due to the linearity pl)), only need to prove that Equation payment rule, i.e., Byy [bb Ee (C1, ..., Ck-1 1, CR2 Ck+l, ..., [k)
7 gives unbiased estimator of the optinal bbe
Fa yu yu(,i,..., Ck- th CR41s ".. CK dlg.
Tt holds that
T- [lbk (C)) G,,,, -,.,,.., (bk- Txk L co
be
1 dle
ye( C Cx+1, 1..tt......
-.,,... Tx)
by Ck
bb L
ya Ix-1, tu: )dt
where the first equation holds because ō follows an uniform distribution over [Ck, bs}-
Therefore, we know that we give an IC,IR and coppimal mechanism with only one sample, namely yu. Cx-1, Ck2 ...,,)) With the proof of Proposition 2.1 we know that the computational com- plexity ((2) for general cases and O when max, ( Take
< for any [K].
P.
Nonetheless, the variance of the payment increases from O(e²) to O(1) as we decrease the mumber of samples when using the Monte Carlo method from (
) to constant. To better balance the tradcoff between computational complexity and variance, i.e., uncertainty, we may need n samples to estimate
f Qx-,,,, Cc+1...., .x.......ddddt Ex)d yielding (()) variance and ((5)/((²) complexity.
C.3.4 Proof of Corollary 2.2
From the constraint yk 1/Px EK Pacy Put// > du, we know that when ys 0, the corresponding slope is infinity. Besides, from the knowledge of convex optimization, we know that (uc1, uCk) is associated with the corresponding subgradient. However, due to Assumption 2.2 ucs is upper bounded and finite. Therefore, the solution of y is strictly positive in all components given usk + 0 and 7/k # 0, which ends the proof.
Since the feasible area is an intersection of K convex sets, when the optimal y is located on the boundary of the k-th set, the k-th demand constraint will be tight, Le., binding. Otherwise, it will be loose, i.e., non-binding. Then, we know that although all domains still exist, some will experience overproduction and it completes analysis.
C.4 Extension to Multiple Human Creators in Each Domain
We conclude that the extension will retain the convexity of the optimization problem, hence preserving the validity of Theorem 2.1 and Proposition 2.1
We first write down the revenue-maxinn maximizin problem as Problem [ 24 (ct, ph resp.) for the i-th creators where i [ng]. We assume these n;
We assume there are ny human creators
in the k-tt domain and humans have i..d. cost function. We stack of and we use y to denote (zt) as before. With these
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preparations, we have the following optimization problem,
max
Ee
(c)
x.P
k=
nk
K
23
s.t.
Σ (c
+ Σ
Ugk
Σ
2)()) 2 du for any & & [K]
1=1
,=1
==1
E_ E- (1 (c)) + 2 -cc.ol-
E_ E- ( (x) + +r((e)
20
(c) 2 0 for any (k: i) E [K] x [mx]
where (k; means laking expectation over all other human creators.
Since the IC and IR conditions hold individually, we know the optimization goal becomes s(c) Falci) accordingly. We are now ready to prove that the constraints still in- argmin, Ex Einl + Jute duce a convex set.
By replacing (xt) by vee the last constraint turns to y 0. The first constraint is now (i Σ Pykk 1 (/osjss 2 dz- Note that Ps 2 for any h E [K] and 7ik E [0, 1]. We know that E 1/Ax 1/PM + Σ/= Hyx (1)
is a concave function with respect to y. Therefore, the area corresponding to larger than de convex set. It soon holds that y belongs to convex set because the intersection of convex sets is still a convex set.
We conclude that even with multiple human creators in each domain, the revenue- maximizing problem is still a convex optimization problem. We can find the optimal mechanism by solving
Fa(ch)
v(c) argmin
Ich
[x(c))
K
ny
s.t. 1/p
+
Mysk
( Messts 2 dg for any E [K]
j=1
v 0 for any (k, i) [K] [na].
Additionally, the proofs of Theorem 2.1 and proposition 2.1 still hold as they are applicable for general convex optimization problems, so the optimality and the complexity results remain valid. Moroover, we can choose a different e for each domain and then allocate assignments among human creators, showing the equivalence of considering only one content creator.
Morcover, we remark that we assume creators in domain & share common Pk to reflect the similar cco- nomic characteristics of creators within the same domain. Matherattcally, our results do not rely on the homogeneity of Pk. Suppose that the i-th creator in domain & has cost (x))), By defining v = (at the optimization problem can be rewritten as
K
Fu(ck) La [a(c))
y(c)= = argmin
Jok +
live
s.t.
1/h
( Velgre 2 de for any E [K]
V 2 0 for any (k, i) [K] x [nk],
which still preserves convexity after the transformation. This illustrates the generality of our model.
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C.5 Omitted Proof in Section 2.3
C.5.1 Proof of Theorem 2.2
From the proof Theorem 2.1 we know that to satisfy IC, the payment rule has to be px(c) = ckskk + f y)(c_,,(e)dd (x)d + C' where C- means all costs except the k-th entry and C' is global constant. To guarantee the participation transformation yı (c) (x(()), From the implementation of Algorithm [ we know that for any positive virtual cost, (-)-) is larger than zero letting alone c. Thus, it holds that Jer yu(c_a, t&)dtt + C 2 C. is not negative, we know that the IR constraint is satisfied. Considering the corner case that c b, we find that the integral zero, so we need that C = is the optimal choice. This shows the feasibility and the optimality of our payment rule.
the IR constraint, we need that Ps(c) casl because of the parameter
be no smaller than zero. Combining these two results, we know
As we have shown that for every allocation rule, we can find corresponding payment to motivate its implementation, we know that we only need to set the virtual cost equaling to the true cost noting that we hope to obtain the lowest social cost, namely ming E Caa% (c) = miny Σx C+3)ec)). It yields the optimality of Algorithm | as it satisfies both IC and IR, and achieves the highest social welfare while fulfilling all demands. Since we only need vea (-) to be increasing to obtain the validity and eg of course increasing, it holds that Ma is valid.
As for the complexity, we can inherit the results of Proposition 2.1 directly the main computational time lies in the subroutine Algorithm 1 So, the computational complexity is polynomial and it finishes the proof.
C.6 Omitted Proof in Section 2.4
C.6.1 Proof of Lemma 2.3
In order to prove that 1e v(t) dt is independent of the integration path for any c, we first write down how decide v(-). From the implementation of Algorithm 1
we know that
y(c)= = argmin e.y
y
s.t. 1/pl
Uik U ne/pi 2 dx
yb 0 for any k E [K].
2019). we know that a sufficient condition is Vxy(c)= 0. On the other hand, if can find potential lunction (e() such that VV((c) y(c), it holds that
From the knowledge of analysis (Griffiths & Schroeter
V V x = v (c) due to V = 0.
We use I to denote the constrained area that I = ly : yk 1/0x + EK Payy T0L/P 2 de and y 2 for any k e []]. From Lemma C.3 we know that is a convex set. Note that e. y is a lincar function, then we can define (c(c)= = minyez y. Let's now detail when we change to c + de, how will y change. There are two possible outcomes. If y()) is a non-smooth point, then y(c + +5)) y(c) (ref. Figure 5 Otherwise, y(c) is on smooth segment, and we know that c Vy(c)- de = 0 (ref. Figure G). Since d. can be arbitrary, it holds that c. Vy(c) = 0. Therefore, we know that Lf + 5c) - (c() de s(c) in the first case and ((c+ + 5,) - (c) + 5, y(c) + e Vy(c) 6, = ó, v(c) as 5, - 0 in the second case. We know thereof that (V((c) = v(c), showing the path independence, Since we have proven the path independence, holds that the payment rule p(-) is well-defined which ends the proof.
C.6.2 Proof of Theorem 2.3
From the revelation principle (Roughgarden 2010), we know the existence of mechanisms satisfying both IC and IR. Let's analyze the IC condition first. It holds that p(c) p(c) c v(c) y(e). Since ē can approach from any direction, know that Vp(c) Vy(c). In practice, applicable mechanisms are
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y2A
A
I
y(c)
I
+ dc
C + c
C
y(c 66)
yc)
V1
V1
Figure 5: When y(c) is vertex.
Figure 6: When s(c) is not vertex.
usually smooth, so we assume that these gradients exist. However, as we will see below the payment rule is indeed a constant, this assumption is not mandatory.
Consequently, holds that
p(c) = p(()) - (1)))-(((vvv0)) d't-vv())dd=-cc-e
c. + [[' v(t) v'v0)=tt CC,
where the second equation holds due to the integration by parts formula and C here is a constant.
We now know we can use this kind of payment rule to induce any allocation rule, which is thereof applied to social welfare maximizing platforms. Since contains family of mechanisms, we use the IR condition find the one with the highest revenue. Recall that need to guarantee p(c) c- y(c) > 0 for any c. Since there are constraints that (-)-) 2 0 for any k E [K], it holds that C' 2 maxe[- Je v(t) df). When <= 6, we know that the right-hand side takes the maximum zero. Therefore, we know the smallest C is zero which is optimal.
The last question is that we only show necessary condition and need to prove the IC condition actually holds. From the proof of Lemma 2.3 we know that Vy(c) = 0. Therefore, constant, say 6 vb).. Then, only need to show that y(c) S c y(C). Note that from Algorithm 2 we find y(c) to minimize e y. Thus, it holds that e y(c) = argmin yer S c y(C) as y(C) is also feasible candidate.
know that p(.) is in fact
Therefore, we show the payment rule in hm satisfies both 1C and IR. Since the objective function is Σ. Crll= = E C&yA = y when c c= and y/ = o we know that the mechanism M4 achieves the highest social welfare. Moreover, since M4 has the same allocation rule as M2, we know it's valid immediately.
Note that most of the computational resource consumption comes from calls to Algorithm D hence we know Algorithm 2 is a polynomial algorithm, which ends our proof.
We finally give the following remark. Observe that from the proof of Theorem 2.3
we don't need Assump- tion [2.1 anymore unlike Theorems 2.1 and 2.2 It may suggest the potential success of high-dimensional mechanism design problems beyond cycle monotonicity (Lavi & Swamy 2007). which is of independent interest for future work.
C.6.3 Discussion on Mechanism M3
The difficulty of Mechanism M3 stands in sharp contrast to Ma, which maximizes social welfare. Un- der welfare maximization, one can leverage a potential-function-bee independent payments. However, this approach docs not apply to Mochanism M3. Specifically, for M3 we require path-independent allocation rule that satisfies y(c) argmine y f Vy(t) dt, which is
formulation (cc) to construct path-
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fundamentally challenging because the linear objective and the integral term are tightly coupled. Moreover, enforcing path independence, without access to a potential function (The existence of a potential function implics that the field is conservative, which naturally leads to path-independent payments), makes identify- ing such an allocation especially difficult. Specifically, there is no clear evidence showing that naively derived y(c) = arg min c. y y Vy(t)- dt naturally satisfies the integral path-independdenee property.
This comparison highlights an important advantage of solial-welfare maximization: it enables simple and practically implementable mechanisms. Meanwhile, our experiments show that the welfare-maximizingg mech- anism only incurs minor revenue loss, further supporting its practical relevance.
C.7 Omitted Proof in Section 2.5
C.7.1 Proof of Theorem 2.4
First, from the implementation of Algorithms 2 and 4 we know that they achieve social welfare optimum the demand constraints are unavoidable. Therefore, we know that SW2 SW4 and they are larger than both SW1 and SW3, yiclding SW₂ = -Wa 2 mSx
In the meanwhile, since M1 and M2 have the same constraints, then the two solutions are both feasible for each optimization problem. Thus, since Mi maximizes the revenue of the platform, we know that Rev 2 Revz
Now, let's compare Reez and Rev. Since they have the same allocation rule, we only need to compare Ex let yu vu(c-ki (,dd)d and Je u() - dt. Since we know the latter integral doesn't depend on the integration path, we choose a path as follows. We first go from to (bi, Ca., and then from (bi, c2, (x) to Ph, (b), b2, ca, "3.....k) etc. The k-th road is from (bI,..., bb- Cke ..., ck) to (bi,.., ..., bb, Ik+,. CK). Therefore, it holds that 1e y(t) dt- Ex foy (b....... b the ck+1. (R, dx)dtt. From the implementation of Algorithms 2 and 4 we know that yu (bi,.... be. Le, I........ cx) 2 Ms(C,, ..., &R Ck+ls CK) as the objective function is linear and the feasible region is convex set, yielding Σn Ser 1hx (olcck;tt)dde S Se v(t) dt. Thus, we know that Ex px(c) (c) in M2 is no larger than Ee(p(c)) in M4 and Revz Revs.
Note that M3 and M4 have the same constraints though different optimization objectives. From the above proof, we know that if the allocation of M3 is xf-), it holds that p(c) 2 e. s(c) + S. v(t) dt where (x(()) xx(+).. Here, we also need that fe y(t) dt is independent of the integration path.
Consequently, as Ms has one more constraint on the integration path than M1, the allocation of M3 is also feasible for the optimization Problem I Let's 's assumee the corresponding allocation rule is associated with y()).
Notice that the payment rule of Ms holds p(c) 2 c. y(c) + S.e v(t) dt, we only need to show that c. y + Ie y(t) dt 2 c y + Ex l (t,,,, cd2) Since Ms is a valid mechanism and has path independence, we choose the following path which goes from the first coordinate to the last one in order. More specifically, the k-th road is from (bi,-.,b h 1,, Ck. C.+1, (x) to (b,,... b be, Ck+1,-.,,Cx cx) along the k-th coordinate. Due to Definition 2.1. it holds that (bi...... .... bb te, Ck- 1, .... cx) 2 (n c- and then we know f y(t) dt 2 Ex thh yn (la, cdddl similar to the proof of Theorem 2.4 yielding p(c) 2 e y + Σx 1th yu (4,, co)dda.
Since the payment of M1 is the minimum of y + Ex Shr yu (.tt, over all possible allocation rule z(-) or equivalently v(-), it holds that p(c) is no smaller than the payment of Mi. Therefore, the expected payment of Ms is at least as large as the one of MI. Since the platform's revenue is equal to the negative of the payment, it holds that Revy < Rev,, which ends our proof.
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Omitted Details in Section 3
D.1 Timeline of the Three-Layer Platform- Braern---Booer-reeatee Markets Markets
We use the following figure to show the timeline in three-layer markets.
The platform announces (z, t).
The broker announces (x,p).
The creators generate x x(C) and get p(c).
The broker decides to accept/reject (z, t).
Human creators report č.
The broker gives the platform units of content and gets t.
Figure 7: The timeline of the three-layer market.
D.2 Omitted Proof in Section 3.1
D.2.1 Proof of Theorem 3.1
First, as notice that the final allocation in both stages is constant, we know that Mg is valid.
We now begin lo consider the second-stage mechanism design. Note that in this stage, and are fixed constants. Therefore, changing the objective from E,[ Σx Pk 2c; to Re(- 1--ppeees pk(c; 2, f)] won't change the optimal allocation and payment rule. Comparing Problem [and Problem ! we know that the only change is the demand constraint is now zy<(;; 2, 2 2k: parallel other axes. Since we know 24 2 0, we can certainly omit F& O. From the proof of Theorem 2.1] we know that this optimization problem equaling to minimizing Pe(cs)
miny [e + [e(es) yuk subject y/ 2 zu where yh = x Since C2 + Talen)
Pl(cx))
1/pa
0, it holds that the optimal
for any c. Recall that due to Assumption have validity directly. In the meanwhile, we know that Pa(c 2, t) C2y 2), J ya ( 2-12, using the fact that Ve(c; t) = 2h for all c, we know that pa(c 2, t) = buyu 2, = by Pt. Since by 2 ck the IR constraint is preserved.
V)(G2, is PP hence (z(G
2.3 C₁
Fe(ox) is increasing, Jalen)
The third step is to find the optimal first-stage mechanism. Since need to satisfy Eglt ExPiic 2 and from above know that p (c; 2,t) (c; = buzf*. So, know that t 2 2x As the objective to maximize Ee[-t], equals to minimizing ΣK bawn where we use wa denote =f Now we turn to the constraints. It holds that w//n ΣX Haaa?ht/ 2 dy and 2 0 for any k [K]. From the implementation of Algorithm 1 we know that we only need to replace vcy (2) by be, and is the output y of Algorithm J
Therefore, we prove the optimality of the sttgn mechanism design using 2,(c;2,t) 25 and 2,) = bezfa Also, we know that the allocation rule in Ms optimal. Since can be any num- ber no smaller than Σk beaf". we know that the payment rule in the first stage is optimal as well.
Finally, since the time complexity mainly depends on the call of Algorithm | we know Ms is polynomial-time from Proposition 2.1 immediately.
D.3
Omitted Proof in Section 3.2
D.3.1 Proof of Theorem 3.2
Note that the second-stage optimization problem is the same the one of Theorem 3.1 We only need to replace the first-stage problem with Problem 0
Now, the objective is to minimize E, Σ cx (x( 2,
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Since we know that 2;(c;,, = 24, it holds that we need to minimize EI Ca cx2 Using we to denote of" it holds that the objective is now [[c] w under the constraints w + + Msaw, f 2 dy and wi 2 for any & E [K]. Replacing (C) by E(cx] yields the optimal w immediately.
K
Since need to guarantee 2 Σn 2f that the broker will participate in the market, set Σ b. w, which is the lower bound of feasible t. Therefore, it holds that (2,t) in Ma is the optimal first-stage mechanism.
Similarly, we know that the corresponding (z. p) is the optimal second-stage mechanism. Since the allocation is fixed in each stage, we know that MM valid. As for the computational complexity, it's polynomially dom- inated by the one of Algorithm Therefore, we know it's polynomial-time algorithm from Proposition 2.1 which ends the proof.
D.4 Omitted Proof in Section 3.3
D.4.1 Proof of Theorem 3.3
From Theorem 2.4 we know that Reve 2 Revg. Also, in the proof of Theorem 2.2 we know that the payment M4 is fact constant b- y(b) where y(b) is the output of Algorithm 1 with input From the implementation of Algorithm 3 we know that the payment of M3 is also b- y(b) thereof Rev, = Revs. Note that Ms and M& have the same second-stage optimization problem, therefore MMs is also feasible for Type I platform. Due to the optimality of Ms shown in Theorem 3.1 we know that Revs 2 Reve. Hence, it holds that Revy 2 Rev, = Revs 2 Revs.
Similarly, as Ms is feasible for a Type 2 platform, we know that SW, 2 SW, because of Theorem 3.2 Note that M2 and M4 achieve the highest social welfare for each e while SWs only uses the optimal allocation with respect to R[c]. Therefore, for each e, the allocation of M2 and M4 is at least as good as the one of Mg. We then know that SW2 SW4 2 SW,. To sum up, it holds that SW2 = SW, 2 SWo 2 SWs.
As result, we've shown that the three-layer market will lead to losc-lose situation that
max (Revs, Revs) & min Revz, Revi)
and
Sax & min min, SWa).
D.4.2 Proof of Corollary 3.1
We only need to construct mechanism based on Ms which is feasible for Problem We choose the second-stage mechanism in Algorithm 3 as candidate. We know that the total payment of Ms is now Σx buzk* and the allocation rule is constant vector z. Then, we choose p(c) = Ek bazt and x(c) = for Problem
For Problem 3 since we know also satisfies demand constraints all domains, holds that xx +
( h 2 du also holds. For the IC constraint, we know that p(c) -kk cxa (c) Ek buzt
ck2f" = =(()) Ek aaff (7) preserved. For the IR constraint, it holds that p(c) -EK Gat PN (c)
Ex b
Pk -K C12lt =
=x)=f* 2 0 as by 2 cp. Finally, xx(c) 2 for all ks E [K] as 2 2 0
entrywise.
Therefore, we find feasible solution to Problem 3 Since Rev, corresponds to the optimal solution, we know that Rev; 2 K bezl" Revs, yielding lower bound for Reva, and it finishes the proof.
E Omitted Details in Section 4
We conduct all the numerical experiments written in Python 3.11.7 running laptop with an Apple M2 CPU, and we provide more omitted details as follows.
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M Hmppptanny
" of Aomnapob
My W M
-20-
socal Veltere
-10-
Revenue
-25
-13
-30-
-20
-40-
SW1
-25
5202
-15
100
00
80
100
Tias
Thats
(a) Comparative platform revenue volatility for M1 and Ma (b) Comparative social welfare volatility for Mr and M2 over over 100 trials.
100 trials.
Figure & Volatility comparison between Mechanism and Mechanism 2.
Experimental evaluation of revenue and welfare among mechanisms. For series of experiments to compare the performance of Algorithms to and 5 we randomly choose js following Unif(O, 0.5), 7 following Unif[0.8, 0.9), d following Unif[2, 5] and p following Unif[1,1.5] To better visualize our results, we still set K- We introduce asymmetry that e follows N(00, 1) truncated by [0, 10] and C₂ is sampled from Unif(o, 1]. We follow Equation (7) and use 100 trials to take the average in order to reduce the variance of our payment rule. Finally, sample the cost vector 1000 times and compare the corresponding revenue and social welfare. Here, compare the resulting revenue and social welfare under different market structures and objective choices, holding the distribution of costs e (i.e., market randomness) and other hyperparameters fixed across all mechanisms. Therefore, the differences observed are solely caused by structural changes (e.g., two-layer vs. three-layer, revenue-maximiiing vs. welfare-maxiiizznng). which makes the comparison meaningful.
We first give the following Table 1 to show the average revenue and social welfare of M1, M2, Ma. Ms and Ms. and we use Theorem 2.4 and Corollary 3.1 to bound the ones with M3.
Table 1: Revenue and social welfare of our mechanisms.
M1
M2
M3
M4
M2
Mg
Revenue
-15.96
-20.28
1-30.68,-15.9
15.966
-30.68
-30.68
-34.21
Social Welfare
-5.24
-4.79
< -4.79
-4.79
-9.58
-4.96
Note that the revenue is the negative payment, and the social welfare is the negative total human cost, so they are both less than zero. The experimental findings corroborate Theorems 2.4 and 3.3 demonstrating that the rankings of revenue and social welfare observed the experiments are consistent with their theoretical rankings.
We visualize in Figure 4 the distributions of revenue and social welfare across different algorithms and provide further explanation and economic insights. We use a red line to represent the mean and an orange one for the median. Moreover, we draw the first quantile, the third quantile, and the extension with a 1.5 interquartile range in Figure ! We find that compared with mechanism M2, mechanism Mi has more outliers and extreme points. We also plotted the specific Rev,, Revz, SW1 and SW2 for the first 100 trials in Figure which clearly demonstrated the high volatility of mechanism Mi. may hint that the social welfare maximizer can enjoy more stable market, say the social welfare and even the revenue are steady, compared with revenue maximizer. This could perhaps explain why some companies turn to maximizing social welfare. Moreover, in the layer-layer market, data brokers and online platforms first establish contracts,
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which are independent of the realization value of cost c, resulting in less volatility. This is also clearly reflected in the box plot.
Revenue comparison
Social Welfare comparison
-10
Social Weltare
Revenue
-20
(with)
(with) Mean (without) with GenAl
(without)
-40
with GenAl
2
2
Trajectory index
Trajectory index
Figure 9: Visualization of Rev and SW with and without GenAI (lower values mean worse performance).
With and Without GenAL. Finally, we provide more details when studying the gap between the presence and absence of GenAL. Recall that when = 0 is equivalent to expelling GenAl out of the market. We adopt the same setup as in the evaluation of the mechanisms. The experiment is repeated 10 times, and the results are reported Table 2 Besides, we include bar plots to visualize their gaps Figure g
Table Revenue and social welfare with and without GenAI in 10 trajoctorics.
Rec
-10.88
-16.65
-7.98
-34.49
-15.55
-15.24
-15.88
-13.18
-24.51
-10.87
with GenAl
SW
-2.94
-5.54
-1866
-9.22
-4.47
-5.01
-4.58
-3.42
-4.81
-2.86
Rev
-21.82
-21.95
-32.08
-42.91
-36.93
-20.57
-35.46
-35.83
-23.68
-37.80
without GenAl
SW
-3.86
-4.20
-4.93
-7.13
-6.54
-3.40
-7.49
-6.15
-4.84
-8.02
The average increases from -30.90 to -16.52 while the average social welfare increases from -5.66 to -4.47. Therefore, the revenue and social welfare increase by 46.54% and 21.02%, respectively. This indicates that the application of GenAI moderately improves production efficiency, while simultancously incrcasing both the platform's 's revenue and overall social welfare.
33


Reviews:
[{"review_id": "n0lDE8BOGD", "reviewer": "Reviewer_ELEV", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies nonlinear procurement mechanism design in content markets where online platforms employ both human creators and generative AI (GenAI). The key novelty is modeling data transferability, the idea that content from one domain can improve GenAI performance in other domains. The authors analyze two market structures: a two-layer (platform-creator) market and a three-layer (platform-broker-creator) market, and derive optimal incentive-compatible and individually rational mechanisms for both revenue-maximizing and welfare-maximizing platforms. The main findings include: (1) human creators remain indispensable even with advanced GenAI, (2) overproduction occurs in domains with high data transferability, and (3) for three-layer markets, the introduction of intermediaries can distort incentives, reducing both platform revenue and social welfare. Numerical simulations validate qualitative insights.", "claims_and_evidence_yn": "Yes", "claims_explanation": "**Strengths:**\n\nS1. Novel and timely problem formulation.  The paper tackles the economic architecture of the GenAI ecosystem, a highly relevant topic for the TMLR community as the field grapples with fair compensation for data creators and the sustainability of data markets.\n\nS2. Solid theoretical contributions. The authors derive closed-form optimal mechanisms for five of six settings (M1, M2, M4, M5, M6) and provide bounds for M3. The construction of a potential function to handle path independence in the union setting (Lemma 2.3, Theorem 2.3) is technically elegant and may be of independent interest beyond this specific application.\n\nS3. Clear economic insights. The paper provides interpretable results: Corollary 2.2 shows that no domain completely disappears due to diminishing returns and increasing marginal costs. The \"lose-lose\" theorem (Theorem 3.3) provides a clean characterization of inefficiencies introduced by data brokers. The surprising result that $Rev_4 = Rev_5$ offers interesting economic intuition about market power.\n\nS4. Complete treatment of multiple market structures. Analyzing both revenue and welfare objectives across two-layer (with/without union) and three-layer settings provides a comprehensive picture. The comparison theorem (Theorem 2.4) nicely organizes the relationships.\n\nS5. Computational tractability. Establishing polynomial-time algorithms (Proposition 2.1) makes the mechanisms practically implementable, which is often overlooked in theoretical mechanism design work.\n\n\n**Weaknesses:**\n\nW1. Strong modeling assumptions that may limit applicability. The representative-creator assumption (one creator per domain) is defended via aggregation (Section C.3), but the aggregation argument assumes homogeneous $\\rho_k$ within domains. In practice, creators have heterogeneous production functions.\nThe assumption that $\\mu_{ik}$ and $\\gamma_{ik}$ are publicly known is strong—these transferability parameters are difficult to estimate in practice and may themselves be private information.\nThe independence assumption (Assumption 2.1) rules out correlated costs across domains, which seems unrealistic when creators face common market conditions.\n\nW2. GenAI modeling choices are debatable. The assumption of \"negligible\" marginal cost for GenAI relative to human creation ignores substantial compute costs, especially for large models. As of 2024-2025, training and inference costs for frontier models are significant. The model assumes GenAI quality scales with human data via a simple power law, but recent work suggests more complex relationships (data quality, distribution shifts, etc.).\n\nW3. Missing discussion of dynamic aspects. The model is static, but in reality, GenAI capabilities improve over time, potentially changing the entire market structure. How robust are these mechanisms to evolving $\\gamma_{ik}$? Creator learning and strategic behavior over multiple interactions is not considered.\n\nLastly, mechanism M3 remains unsolved: The revenue-maximizing mechanism with a creator union lacks an explicit solution. Given that unions/collectives are increasingly relevant in practice (e.g., writers' guilds, content creator unions), this remains a gap.", "audience_yn": "Yes", "audience_explanation": "This is an interesting paper with strengths shown above.", "requested_changes": "It would be helpful if the authors could provide a discussion of the weaknesses outlined above.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "yBozUEA08O", "reviewer": "Reviewer_k4Hf", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies nonlinear procurement mechanism design in markets where a platform acquires content from human creators and GenAI, introducing cross-domain transferability of data. The authors analyze six mechanism variants (M1–M6) across different market structures. A core contribution is to capture how content in one domain enables GenAI to generate content in another domain.", "claims_and_evidence_yn": "No", "claims_explanation": "The claims are partially convincing to me. For example, while I am convinced on items such as:\n\n- I can see that there can be information flow from different domains that boost the productivity of AI systems.\n\nI am not sure that I am convinced:\n\n- That the transferability is absent in other goods. For example, regarding \"R&D spillovers\": Innovation in one sector increases productivity in another (e.g., semiconductor advances benefit healthcare imaging. Another e.g., is the skills learned when a company builds a car engine is transferable to building jet engines --- a famous instance is Rolls-Royce). In a broader sense, once can also say technology transfers (in between countries), when considered as a \"good\", do have similar effects. \n\n- I am also not completely convinced by the argument $\\rho < 1$: This is because one can also argue that as generative AI continues to advance, the effective cost of producing novel ideas decreases. In my own experience, I can now explore many directions in parallel and quickly identify the most promising ones—something that would have required substantially more time before.\n\n- Scaling laws look like a plausible assumption, but not validated in the context of cross-domain content production.", "audience_yn": "No", "audience_explanation": "TMLR’s readership is unlikely to find this paper aligned with the venue’s core interests. TMLR focuses on empirical machine learning, theoretical work grounded in learning algorithms, and analyses of artificial or biological learning systems \n\n(https://jmlr.org/tmlr/editorial-policies.html#:~:text=Transactions%20on%20Machine%20Learning%20Research,artificial%20or%20biological%20learning%20systems;),\n\nwhereas this submission is fundamentally an economics/game-theoretic paper centered on pricing digital goods, platform revenue optimization, multidimensional incentive-compatibility constraints, and transferability as an economic property. Although the authors motivate their work through GenAI, the actual contributions lie in mechanism design and economic modeling rather than in machine learning theory: there is no learning algorithm, no optimization method beyond a convex reduction, no ML benchmarking, no generalization insight, no modeling contribution to GenAI systems, and no theoretical advance in ML. The connection to machine learning is therefore minimal and largely thematic. For these reasons, this work may not fit TMLR’s intended audience and would be more appropriate for venues in computational economics, econometrics, operations research etc.", "requested_changes": "If this paper were to be published in TMLR, I recommend:\n\n**(a) Add a detailed background / primer**\n\nMany readers will not be familiar with:\n\n- virtual costs,\n- reverse hazard rates,\n- incentive compatibility (IC)  (does that simply mean \"the utility the creator gets when reporting truthfully must be at least as high as the utility they get by not. In the notation, diverging from truth is denoted as $\\hat{c}_k$, if so, such an elaboration could be useful)\n- path independence,\n- Myerson’s analysis: I am not sure if the readers of TMLR will be familiar with this.\n- convex procurement formulations,\n- “valid mechanisms” monotonicity conditions.\n\nThese concepts may be standard in economic theory but not in ML. Since the audience of TMLR is ML focus, it would be nice to have a background section where you clarify these.\n\n**(b) Clarify economic meaning of parameters**\n\n- $x_k$ is defined as the amount of content created in domain k. Then, what does $x_k^\\rho$ for $\\rho < 1$? Is $x_k$ assigned to creators by the platforms?\n- What are $\\tilde{c}$?\n\n**(c) Figures and results**\n\n- Figure 4 is particularly unclear to me. I am not sure what the red and orange lines stand for. If the punchline is \"both revenue and social welfare increase in the presence of GenAI.\", which I see at Table 2, then it would be useful to see plots that support this claim (perhaps present the difference, rather than the objectives)\n\n- I am note sure what \"comparing the **performance** of Algorithms 1 to 3, 4 and 5\" means in this setting. These are different models/assumptions that can't be verified. In other words, as far as I understand, a ground truth (held-out) data does not exists that would allow you compare the performance of these approaches. So I am not sure what \"performance comparison\" means in this setting.", "broader_impact_concerns": "I do not see any ethical implications that would require adding a Broader Impact Statement.", "additional_comments": "Although, I can see that M1 is convex and has a single optimum w.r.t. $y$, it is still non-linear w.r.t. $x$ and can exhibit multiple solutions. Is that correct (If I am mistaken please correct me)? If so, how does that affect the analysis?", "extra_scores": {}, "extra_text": {}}, {"review_id": "HrLag83f7W", "reviewer": "Reviewer_HWUu", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper develops optimal procurement mechanisms in markets where platforms source content from both human creators and GenAI, under nonlinear production technologies and cross-domain transferability. Its main contributions are:\n1. Two-layer market mechanisms: Derivation of optimal revenue-maximizing (M1) and welfare-maximizing (M2) mechanisms, extending the classical analysis to incorporate non-linear terms arising from GenAI creators. Also, formulating the problem of designing creator-union mechanisms.\n2. Three-layer market: Providing optimal mechanisms for both objectives (M5, M6), and identifying a “lose–lose” effect, where brokers reduce both platform revenue and welfare.\n3. Simulation results supporting qualitative predictions.\n\nSome noticeable weaknesses include:\n1. Modeling assumptions (e.g., functional forms of GenAI transferability) may be strong or narrow.\n2. Revenue-maximizing multidimensional case (M3) remains incomplete and open, as pointed out by the authors.\n3. Simulations are synthetic; empirical grounding is limited.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper’s primary theoretical claims are backed by formal derivations with stated assumptions; the mechanisms (M1~M6) follow from well-defined optimization problems, convexity arguments, and standard incentive-compatibility constructions.", "audience_yn": "Yes", "audience_explanation": "The paper addresses an increasingly important topic at the intersection of mechanism design, AI economics, and content-generation markets. The paper provides new theoretical mechanisms tailored to environments involving GenAI transferability, which is a novel modeling consideration likely to interest both ML researchers and economic mechanism designers in the TMLR community.", "requested_changes": "1. Justifying the cost of GenAI is \"negligible,\" compared to human effort. This might be true only in some domains, at least in the present.\n2. More discussion around $\\mathcal{M}_3$ can be provided, for instance, in Appendix. Difficulties and failed attempts are illustrative enough to provide more context on the exact challenges.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)