Market Games for Generative Models: Equilibria, Welfare, and Strategic Entry
Authors: Xiukun Wei
Min Shi
Xueru Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: Generative model competition
Nash equilibrium
Welfare analysis
Best-response training
Primary area: alignment, fairness, safety, privacy, and societal considerations

Abstract: Generative model ecosystems increasingly operate as competitive multi-platform markets, where platforms strategically select models from a shared pool and users with heterogeneous preferences choose among them. Understanding how platforms interact, when market equilibria exist, how outcomes are shaped by model-provider, platforms, and user behavior, and how social welfare is affected is critical for fostering beneficial market environment. In this paper, we formalize a three-layer *model-platfrom-user* market game and identify conditions for the existence of pure Nash equilibrium. Our analysis shows that market structure, whether platforms converge on similar models or differentiate by selecting distinct ones, depends not only on models’ global average performance but also on their localized attraction to user groups. We further examine welfare outcomes and show that expanding the model pool does not necessarily increase user welfare or market diversity. Finally, we design and evaluate best-response training schemes that allow model-provider to strategically introduce new models into competitive markets.

Published as a conference paper at ICLR 2026
MARKET GAMES FOR GENERATIVE MODELS: EQUI- LIBRIA, WELFARE, AND STRATEGIC ENTRY
Xiukun Wei, Min Shi, Xueru Zhang The Ohio State University fwei. 11ei.1418, shi. 1532, zhang 12807]eosu. edu
ABSTRACT
Generative model ecosystems increasingly operate as competitive multi-plane talti-plaffom markets, where platforms strategically select models from a shared pool and users with heterogeneous preferences choose among them. Understanding how plat- forms interact, when market equilibria exist, how outcomes are shaped by model- providers, platforms, and user behavior, and how social welfare affected crit- ical for fostering beneficial market environment. In this paper, we formalize a three-layer model-platform-sser market game and identify conditions for the exis- tence of pure Nash equilibrium. Our analysis shows that market structure, whether platforms converge on similar models or differentiate by selecting distinct ones, depends not only on models' global average performance but also on their local- ized attraction to user groups. We further examine welfare outcomes and show that expanding the model pool does not necessarily increase user welfare mar- ket diversity. Finally, we design novel best-response training schemes that allow model providers to strategically introduce new models into competitive markets.
INTRODUCTION
Generative models are no longer developed in isolation. They now operate within competitive, multi-platform markets, where platforms strategically deploy models attract heterogeneous user groups and compete for market share. For example, Microsoft Azure and Amazon Bedrock compete to license foundation models to enterprises (Buchanan) 2024 Janakiram 2023). Canva and Adobe Firefly compete to attract designers by integrating state-of-the-arr generative models (Newsroom) 2024 Weatherbed 2024), and platforms like Midjourney and Stability AI compete directly for end users seeking creative tools (Staff) 2025a b). Understanding the behavior of such markets is crucial for guiding governance, policy, and the design of trustworthy Al ecosystems.
Prior work has largely focused on two-layer market, where model developers also operate the de-
livery platforms and offer services directly to users (Binav & Rosenfeld [025] [agadeessnn et al., 2023b [Raghavan] 2024 Taitter & Ben-Porat, 2025). Within this setting, a substantial body of literature examines discriminative scenarios in which (binary) classifiers are trained and used by end users. For instance, Einav & Rosenfeld (025) show that platforms often prioritize capturing user share over maximizing predictive accuracy. leading to equilibrium states that deviate from so- al. (2023b) demonstrate that even when individual
cially optimal outcomes. Similarly,Dagadees classifiers achieve lower Bayes risk, competition can perversely reduce overall social welfare. By contrast, the literature generative model markets remains sparse, with only few recent studies analyzing how competition shapes overall welfare outcomes (Taitler & PoraP 2025) Raghavan 2024]. Specifically, Taitler & Ben PoraP (2025) identify a counterintuitive phenomenon: adding more models can paradoxically decrease user welfare. More recently, Raghavan (2024) find that competitive pressures often reduce diversity, though stronger competition can partially mitigate ho- mogenization, and that models performing well in isolation may fail in competitive environments. Collectively, these studies highlight that, in generative markets, improvements in individual model performance do not necessarily translate into greater welfare or diversity.
However, the generative ecosystem is increasingly structured as a lhyee-layer market FFalaah et al.) 2024): ; model providers develop models and license them to platforms, which in turn deliver services
Corresponding author.

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to end users. Unlike the two-layer setting, where model developers both build and operate the deliv- ery platforms, in the three-layer market, platforms act as intermediaries: they decide which models to adopt and deploy, ultimately shaping how users experience generative AI. For instance, Azure OpenAl Service (Azure, 2025) supplies GPT-family models through Microsoft Azure, enabling en- terprise clients to embed them into wide range of applications; Cohere (Cohere 2025) provides large language models as APIs for enterprises, serving platforms rather than end users directly; and Canva (Canva 2025) integrates external models such as Stable Diffusion and Leonardo Phoenix into its design suite, allowing millions of users to access generative capabilities without ever selecting the underlying model themselves. all these cases, platforms are the direct consumers of models, while users experience only the models that platforms choose.
In this work, we formalize the market as three-layer model-platform-user game, in which hetero- geneous users choose the platform that best aligns with their preferences, while platforms strategi- cally adopt the models from providers that maximize their market share (Fig. [T.. We then conduct a rigorous analysis of how platform-level competition influences user welfare, diversity, and equilib- rium outcomes. Our main contributions and findings are summarized below:
1. In Section we formalize the plofel-plafoorm-user game and show that when users make hard selections on platforms, the resulting game among platforms may not admit pure Nash equilibria. In Section 3 we identify conditions for the existence of pure Nash equilibria. Crucially, we analyze market structure equilibrium and derive conditions for both fully differentiated equi- libria (all platforms choose distinct models) and homogeneous equilibria (all platforms converge on the same model). We show that market structure is determined not only by models' average performance but also by their deviation advantage to heterogeneous users.
3. In Section 4 we analyze market diversity and user welfare. We show that equilibrium may not achieve the socially optimal outcome that maximizes user welfare, and that increasing competi- tion (e.g., adding more platforms or models) does not necessarily improve user welfare market diversity. This finding aligns with recent observations of growing homogenization in generative
model markets (Zhang al. 2025 Wu et al. 2025).
In Section we take the model providers' perspective and design best-response training schemes that allow provider to introduce a new model effectively into the competitive market.
5. In Section
we conduct experiments on both synthetic and real data to validate our theorems.
Model Provider G
Q: How to train a new best-response model g against existing ones?
Choose Ø depending on Utility
? Q: How do platforms choose which models to use? How about market?
Platform Player
Choose
depending on Score
User Group
Q: Hlo about social welfure?
8
The three-layer model-platform-uue market structure. Model providers develop genera- tive models, platforms select models to deploy, and heterogeneous users choose platforms.
Figure 1:
2 PROBLEM MODEL
Consider three-layer generative model service market as shown in Fig. I which consists of:
Model Layer: A set of generative models G = {gi...... .aa.......ee ..M trained by M model providers. data distribution over domain X.
Each model 9;. i M [1,..., ..I.....M.... is characterized
Platform Layer: A set of platform-player I = {1..... .N. From the provider, each platform i selects model f: MI serve its users. Denote the platform selection f (fi....ffy
categorized into types θ {01.... 0x) S Rd with distribution ( (To Jece such that Eoce मo = 1. For each user type 0g. let ro, (a) be the underlying reward function indicating their preference over generated content x X.
User Layer: A population of heterogeneous
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User's choice of platform. Given platform selections, users of type choose platforms interact based on the generation ability of selected models. Let (,(0) := Ee [re(z]] be score measuring the quality of contents generated by model 9; for user type 0; higher S indicates better alignment with user preferences. Then, the probability for users of type selecting platform i is given by:
if Sa & arg max S,;(0) if fa E arg max S):(0)
P0)
(1)
(aIN ())
That is, users select the platform with the highest score, breaking ties uniformly at random. This hardmax choice model is standard simplification commonly used in prior work on platform or model selection (Jagadeesan et al., 2203b a Mansour et al. (018). In Section analysis and results can be extended to the softmax user choice model.
[S] we discuss how our
Platform's choice of models. Each platform E [N] strategically selects a model f from the model-provider to compete for market share, in response other platforms' selections:
fr = arg max f(, F.E
(2)
Ta (0) Sy, (0) is the utility function capturing the market share of platform. Denote profile f = (fi, (N) as the strategies of all platforms, and f-r strategies of all excluding the platform i. We consider normal-form game G(G, II, θ) in which each platform chooses a model to maximize its utility.
where
Definition 2.1 (Nash Equilibrium). We say a strategy profile f = -fi. FR. is a pure Nash equilibrium (PNE) if no platform can improve its utility by unilaterally deviating, i.e., : Z U.(f:f Vi. A pure Nash equilibrium is fully differentiated equilibrium if all platforms choose distinct models: [[ : JN J N. A pure Nash equilibrium homogeneous equilibrium if all platforms choose the same model: [[/r "[/f.
=
By Nash's s classical theorem, every finite game admits at least one mixeul-strategy Nash equilibrium, where each player i randomizes over actions according to probability distribution (Monderer & Shapley 1996). However, when users make deterministic selections as in Eq. [ we show in Propo- sition 2.2 that pure Nash equilibrium may not exist.
Proposition 2.2. [Nonexistence of PNEJ Consider the game G(G, I, O) with finite sets of platforms I. models G, and user types θ where each platform i chooses a model ft E Ml based on Eq. 7 The game may not admit pure-strategy Nash equilibrium f
To prove Proposition we provide a counterexample in Appendix Nash equilibrium can be examined via best-response dynamics: starting from any profile f. plat- forms sequentially update their strategies as best responses to the current strategies of others. When a pure equilibrium does not exist, this process fails to converge and instead enters cycles.
The existence of a pure
Definition 2.3 (Best-Response Cycle). A best-response cycle is a finite sequence of strategy pro- files (2), f f(L) in best-response dynamics such that: (i) each step I, only one platform changes its strategy, and it does as a best response to f(t), and (ii) the sequence eventually returns to the initial profile, i.e., f (I+1) = (1).
User welfare & market diversity. Next, we introduce metrics we will use for analyzing the market. fN) with fr E M. define
Definition 2.4 (Coverage Value). For a strategy profile f = the coverage value of f as V(f)= Eece To maxigreN Sj, (0), i.e., the expected quality users receive from their best available platforms.
Definition 2.5 (User Welfare). Define user welfare W as W = V(S*) if the game I, admits pure Nash equilibrium f"; and W : + ((f(t) if enters best-response cycle. Definition 2.6 (Social Optimum Welfare). The social optimum welfare is defined as the highest coverage value achievable by any strategy profile. That is, Wopt max V(f).
Definition 2.7 (Herfindahl- Hirschman Index (HHI) Diversity). For strategy f = (fi,.... fN) with ft E M, let (M1, P2,.... , AN) be the market share vector platforms, where μ4 Loce (0).
HHI diversity is defined D (f) EN 12.
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HHI diversity measures how evenly users distributed across platforms. Its value lies in [ 1], taking value when users are evenly split across all platforms (maximum diversity) and 1 when all users concentrate on a single platform (no diversity).
Definition 2.8 (Support Diversity), For strategy f = (fi.....fNv with f. E M,, the support diver- sity is defined Suupp ff == Hm M Ei II m
Support diversity measures the number of distinct models adopted by platforms, taking integer val- ues between 1 and N. Larger values indicate that more models are represented in the market, reflecting greater model diversity.
Objectives. With the platform selection game and evaluation metrics in place, the remainder of this paper aims to address the following key questions about competitive generative model markets:
Under what conditions do pure Nash equilibria (PNE) arise, and when platforms converge to differentiated versus homogeneous structures?
How does the user welfare depend on the number of platforms and models, and how does it compare the social optimum?
From the model-provider's perspective, how design new generative models that can successfully enter the market and be strategically adopted by competing platforms?
3 EQUILIBRIUM ANALYSIS AND MARKET STRUCTURE
Proposition 2.2 showed that a pure Nash equilibrium (PNE) may not exist in the platform selection
game. We now investigate the conditions under which PNE does exist. To facilitate the analysis, we first introduce some basic notations.
Definition 3.1 (Average Score). Let average score of model gj be defined as T, = Eoco To S,
Definition 3.2 (Attraction Term and Deviation Advantage). For strategy profile f = (f fN) with fi E Mi, define A (0) := arg maxi SiNN Sy, (0) as the set of maximizers a user type O, and let A (0) == s (0) be the number of platforms tied for the maximum. Then, the attraction term for fo in strategy defined as
N () ()) if E Ay(
(0:(f)) =
(0)
(3)
(0),
otherwise
The deviation advantage for fa under strategy f is defined as
(f)
Σ T (0:
(4)
Intuitively, the attraction term 2,, (0 f) quantifies how much fi benefits when it is among the winners for type O and how much it loses otherwise. Aggregated across all user types, the deviation advantage 5,, (f) represents the net gain of fa relative to its competitors under the current strategy. Proposition 3.3. [Utility Decomposition.] The expected utility U, f-s) of platform in Eq. can be decomposed into T', and , (f) as:
Us (fa f-) = N (T + (f).
(5)
Lemma 3.4. [Existence of Equilibrium] Consider the game G 11, θ) with finite user types Q, and N platforms choosing from M models G, where M 2 N. A fully differentiated equilibrium f (Ji...., 1%) exists if and only if for every platform and every alternative model Jū E G (
Ty: Ty, 2 (f,, 85; (f*)
homogeneous equilibrium f (ft .f...... . fz), ft m exists if and only for some m M,
T. T 2 8g (f" U f&) 5m (f*)

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Scenario A: fully differentiated equilibrium
Scenario B: homogeneous equilibrium
a = T0
0.5
-0 = 0.5
(0)
(2)
S (
S2(
A 0.90
0.85
0 A
0.60
0.70
B
0.35
0.80
0 B
0.65
0.95
f (
(g2. 92)
Figure 2: Two scenarios with the same average score gap [T2 Ti. 0.20 but different deviation advantage (ō), 62) for N and M 2. resulting opposite equilibrium outcomes. In Scenario A, although 91 has a lower average score (7) < 72), it is still chosen in equilibrium because its strong advantage on the high-weight type 4 satisfies the differentiation condition. Removing this type- specific advantage in Scenario B breaks the condition. leading to homogeneous equilibrium on 92- These scenarios demonstrate that market structure is not determined solely by average performance; strong local advantage in high-weigh user segments can sustain model's presence in equilibrium. Full calculation details are provided in Section C.4
The proof is given in Section Lemma 3.4 shows that market structure determined by the balance between average performance 7 and the deviation advantage 5. When average performance uniformly strong across models, platforms tend to converge on the single best model, where performance alone cannot sustain multi-model entry. In contrast, when models differ in whom they serve best, even uniformly weak models can secure adoption as platforms specialize, yielding a differentiated market. To illustrate this, provide example in Fig.
Lemma 3.4 also implies that market structure depends on the user distribution. Building on this, Corollary 3.5 shows that when the user distribution is centralized, i.e., single user type constitutes a large fraction of the population, the market tends to converge to a homogeneous structure.
Corollary 3.5 (High User Centralization + Homogeneous Equilibrium). Assume there exists a dominant type O* with fraction z and model satisfying: Vj * m. Sm( Sy(0)) 2 > 0 and v0 + 0 + m, [S,(0) - S. (n)) T. r. Hă is sufficiently large and satisfies
0 2 1
1 1+
then the homogeneous strategy f (m,... m) is a pure-strategy Nash equilibrium.
Intuitively, P measures strength of the "majority advantage" model, I measures the maximum performance difference outside the majority group, and measures the "relative strength of minor-
1.0
0.8
ity variation to majority advantage." From Corollary 3.5 increasing Th effectively lowers the threshold of p needed for homogeneous equilibrium. So even small quality advantage p on the domi- nant user type can outweigh potential gains from minority types (bounded by I'), allowing single model to dominate the entire mar- ket. illustrate the parameter regime in which Corollary 3.5 ap- plies, Fig. 3 plots the т against _ the shaded region shows the pa- rameter range where the homogeneous strategy f (m,... ... ..) is a PNE.
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
T/P
Figure 3: T versus
I p
4 USER WELFARE AND MARKET DIVERSITY
Proposition 4.1. [Coverage Value Calculation] Given strategy profile f (Ji,..., SN), the 2.4 can be written as:
coverage value in Desinition 2.4
v()
(Tr
+8)
N 1=1
where T and are defined in Definitions 3.1 and 3.2
respectively.
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Scenario A: fully differentiated equilibrium
Scenario B: Add new model 9s
a = T0 = 0.5
T0 = -0p = 0.5
(0)
(2)
Si
S₂(
Sa
A 0.90
0.85
0 A
0.90
0.85
0.91
0B
0.35
0.80
0 B
0.35
0.80
0.77
W- V(f*)
(1, 0.85
W = V(f*)
V(3 = 0.84
Figure 4: An example where enlarging the model pool decreases welfare. In Scenario A, with models gi and d2 the equilibrium is fully differentiated with the welfare W = 0.85. Adding new model f3 in Scenario B shifts the equilibrium to the homogeneous (93, 9a). where welfare decreases W 0.84. It pulls both platforms toward homogenization, thereby sacrificing the welfare of minority types. The calculation details are provided in Section C,9
Proposition 4.1 with proof in Section provides a closed-form expression for the coverage value of any strategy profile f. showing that the sum of individual platform utilities equals the coverage value, i.e., V(f) = E U.(f). However, under competition, self-interested platforms that each maximize their own utility U, do not necessarily achieve optimal user welfare, discussed below.
Lemma 4.2. Let W denote the user welfare (Definition 2.5g achieved under the game ( I, O), and Wopt the social optimum welfare (Definition 2.6 Then, it always holds that W S Wopt-
Note that the equality in Lemma 4.2 holds only in the degenerate cases: when f E aaggmass V(f) or when every (t) the best-response cycle attains the maximum welfare value. Such situations rarely occur in competitive markets, highlighting the misalignment between platform incentives and user welfare as the social objective. We provide an example in Section
Next, examine the impact of the number of models and platforms on the market. Intuitively, enlarging the model pool G or increasing the number of platforms N might be expected to promote competition and enhance user welfare and market diversity. However, that neither approach is reliably effective. As illustrated in Fig. 4 expanding the model pool can introduce a uniformly strong model, pulling the market toward homogenization and reducing welfare for minority users, as shown in Corollary 3.5 Similarly, adding platforms can be counterproductive: strategic interactions may induce best-response cycles lead platforms adopt weaker models to avoid competition, thereby lowering welfare. These results demonstrate that welfare and diversity are not monotone in competition intensity. Nonetheless, we can identify sufficient conditions under which platform entry does not reduce welfare or diversity, as detailed in Proposition 4.3
counterexamples show
Proposition 4.3. Consider a game (G, O) with an equilibrium f Let Ū := I= be another game with one additional platform added. Suppose there exists model h E G and an incumbent equilibrium strategy f from f such that the extended profile = (ff",N, h) satisfies the best-response conditions: (i) the best response to f is h; (ūi no incumbent platform platforrm has a profitable deviation against f. Then ff is an equilibrium of the game G. Furthermore, the user welfare and market diversity in G are at least as high as in G. i.e. W W and Dsupp 2 Daupp-
5 FROM HARDMAX TO SOFTMAX USER CHOICE
Our main analysis adopts the hardmax user choice rule in Eq. I where each user deterministically selects the platform whose model achieves the highest score Sy. (0) for their type 0, breaking ties uniformly. This assumption makes the analysis of strategic interactions more straightforward. In practice, however, users exhibit noisy and heterogeneous behavior rather than perfectly rational best responses. A natural extension is a softmax choice model. Given profile f = (J1,.... and user type 0, we define
pfoft (0) = S (0)/T
(8)
1 S (0)/T
where 7 2 is a temperature parameter controlling the level of randomness in user choice. When 00, users nearly indifferent and split across platforms almost uniformly. As decreases,

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users concentrate more higher-scoring platforms, and in the limit 0 the softmax model converges to the hardmax rule.
Indeed, our negative result on the existence of equilibrium (Proposition 2.2 extends to the softmax user choice model. The platform game under Eq. 1al.8rrema remmins a finite normal-form game, and pure shows
Nash equilibria may not exist. Proposition C.7 in Appendix Chows that any instance with no PNE under the hardmax choice model remains without a PNE for all sufficiently small temperatures in the softmax model. We also provide an example in Appendix [C.10 demonstrating that, for fixed T > 0, the softmax model still admits no pure Nash equilibrium.
The utility decomposition (Proposition 3.3) and the existence of fully differentiated and homo- geneous equilibrium (Lemma (.4] also extend beyond the hardmax model. As shown in Ap- pendix C.10 an analogous decomposition holds under softmax choice once the deviation term redefined as (f) ( in Eq. 21 With this modified deviation, Lemma 3.4 carries over directly: the equilibrium conditions retain the same form and can still be expressed as inequalities involving T and i (f). This highlights that that market segmentation not determined by average model per- formance alone; a model with lower average performance may support a differentiated equilibrium if it performs particularly well for certain user types.
Finally, our notion of welfare is largely independent of the choice rule. The only change is in the relation between coverage V (f) and platform utilities: under hardmax, Σ, U.U.)) = ( whereas under softmax Σ. UT (f) V(f), typically with strict inequality, which further increases the misalignment between platform incentives and user welfare. Nevertheless, since the definition of coverage and welfare is determined only by the available models and their scores, all comparisons between V V(f) and W Wapt including Lemma 4.2 and and its Proposition 4.3 remain valid.
DESIGNING COMPETITIVE MODELS FOR PLATFORM ADOPTION
In this section, we shift focus to model providers. Consider a single provider aiming to learn pa- rameters o for an entrant model Is that will be adopted by rational platforms. The provider seeks to maximize an adoption-weightee quality objective:
max F(Ø)= = T
GEG
where de E [0, 1] represents the adoption probability that users type 0 would choose Is when competes against incumbents, and (0) Ex [z)] is the expected quality that user type receives from 9d.
To calculate dg. suppose
estimate the user distribution {eee and the score ()) of incumbent E M. Let S(0) = maxjeM (0) be the best opponent score for user type A. hard adoption rule would set ae = when (0) > S(O) and 0 otherwise. As this non-differentiable and unsuitable for gradient-based optimization, we adopt Bradley-Terry [Badley & Terry (1952) soft gate on the margin Ag := (0) - 5(0) ) and define the adoption probability of type attracted by the new model as ag = a (3Aa)). where (z)) = Ite = Here, B controls the softness, as 3 + O, dg approaches the hard adoption.
We provide two solutions to solving the above optimization: 1) training data resampling; and 2) direct-gradient optimization.
Training Data Resampling. We first adopt a resampling- scheme that biases the training data distribution toward user types with higher payoff weights ag To (on)". S(0), where emphasizes user types for which the entrant more likely to outperform incumbents. Each data
point & is then assigned sampling probability wxx). normalized from w(x) Ence (o))(r) where (o(x)) (x:) [0, 1] measures how strongly x: is preferred by users of type 0. Specifically:
Structured data: Each x has an attribute (e.g., class, domain, style) u(x) E U, and type specifies distribution qo (u). We set v0(x:) (u()))). i.e., the probability that matches type 0's preferred attribute. Sampling then proceeds by first drawing u wu) x Eo aogo (u), and then sampling &
D-: u)).

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Unstructured data:
We use the reward itself ve normalize(ro(z)), so data points yielding higher expected rewards for type & are sampled more frequently.
In this method, type weights computed based dg and S(0), and the model gø is trained on data resampled according to these weights. This method alters the data distribution but not the loss function, making it compatible with standard training pipelines while effectively biasing training toward strategically valuable user types. The detailed procedure provided in Algorithm
Direct-Gradient Optimization. and its competitive attractiveness against fixed opponents. Specifically, the training objective is:
We train the model to directly improve both its generation quality
arg min L(Ø) [( (F(()
where L(ő) is the standard loss ensuring that the model maintains overall sample quality, and F(ø) is the adoption-weighted weighted quality objective that promotes competitiveness. The trade-off parameter 2 balances quality and competitiveness. The main challenge optimizing this objective via gradient descent lies in computing the gradient of F(6). Note that the only term depending on é in F(Ø) Eoce To is (²) = Eg (.)). By the chain rule, we have:
VoF()) = Σ To loo + 3o (1 σa) (s) V (0)
E
We next present two estimators for (.). The detailed procedure is shown in Algorithm 2
Pathwise gradient: This estimator applies when both the reward function (x) (e.g.. classi- fier score, probability output) and the generative model (e.g.. GAN (Goodfellow al. 2014). DDPM (Ho et al. 2020). SGM (Song & Ermon 2019)) are differentiable. The model samples I go (E). where { ~ PC( drawn from a fixed prior po and do a deterministic transforma- tion of the noise ç, then
S = Eg Vxro(x) (=))({)
Jego(E)
where (E) the Jacobian of (E) with respect d.
REINFORCE gradient (Williams) 1992]: This estimator applies when either the reward function rx(x) (e.g.. discrete 0/1 feedback) or the generative process (e.g., SeqGAN Yu et al. 2017), MaliGAN (Che al., 2017)is non-differentiable. With moving-average baseline bg reduce gradient variance and let (x) is the model distribution, then
V Ex-po [(ro(x) bb) Vo log(ps(x))
EXPERIMENTS
In this section, we conduct experiments on both synthetic (Section D) and real-world data to provide reproducible prototype for validating the theory
Model Pool. We adopt denoising diffusion probabilistic model (DDPM) (Ho et al. 2020) trained on the full CIFAR- 10 dataset (Krizhevsky 2009), contains 60,000 images from 10 classes C (airpIne == 0, automobile := 1, bird : 2, cat := 3, deer := 4, dog 5, frog : 6, horse := ship := 8, truck := 9). as the base model. To construct preference-oriented variants, we apply Low-Rank Adaptation (LoRA) (Hu al. 2022) fine-tuning with different class-specific subsets. The choice of class groups and LoRA hyperparameters for each variant is summarized in Table I Each variant captures preferences aligned with a subset of CIFAR-10 classes.
User Group. We partition the user population into six groups, each characterized by heteroge- neous preferences over CIFAR-10 10 classes. Formally, for user group 8, we specify distribution of weights 0 {fe)oec, Eeec 0. 1, the details given in Table 2
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Table I: Model Pool
Table 2: User Groups
#
classes
d
Oxg
#(0)
preferences(claass(e.))
T(0)
M1
airplane, auto
4
16
1.0
A
cat (0.6), dog (0.4)
0.18
M2
ship, truck bird, cat
16
1.0
B
dog (0.7) cat (0.3)
0.17
16
1.0
C
airplane (0.5), ship (0.3), auto (0.2)
0.16
M4
cat, dog
8
32
1.5
D
auto (0.6), truck (0.4)
0.16
M5
cat, dog
4
16
1.0
bird (0.4), deer (0.3),
E
0.17
Notes: d denotes the LoRA rank, og the LoRA scaling factor, and ne is the external scale applied during fine-tuning.
frog (0.2). horse (0.1)
(0.2), dog (0.2), airplane (0.15),
F
0.16
auto (0.15), ship (0.1), truck (0.2)
Reward Function. We employ pretrained ResNer20 model (He et al. 2016) with the 92.60% Top-1 accuary trained on CIFAR-10 and held fixed during experiments, assume Pace (c x) is the posterior class probability computed by this model for class c. Then the reward of v for user type calculated by (e Eeet Oe Pace (c x). For every calculation of S, we sample 2000 samples.
Discrete Rest-Response Simulation. The average performance of the five models T, and their user-specific performance (,) shown in Fig. 8 in Section E.1 where conduct discrete best-response simulation by progressively enlarging the model pool (from to models with players) and increasing the number of platforms (from to 6 players with 5 models). At each round, platforms update their strategies by choosing the best response among the available models, given the current distribution of opponents' choices. For each game, we perform three independent runs and track diversity DHHI and coverage value V(f) at every step of best-response dynamics.
round
round
2
player
player 2
player
round 2
0.50
model
0.22
0.14
HHI diversity
models
0.20
0.12
0.45
models
0.18
0.10
models
0.16
model models
0.08
0.40
0.14
models modeb
0.12
W modebs
0.35
0.04
0.10
step
step
step 6
(a) DHHI as model pool increases.
(b) V(f) as model pool increases.
(c) BR with M = 4,N = 3
round
0.18
1.0
0.22
0.16
player
HHI diversity
player players
0.20
0.14
player
0.8
playors player
0.12
player 3
0.6
players plavers
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player
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0.4
player
2 players
0.2
0.10
players
players player
players
0.04
0.02
0.08
0.00
round
round
step
(d) DHm player increases.
(c) V(f) as player increases.
(f) BR with M 5.N
Figure 5: When enlarging either the model pool (a.b) with 3 players or the number of platforms (d,e) with models, the change of HHI diversity ((a,d), where larger values indicate more homoge- nization) and coverage value ((b.e), where larger values are better). (c.f) provide examples of utility trajectories across best-response steps, where filled markers denote the player taking the action at each step. (c) shows best-response cycle and (d) shows equilibrium.
From the Fig. 5a enlarging the model pool does not automatically increase diversity. Only when the newly introduced models sufficiently distinct (add M2 M3) can they enhance diversity. By contrast, strong entrants that are merely close substitutes for existing leaders tend to homogenize the market (add M5), as platforms converge toward the same high-performing options. This reproduces

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the convergence phenomenon widely observed in today's generative model markets. Increasing the number of platforms in Fig. 5d, however, expands adoption opportunities and thus promotes diver- sity. In terms of welfare, adding more platforms Fig.53 (which enables greater choice) improves user welfare and accelerates its growth. However, welfare never reaches the social optimum. The trajectory examples in Fig. [c and Fig- 55 further reveal that early movers often select the "best" model, but are later forced to share its benefits with subsequent players, leaving their utilities subop- timal. In contrast, players who move later sometimes adopt models that less attractive globally but provide relative advantages when not shared, resulting in higher individual utilities.
In Section D we systematically vary the model pool size, the number of platform players, and the user group distributions on synthetic data, with detailed results and figures provided.
Algorithmic Best-Response Entry. The hyperparameters and full algorithmic details are pro- r.2.2
vided in the Section. E.2. Section E.2.2 reports systematic hyperparameter tuning for both meth- ods. Both algorithms are initialized on the full CIFAR-I0 dataset. The result is shown Fig.
MS
M1
karp
M.
ard
Original Pool
Resampling
Direct-Gradient
0.6
0,40
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0L87894
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0.35
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0.97244
BO7TLS
Y18"
E
0.00
0.05
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0.25
0.30
0.30
Liility
Diversity
Wetfare
M
Figure 6: Performance of new models generated by the direct-gradient method M Mgrad and the resam- pling method Maamp- (a) Left: equilibrium outcomes and utilities three players when competing over the original pool, after introducing Mgrad and Msamp- (b) Middle: diversity DHHT and welfare Won under the three equilibrium.
(c) Right: comparison between the best original model Ms and the new entrants Mgrad, Msamp on user scores.
The direct-gradient method achieves stronger performance: successfully replaces the best model in the original pool, dominates the market, and yields higher welfare and diversity. Moreover, it con- verges with fewer iterations (+= 20). However, it requires modifying the model's internal objective, and the diversity of sample classes is decreased shown in Fig.
The resampling method suffers from higher variance due to stochasticity. only approaches the best model in the original pool, while reducing welfare. It is also more computationally demanding (s 10 resampling with 50 iterations each). However, the method has the practical advantage of being alug-and-playy it can be applied any model without altering its loss function.
CONCLUSION
In this paper, formalize generative Al markets a three-layer model-plattorm-uuse game. From the platform perspective, we characterize conditions for both fully differentiated and homogencous equilibria, showing that the market jointly shaped by average model performance and user-specific deviation advantages. From the user perspective, we demonstrate that enlarging the model pool or increasing the number of platforms does not necessarily translate into higher welfare. From the model provider perspective, we propose training schemes that strategically facilitate entry into competitive markets. Together, these findings highlight inherent paradoxes in generative Al markets and point to design principles for more socially aligned ecosystems.
ETHICS STATEMENT
This work analyzes competitive generative model markets using both theoretical modeling and em- pirical experiments. Our analysis is abstracted from specific settings and does not involve sensitive
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personal data, human subjects, system manipulations. Our findings raise broader ethical impli- cations. The results show that competition among generative models may reduce diversity and user welfare, highlighting the need for responsible governance and transparent platform practices. The proposed training schemes are designed to advance understanding market dynamics, not to pre- scribe adversarial practices. Overall, this work aims inform policy discussions and support the design of more socially beneficial generative ecosystems.
REPRODUCIBILIIY STATEMENT
We ensure reproducibility by including all definitions, propositions, and proofs, and by detailing model pool construction, user groups, reward functions, and evaluation metries. Hyperparameters and training procedures are documented in the appendix, and code is released to support replication and extension of our results.
ACKNOWLEDGMENTS
This work was funded in part by the National Science Foundation under award number IIS-2202699, IIS I18-2418995, IIS-2301599 and CMMI-2301601.
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A
RELATED WORK
Recent advances in machine learning has led to the emergence of markets in which multiple models simultaneously operate and compete for user adoption. In such competitive environments, the strate- gic interactions among providers critically shape both market outcomes and overall social welfare.
Competition in Predictive Models. Competition among classification models has been extensively studied through the market design and strategic learning. A key insight is that in competitive mar- kets, maximizing classification accuracy alone does not guarantee higher adoption or improved so- cial welfare. For example, recent work Einav & Rosenteld (2025) formalizes the accuracy market, where multiple classification providers compete for users, showing that optimal strategies must ac- count for rivals' actions rather than accuracy in isolation. Similarly, Ben- Ben-PPraa & Tennenholtz (2017) study the learnability of optimal responses in competitive regression, and further establish that pure- strategy equilibria exist and that competition may induce strategic misprediction -Ben-Porat& & Ten- (2119) Further, Further,Jagadeese. Jagadeesan et al. (2023b) demonstrate that even when individual predictors achieve lower Bayes risk, strategic competition can paradoxically reduce overall social welfare. Ex- tending beyond classification and regression, Yao et al. (2023) analyze how top-K recommendation performs under competing content creators, showing that user welfare losses remain bounded, while Yao et al. (2024b) propose platform interventions that directly optimize user welfare in such com- petitive recommendation environments. Overall, these results show that accuracy must be assessed in the context of competition, entry, and social welfare.
Competition in Generative Models. As model capabilities improve, research has increasingly focused on competition among generative models. |Raghavan (2024) show that equilibrium under generative AI competition tends toward content homogeneity, even when models perform well in isolation, while stronger competition can counteract this effect. Empirical studies further suggest that generative Al usage in areas such as peer review (Ebadi al. 2025 Kankanhalli [024], writ- ing (Doshi & Hauser 2024), and creative generation (Wu et al.) (2255 often associated with reduced output diversity. Beyond direct competition between models, recent work also investigates inter- actions between generative Al and human participants. At the creator level, Yao et al.] (2024a) model competition between human creators and generative AI using a generalized contest frame- work, showing conditions for coexistence, conflict, even the absence of stable equilibria; while at the platform level. Taitler & Ben-Porat (2025) demonstrate that generative Al can paradoxically reduce overall welfare in human-driven platforms, echoing Braess's ' paradox.
Strategic Model Behavior. Another line of work studies how learning systems are shaped by strate- gic environments and feedback-driven dynamics. For example, Wei & Zhang (2025) show that when models learn from feedback, they can infer latent human preferences from the resulting data and manipulate the feedback signals, inducing undesirable shifts in downstream output distributions. Related concerns also arise in performative prediction (Perdomo et al., 2020: Jin et al. (0026) and strategic learning (Zuo al. 2024; Xie Zhang, 2024b at Jin al.,
2022 Zhang et al. (0022)
where model deployment endogenously alters the data-generating process, inducing distributional drift and welfare loss.
Unlike prior studies that focus on two-layer market, our work formalizes a three-layer model- platform-user game. Under the assumption of deterministic user choice, we show that pure Nash equilibria may not exist. Building on this observation, we characterize the conditions under which equilibria arise and analyze how the resulting market structures shape welfare and diversity as the pool of available models expands. Moreover, depart from prior work by adopting the perspec- tive of model providers and proposing best-response entry training schemes that allow entrants to strategically introduce models. This moovidrr-cettrin view and the resulting entry dynamics are largely absent from existing studies of model competition.
B DISCUSSION
Platforms with Multiple Models. For tractability, the current framework assumes that each plat- form selects single model. However, it can be extended naturally to accommodate multiple models per platform. In this setting, each platform's 's strategy would be set of models M,, and user choice could depend the higfestg model in that set for their type, S,) maxjeM, S,(0),
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expected score 5,(0) = (0). The three-layer market formalization remains before, with platform payofts computed using (,) instead (,), and best responses now taken over model mixtures rather than single models. Then, many of the existing analysis, such as the utility decomposition, equilibrium characterization, and welfare analysis, can be generalized to this setting.
Partially Overlapping Markets. Our framework designed for settings where platforms offer comparable services and draw from a shared pool of models M. In such markets, platforms face similar types of demand (e.g.. overlapping mixes of coding and translation tasks), and their strategic decision is which model from this common pool to deploy in order to attract groups. By contrast, different platforms specialize in largely disjoint services (e.g., one focuses almost exclusively on coding while another focuses almost exclusively on translation), then their effective model pools may overlap only partially or not at all. In that case, they are not competing for the same user base in the sense of our model: user might naturally use both services for different tasks, and platforms no longer face a shared competitive environment. Extending the framework to this multi market or partially overlapping-marte environments is interesting direction for future work.
Dynamic Interactions. Our analysis focuses on first-round interactions in static, complete- information setting. In practice, user behavior and platform strategies evolve gradually: the data distribution shifts under repeated deployments, and models retrained iteratively. Incorporating these multi-round feedback effects into our three-layer framework is left for future work.
C PROOFS
C.1 PROOF OF PROPOSITION
2.2
Proposition 2.2. [Nonexistence of PNEJ Consider the game G(G, I, O) with finite sets of platforms I, models G. and user types O, where each platform i chooses model MI based Eq. 7 The game may not admit a pure-strategy Nash equilibrium f
Proof. We provide constructive counterexample here. Let θ = {0 1z 0,) with uniform
weights x0a)) 3. Let (g), 92 933 and define scores:
S (0)
(2(
Sa
0 ,
0.2
0.1
0
0
0
0.2
0.1
0c
0.1
0
0.2
Then when there are two players, the payoff matrix is
f
(f1,
91
92
93
91
(0.05, 0.05)
(0.1, 0.067)
(0.067, 0.1)
92
(0.067, 0.1)
(0.05, 0.05)
(0.1, 0.067)
g3
(0.1,
(0.067, 0.1)
(0.05, 0.05)
On the diagonal (gk. 9 each platform gets 0.05. Against gk. the unique best response is the model that yields 0.10, so any diagonal profile is profitably deviated from. Off the diagonal, the player receiving 0.067 can switch the third model and improve to 0.10. Hence no profile mutual best response; therefore no PNE exists.
C.2 PROOFS OF PROPOSITION
3.
To illustrate the intuition, we first consider the case with two platforms N 2.
The strategy profile f (fi, fy), in this case, the attraction term Definition 3.2 simplifies to:
Sr. (0)
if S (0) S (0)
(1)
0
tie
(10)
if S (0) < S (0)
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which measures how much user type 0 strictly prefers ft over fy- Accordingly, the deviation advan- tage is:
dij Σ n Zz))
(11)
Proposition C.1 (Utility Decomposition for two platforms.). Suppose N = 2, let i,j E ll be the two players who choose model and Jj under strategy f. Then the expected utility is
[ Tr
if = 2 (T + +,g)
U. (Jo, 1z)
(12)
1
if 1
Proof. Consider the case with two platforms (N 2) and strategy profie f = ff), let Win,
{0: Sy, (0) S₁, (0)), Tie - (0: Sy, (0) S (0)), Loser; = {0 S (0) (0)}.
With Eq. D type 0 is fully assigned to the winner, split evenly on a tie, and assigned zero to the loser. Therefore, the utility of the platform choosing 9i is:
Un(i,
Σ no p0) (0) OEO
Σ OEW in.
(0)S,, + 1 Σ
(0)S,,
BET
By definition,
T: = Σ (())
Σ (0))
Σ
T(0)S;
Σ
x(0),,
OEB
BEWin:
OETiE
Le
81g (0)),,,
Σ
T(0)S},
Σ
(()),,,00)
iw
LELoser:
Adding these two:
T₁ + 8.j =2 Σ ((0)S,,0)) + Σ (0)),,
OEWin:
BETiE
Then
(T + 0.z) = Σ (0)S),
1
Σ π (0)S u,
2
Win:
T
both platforms choose the same model, then users are split evenly, so
U Σ (0)S,, (0)
1 Tí
U4
Therefore, for N 2:
TT, (T + 8ig)
if
U. (1i,))))
if
Proposition 3.3. [Utility Decomposition.) can be decomposed into Tr, and 6,, (f) as:
The expected utility U, (fr: f-i) of platform in Eq.
U, (faff-))
N (T + (f)
(5)
Proof. Fix a strategy profile f = (f1,..., ,i....,f'' fN) and a model f. chosen by player i. Recall that let M(f) {fi...... .N denote the set of models used in this strategy. For user type 0, define the set maximizers y := arg maXseM(s) (e) and let Ay(0) == [A(()) be the number of models tied for the maximum.
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Under the rule, the share of type 0 that is alloccated to plattoom using fa is
fi e BeArlo (0) f / 1 ()))
1r16
pa (0:0:f)
Hence the expected utility of player i equals
f-) = 9EG
We now prove the followingg per-yye ademitig
N. @ )) 55 (0) Sr, (0) 1/ (0:):))
Y0 10E6
(13)
where 2,(6:3 f) is defined in Definition 3.2
Casee 1: fy & ()) Then pj (0; f) f0:
is O. By Defimition
0, the lent-hand side of Eq.
13
3.2
(-(0:)) f) = -S,, (0) hence S/, (0) + (, f) = S (0) --,,,,,,5
S/, (0) = U. Thus Eq
13 holds. .
Case 2: j E A (0)) Then py (0; f) = _ Again by Definition 3.2
| (000 1,10) S (0)
N - Ar(e)
Therefore
N - A
Sj, (0) N (p:00 1101-115,,0 (1,
51, (0) - 2, (0: 77 =
+
(0) = Ay(0) )
N
So Eq. also holds.
When we have Eq. 13 sum both sides over O with weightts T(0) and divide by N:
Σ 0 1: () = N BEE
I Σ T0)) (S₁, (0) 2, (0
1 N (Te, + 86 (f)
where 6,(f) = Eo -/002//0011 (0:
U U.( = E Eoce (())p: (0 f)5 (0), we obtain:
Since Sincebu
(0),
= N (T, + (f)
1
PROOFS OF LEMMA 3.4
C.3
with We first consider the case with N 2.
Lamma C.2 Condiiionn for Equilibrium for two platforms). (ifformss)
2 plcasformm
Consider geame with N - players choosing Retweern M models G with finite users' type space θ with weights (0) 2 0.
with # j is a fully snttereetii e
Eoce (0)) The utility of each player is defined in Eq. 2 A Stratees f (ft = gi. f3 = 94)
platibbruun iff
T: + dus z max(Ty: maxs& (Tr + 8xy3}
T, + 6j Z max(Ti maass7 TT
(14)
TT + dms)}
When M = 2. the condittoo i Decomeer
86s S T, TT S 65z
(1S)
A Strauees 5 - - (3)) is a Commegenni
prnniiiiiiiiii where all f = a7 for some nn E Ml iff
In E M s.A.
Tm. Tp 2 dam. vk E M. \ fin)
(16)
Whennn When M- = 2, the condiiiion becomes
T T > dg
T: - T3 > 85
(17)
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Proof. First, let's consider that there is only two models, N 2 and M 2. Using Proposition C.I we obtain the utility of each model, from which the payoff matrix can be derived.
f
g:
gi + d()) 7 (T, + +gji)
gi 9,
TT, T
(5
(T) + 6ge), (T + &,z))
(1 2T)
Suppose players choose different models. This is a pure strategy Nash equilibrium if and only if neither player wants deviate, that
[T; + 86)) 2 Tj (T, + 6gx) 2 T
2 Tj Tz 2 T
=
Then the condition is:
8ij T; T3 V وي
Suppose players choose the same model. This is pure strategy Nash equilibrium if and only if neither player wants deviate:
1
2 (T) + 8j¹)
Z
+
T3 0jx <T
or
(T: + 6.y)
< <
=
Ti + 6,5 <;
2
Then the condition is:
T - T: 83j
T. T 8jز
Second, let's consider that there is more than two models. The payoff matrix is:
f
9i
9j
9k. [T. d(x), (T: + dkz)
(1 1 TT)
(T 5ty), ( (T + 5,:))
gi
(H + 0zz): (T) + 6t))
(
(ET 55k)) 2 (Tx + 5kz)
9j
(T + 6k)) [T, (g))
(L + 8kj), 2 (T + 85k))
gk
Suppose players choose different models i. This is pure strategy Nash equilibrium if and only neither player wants to deviate, that is:
(T, &,)) T
1 9 (T. + 8,y) 2 (Th 5kj) Ak M 1 (T, + 6n) 2
=
Tí
1
2 (T 6n)) 2
(Th 85k)) V M
Then the condition is:
T + d.j 2 max (T,, maxky (Tr. : + dzy}) T, + 5, Z max (T,, max / (Tx + dx:
3i + M s.t.
Suppose players choose different models m. This is a pure strategy Nash equilibrium if and only if neither player wants deviate:
2 (T, 551) < Ti
+ T3 65 <T Vijj
Then the condition is:
3m E M s.t.
Tm - T3 2 8pm
Wk M {m}
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Lemma 3.4. [Existence Equilibrium] Consider the game G 1) with finite user rypes O and N platforms choosing from M models G, where M 2 N. A fully differentiated equilibrium f (fi ....... f&) exists if and only if for every platform and every alternative model f; E G (
T3 T₁, 2 (f -5;(f²))
(6)
homogeneous equilibrium f ifi..... .f)) fi exists if and only for some m M
Tm - T 2 5y (f-m U 6m (f*)
(7)
Proof. Fix a candidate profile == let f .Ji...., -. 9T-....
= (fi..... f.). For any player i and any deviation g E G (ff).
f" (o).
By definition pure Nash equilibrium, f is PNE iff for all i and all such g.
U:(f") z U.(f'
Using Proposition
3.3
we have
N U.UU()) = Ty. + 65, (f*)
N N.U.()) = + 0s(
Therefore:
T + of)
Conversely, it holds for all i and all g * ft, then the above inequality reverses to Uf( 2 Uf() for every deviation, so player profits from deviating and f is PNE. This completes the proof of the fully differentiated case.
The homogeneous case is similar, with ft m for all i; plugging m into the same inequality obtain the desired results.
C.4 CALCULATION OF THE EXAMPLE IN FIG.
Scenario A: Two user types O 10 On} with equal weights =(0x) +(0B) 0.5. The model
scores are:
(0)
S₂(0)
0,
0.90
0.85
0R
0.35
0.80
The average scores are:
T = 0.625,
Tg = 0.825,
T3 - Ti = 0.20
The deviation advantages
612 = } (+0.9 0.35) = 0.275,
52 1 1-0085 + 0.80) = 0..25
The differentiation condition - < Ti T2 S 82i becomes:
-0.275 < -0.20 < 00.025
which holds. Hence, by Lemma C.2 the cquilibrium is full differentiated: the two platforms select
different models, even though T
T2.
The payoff matric of this scenario is:
91
g2
91 (0.3125, 0.3125)
(0.45, 0.4) (0.4125, 0.4125)
92 0.45)
So the equilibrium is (91, 92) (g2, 91).
Scenario B: We keep Ti = 0.625, T2 = 0.825, and Tz
Ti = 0.20, but change the type-level
structure to weaken 91 's advantage:
(1)
S₂(0)
0
0.60
0.70
0
0.65
0.95
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The deviation advantages are now:
812 {{ (-0600 0.65) -0625, 821 2 -- 0.95) = 0.825
The differentiation condition - 612 - Ti T2 S 621 becomes:
0.625 - -0.20 < 0.825
which fails. The consolidation condition T2 - > 812 or Ti T2 > 821 holds since 0.20 > -0.625; thus, the equilibrium is homogeneous 92-
The payoff matric of this scenario is:
92
91
0,,122,0.3122
(0, 0.825) (0.4125, 0.4125)
92 (0.825,0)
So the equilibrium is (02.9))-
92)-
C.5 THE PROOF OF COROLLARY 3.5
Corollary 3.5 (High User Centralization = Homogeneous Equilibrium). Assume there exists dominant user type O* with fraction z and model m satisfying: Vj + m, Sm 50(0)) Sz(0') 2 p > and v0 + 0" j F m, (S,(0) Sm (0) S P. If m is sufficiently large and satisfies
T 2 1
1 1+2
then the homogeneous strategy f (m,.... m) is a pure-strategy Nash equilibrium.
Proof. We use the utility decomposition Proposition 3.3 N UUU( TT, (,)
Suppose all players currently choose m, consider a deviation by a single platform to some k + m. Let A denote the utility gain from this deviation, then
A [h + 8n (f,, U [k))] - [T. + 8m(("))
The A is consists of three parts:
Loss on the dominant type 0": Under f", each platform receives a N share of 0*'s contribution Sm. (0*). After deviating to k, the deviator's share on 0* becomes 0 because strictly wins there.
Üsing the margin Smm(0)) 5x p. the utility loss from 0* is least
TOP N
AUg.
Gain on minority types where wins: On θ \ (0*), the total mass is zg. Wherever wins
m, the deviator's share improves from N 1. Since k's per-type advantage over is most I, the upper bound gain is
win
N
Additional loss on minority types where A: loses: On those types where remains superior, the deviator's share falls from to 0. Bounding score levels by the same heterogeneity constant T, we
get
Tÿ)r
A A S
N
So the change:
A AU AUlose AUD-
(1 - T
+ (1 - I
TOP
N
N
N
2(1 m²)l" TAP
N
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Therefore, if A 0, 2(1 Tz)r TØp I 0, that is:
P
p+ 2T
so no player benefits from deviating and the homogeneous profile f Nash equilibrium. So the condition is:
P
T
p+ 2r
C.6 PROOF OF PROPOSITION 4.1
Proposition C.3. Consider the case where N = 2 and fix strategy f = (gi,gg). For each user type 0 O with weight (0) 2 0, the coverage value of the pair (i.j) is
1 (T: + 8zg +dz))
V(i,))
Proof. First, for any i ( j.
8ig + dos
Σ
T S,() (0)
(18)
0E6
Fix a type Consider three cases.
Case 1: S,(0) S,(0): Then Zip (0) S;(0) and (g) -S,(0), so (,) + Zy,(0)
(,(0) - 5, = [5;(0) (,))
Case 2: S,(0) (,) Then Zi (0) (S: and ()) S,(0), (,) + Zz(
S,(0) (0) = =S (O))
Case 3: ()) =S, Then (0) 7 (0) 0. hence the sum is 0 = [5:(0) S,(0))
Multiplying by To and summing over yields the claim.
Use the pointwise identity: maxfa, b} 2 (a + b + la bl). Applying it with S.(0) and
b = (0) and summing over
v(i,) Σ (0) max ( (5,()),)(())
T(0) (10) S,(
2
9
1
(0)).,(0)) + Σ (0)S)( + Σ
-0) π (0))s
2
0
0
T
5. b Eq
(T, + T3 + 8zz + 05z)
2
Proposition 4.1. (Coverage Value Calculation] Given strategy profile f = (fiy..,, SN), the coverage value in Definition 2.4 can written as:
1
v(
(T, +8,
N
t=1
where T and are defined in Definitions 3.1 and
3.2
respectively.
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Proof. Fix 0 and recall Ay arg maX)eM)()) Ss(0) and Ay(0) [Ay( By the definition of (0;0;)) f)::
1+ N - -))
Σ JEM(S)
Σ )EAy
S,
A₃(0)
Σ SJ(0)
Ay
(0)
N max (x)
kEM(f)
Multiply by =0), sum over O, and divide by N to obtain
V(f)
Σ JEM(S)
Σ क (0)S,(()) + E= =(0)
Σ
(g)
JEMIS)
N
Σ (Tx. (g)
C.7 PROOF OF LEMMA 4.2
Lemma 4.2. Let W denote the user welfare (Definition 2.5
achieved under the game ( I,)),
and W opt. the social optimum welfare (Defiittion) 2.6 Then, it always holds that W < W opt..
Proof. If = f is a PNE, then W(O) = V(f) < maxy V(f) = Wopt by definition.
is cycle f()),.. ((L), then
- vv Σ max V(f) W Wapt
W(O)
L E=1
since an arithmetic mean is at most its maximum term
Therefore, (OO)= 2 Wopt
C.8 THE EXAMPLE OF LEMMA 4.2
Example C.4. Consider three user types 0A,0n.0c with weights T(OA)
= 0.5, T00x) 0.3,
T0c) 0.2. Their scores for each of the three models gl, f2, fa
91
92
93
0
0.434
0.698
0.760
0
0.828
0.679
0.431
0c
0.343
0.776
0.565
The average scores are:
T 0.534,
Tg 0.7079,
T3 0.6223.
The pairwise attraction shifts d.j are computed as in the model:
812 00.0372
821 = 0.3005,
513 = 00.372
5q 0.3637,
823 0.0099,
8z2 0.1377
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The coverage value of pair (gs: 93) is
Vii,j) =
Σ no max So(g;)
Numerically:
2( 0.7526, V(133) 0.7414, V(2,3) 0.7389
Thus, the socially optimal pair (2) with
W opt 0.7526.
The payoff::
91
92 (0.2484, 0.5042)
J3 (0.2484, 0.5112)
91
(0.267, 0.267)
92
(0.5042, 0.2484)
(0.35395, 0.35395)
(0.3589, 0.38)
93
(0.5112, 0.2484)
(0.38, 0.3589)
00.31115, 0.31115)
Equilibrium check for (92: gs):: the differentiation condition requires:
Th + 823 2 max (3,, T + 612}
+ 632 max (T2, Ti + 812}
Substituting:
T2 + 623 = 0.7079 + 0.0099 = 0.7178
max Ts. Ti 813} max(0.6223, 0.534 0.0372) 0.6223
Ta + 632 = 0.6223 + 0.1377 = 0.7600
max Ti + 812} max(0.7079, 0.534 0.0372) 0.7079
Both inequalities hold, hence (g2, 93) is full differentiated equilibrium with welfare
W V(2, 3) 0.7389.
Although (92, ga) is a valid differentiated equilibrium, yields lower welfare than the optimal pair (91.
W eq = 0.7389 < Wopt = 0.7526
This demonstrates that a differentiated equilibrium does not necessarily coincide with socially opti- mal differentiation.
C.9 PROOF PROPOSITION 4.3
Proposition 4.3. Consider a game G I, ④) with an equilibrium f". Let := I= be another game with one additional platform added. Suppose there exists a model E G and an satisfies the
incumbent equilibrium strategy f from f such that the extended profile ( h) best-response conditions: (i) the best response f is h; (ūi) incumbent platform has profitable deviation against f. Then f is an equilibrium of the game G. Furthermore, the user welfare and market diversity in G are at least as high as G. i.e., 2 W and Daupp 2 Daupp-
Proof. Best-response conditions (i) and (ii) imply is PNE. If f belongs cycle, appending h yields a one-step extension that meets the same no-improvement conditions for that period, so the induced outcome is an equilibrium.
For welfare, by the Proposition 4.1 since M(f) M(f*) U {h), for every type O we have
max 5z( 2 max
Sx(0)
kEM(f)
kEM(f*)
Summing with weights (0)): v(f) 2 V(
If h & M())) and improves some type strictly, then the inequality is strict.
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Example C.5 (counterexample: two - three models). Two user types è [0A.02) with equal weights (0 = π T(0g) 0.5.
Scenario A: The model scores are:
(1)
S2(O)
A
0.90
0.85
B
0.35
0.80
The average scores are:
Ti = 0.625,
T3 = 0.825
The deviation advantages are:
812 1 (+0.90 - 0.35) = 0.275,
821 = -- (-0.5 0.80) -0.025
The payoff matric of this scenario is:
91
g2 (0.45, 0.4)
91 (0.3125, 0.3125)
92
(0.4, 0.45)
(0.4125,0.4125)
So the equilibrium is (g1, 92) (02, 91). and W(O) = -V112) = 0.85
Scenario B: Add new model Is with (6() = 0.91,S;(0g) 0.77 Then T3 0.84 The
deviation advantages are now:
512 1 (+0.90-0.35) = 0.275,
821 2 (-00885+0.80) = -0025
613 = (-0.900-0335) -0.625
531 ++0.91++0.77) = 1.68,
623 = ( (-0.85+0..80 -:0==-0000 0.025,
832 = + = 0.07
The payoff matric of this scenario is:
f
91
92 (0.45, 0.4)
93
91
(0.3125, 0.3125)
(0, 0.84)
92
(0.45, 0.4)
(0.4125, 0.4125)
(0.4, 0.455)
g3
(0.84, 0)
(0.455, 0.4)
(0.42, 0.42)
So the equilibrium is (ga, 9a), and W(O) = V(3,3) = 0.84.
Here, 0.84 < 0.85
Example C.6 (counterexample: two - three players). Consider user types ((0A) = 0.18, 7(0)) 0.17, x (0c) 0.16, x ((0)) 0.16, (02g) 0.17, (0p)) 0.16
The model scores are:
(0)
())
(x)
S,(0)
(0)
(g(
0A
0.030658748
0.208093837
0.32744655
0.298774863
0.154842913
0.020151094
03
0.021978186
0.149636775
0.298145086
0.274754494
0.092761844
0.014372437
dc
0.266589463
0.035725005
0.019578686
0.029395873
0.04788997
0.182804301
0D
0.171553999
0.007992042
0.007932614
0.019235272
0.067757338
0.160182327
0F
0.039888468
0.145473659
0.077957489
0.078738135
0.110034101
0.019024562
Or
0.131 100401
0.089481771
0.136355415
0.132456332
0.095638528
0.136379898
Scenario A: With only platform: the equilibrium is (gs, 96) with user welfare W 0.2148
Scenario B: With three platforms: the cycle is (gs, 93 91) - (gs. Iz. 96) + (g), 93. 96)
(g1.ga 93. (3) and W (V 93. 91) + V(gs, Pa. 9a) + V S3. (a)) /3 (0.2147 + 0.2148 + 0. 0.199571) /3 = 0.2097
Since 0.214 0.210, adding platform may not increase the welfare.
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C.10 EXTENSION SOFTMAX USER CHOICE MODEL
Proposition C.7 (Robust nonexistence of PNE under softmax choice), Consider fixed instance (o, ni {0)()) Let Uhard (f) denote platform ufilities under the hardmax user choice rule Eq. I and suppose that the induced platform game admits no pure Nash equilibrium, in the following strict sense: there exists A such that for every profile f there is a platform i and a deviation ft with:
U.hard (fi: f-i 2 (f.
(19)
Let Utoft (f:T) be the utilities under the saftmax user choice rule Eq. 8 with O. Then there exists To such that for V 0 < To, the softmax game (Uzoft (:5 (;))) also admits no pure Nash
equilibrium.
Proof. Fix a profile f and a type 0. Under hardmax, type user only considers platforms whose model achieves the highest score max; St (0), assigns equal probability to those platforms, and assigns zero probability to all others. Under the softmax rule Eq
S1, (0)/T
soll
Σ (,, (0)7
as -> 0, the largest-score terms dominate the denominator, (0; - (hard(O).
Platform utilities are finite weighted sums of these probabilities:
Usoff (f) Σ
Top: soft (0)
hence UNoft( (f) + Uhardd (f) as r + 0.
Because the strategy space is finite, this convergence is uniform over all profiles f: for any >> there exists To > such that for all 0 < r S To. all platforms i, and profiles f.
U (f) Uhard (f) S
Consider any profile f. By the strict no NE condition Eq. 19 there exist and f with
Uhard f-+) - Uhard (-: A4
Choose = A/4 4 and the corresponding For any 0 < <TST
Upofl (f:: (f:: f-1) U:oft (fr: f-)
[ (f:if -a] 2 A 2: A/2 0
[U.bard +
Thus f cannot be best response strategy in the softmax user choice, Since f was arbitrary, the softmax game has no pure Nash equilibrium for any 0 < r S To-
Example C.8. We provide constructive counterexample here. Let θ {0A.0B} with uniform weights x (0x) = 0.5. Let = (91, 82, 9a} and define scores
(0)
(2
(a
0
0.734
0.148
0.934
0
0.833
0.935
0.534
the softmax user choice is used with 7 = 0.1, then when there are two players, the payoff matrix
is:
f = (f1, f2)
82
91
(0.39175, 0.39175)
(0.47634, 0.34381)
(0.43853, 0.42549)
92
(0.34381, 0.4763)
(0.27075, 0.27075)
(0.45843, 0.47210) (0.36925, 0.36925)
g3
(0.42549, 0.43853)
(0.47210, 0.45843)
Here, the cycle is (ga, 91 (-(s) 92 (g), (2)
The W opt 0.9345, but W (0.8835 0.9345
0.835)/3 0.87067
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We now show that the T + 8 decomposition extends to the softmax choice rule in Eq. 8 We keep Definition 3.1 for the average score T unchanged, and adapt the attraction term and deviation advantage follows:
Definition C.9 (Attraction Term and Deviation Advantage of softmax). For strategy profile f = (fi..... fN) with fi E M, the attraction term for f. in strategy f is defined as
(N 1) (0)/T 1
F (0; f)
(g)
(20)
Σ (0/=
The deviation advantage for fa under strategy f is defined as
8/ (f)
Σ
ne Zz, (0: f)
(21)
B
With this definition, the utility decomposition in Proposition choice:
3.3 continues hold under softmax
1 (Ty, + 0% (f))
U f-i)=
(22)
N
Proof. We use el denote Eyjer (0/T
We have:
(0)
U (fr.fi)) Σ T())
(0)
(23)
ef
OEO
e0, ct (g:
T₁ =
Σ
T(0) (0)
Σ (
(24)
et
AE
AEO
(N
c ())
f (f)
Σ
To f.. (0; f)
Σ To
(25)
ef
is clear that U.D0 (fr, f-i) = N TT, + 5 (
D EXPERIMENTS WITH SYNTHETIC DATASET
Table 3: Simulation model pool.
Model
by
Ajr
Gjr
0.12
(1.5, 0.0)
0.90
1.20
2 0.05
(0.0, 0.0)
1.30
0.35
3
0.08
(3.0, 0.0)
1,00
0.50
4
0.06
(0.0,0.0), (3.0,0.0)
0.70, 0.70
0.70, 0.70
5
0.05
(1.5, 0.6)
1.00
0.40
0.05
(1.5,-0.6)
1.00
0.40
In this section, we design controlled simulation environment to study equilibrium outcomes under different models and user populations.
Generative Models. We consider M generative models G = {6) M each parameterized as Radial Basis Function (RBF) [Broomhead & Lowe 1988) mixture:
R2 Agr exp (- 1-25 1| 13rll3))
(x) b, + Σ
where R, is the number kernels for model j. Pir is the center of the r-th kernel, ojr is its width, Ajr is its amplitude, and by is bias. Outputs truncated (0, 1].
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User Distributions We represent the user by θ [0 where 0 E R has distribution ne- The distribution Te is derived by discretizing Gaussian Mixture Model (GMM) with Q compo- nents, where each component is parameterized by weight w 2 0 with Eg wy mean vector Mg. and covariance matrix Zig.
(u)
Σ Wy Vu (u)). 9=1
continuous samples u drawn from this GMM are then mapped to the nearest discrete type 0x. This construction yields a finite user distribution x0) that serves as input the equilibrium analysis. The variant user groups are constructed by shifting all component means along the z-axis: py + (4) + (dx, 0) where dr controls the degree of population shift or by adjust different weight wig.
Table 4: User distribution parameters.
Weight
Mcan M4g
Covariance Eg
0.6
(0.0, 0.0)
[|0.25, 0.0). [0.0, 0.25]] [[0.25, 0.0). [0.0, 0.25
0.4
(3.0, 0.0)
round
2
round
0.75
0.7
0.70
1.9
0.65
model
HHl diversity
player
HHI diversity
dys -1.6
HHI diversity
0.60
2 models
0.8
players
player
0.6
models
players
players
0.55
model
0.6
dy- +1.2
models
model
0.4
0.40
0.4
0.35
0.2
0.30
step
round
step
(a) DHE model pool increases.
(b) Dum as player increases.
(c) DHHI as increases.
round
round
0.65
0.60
0.60
model
models
0.55
0.50
3 models
models
models
0.55
0.50
- players player
player player
0.55
0.50
dx- +122
dv +0.6
dx- -0.6
dx 0.0
plavers players
step
round
step
(d) V(f) model pool increases.
(c) V (f)) as player increases.
(f) V(F) dz increases.
round
round
2
0.7
M5
HHI diversity
0.7
M2
w, 0.44 w, 0.66
0.6
((L6,
0.6
n, 0.06 00.4
(0.4.0.))
0.5
w,, w, -0.3
0.5
-- Hi.1 W. m.. 1.9---- w.
(0.2.08)
---W, W. 10.6- -0.6
0.3
N.
(0.1.0.)
step
step
Utility
(g) Dun with different w.
(b) V(f) with different w.
(i) Utility with different uc.
Figure 7: Best-response simulations under different settings.
The expected reward of model for type is Sy(0) Exwgy [re(z)]. In theory, 9j and ry are distinct objects, however, in our simulation, we collapse 9j and mo into a single score function implemented radial basis function (RBF) mixture.
Reward Function.
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Simulation Parameters and Results. For all simulations, have model pool with M = models as shown in Table and = 12 user types drawn from the GMMs. The baseline user
distribution uses Q 2 components shown in Table
We conduct four sets of experiments:
Expanding the model pool. With three players fixed, we gradually enlarge the model pool size from to 6. The resulting diversity DHHI and coverage value (ff) are shown in Fig. 7a and Fig. 7 respectively.
Increasing the number of players. With the full model pool available, we increase the number of players from to 6. The resulting diversity Dum and coverage value V(f) are shown in Fig. 7b
and Fig. 7e respectively.
Shifting user groups. With the full model pool and three players, we vary the GMM means used to sample user types by setting dr
f00.6, 0.6, 1.2). The diversity DHHI and coverage value
V(f) are shown in Fig. and Fig.
7f respectively.
Changing mixture weights. With the full model pool and three players, we alter the GMM compo- nent weights (Way Wgz reported in Fig. Fig. 7h and Fig.
{(0.1, 0.9), 0.8), (0.4, 0.6), (0.6, 0.4), (0.7, 0.3)) The results are
We observe that equilibria always exist in this setting, but diversity and welfare vary substantially depending whether the models are sufficiently differentiated. Strong but substitutable mod- els lead to market homogenization, while genuinely differentiated entrants promote diversity and increase welfare.
E EXPERIMENTS WITH REAL DATASET
E.1 DISCRETE BEST- RESPONSE SIMULATION
The models in the model pool are constructed by applying different LoRA parameters to the back- bone network, each trained on different CIFAR-10 10 subsets, summarized in Table. The backbone network itself was trained on the full CIFAR-10 dataset for 200 epochs. During training, we used a learning rate of 10a 1000 diffusion steps, and batch size of 256.
We first provide the average performance T: of the five models in model pool and their user-specific performance (0) in user groups in Fig.
0.3
M1
M2
M3
M4
M5
F
0.16
0.1
0.14
0.0
Avg. Scores
0.12
0.1
0.10
M2
0.08
MI
0.06
0.04
E
0.02
0.00
Models
D
(a) The average performance.
(b) The user-specific performance.
Figure 8: The average performance T, of the five models model pool in Table and their user- specific performance (,) in Table 2
Impact of different user groups As a complementary experiment, we examine how platform choices vary when facing different user groups. Specifically, consider three player choosing
29

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Table 6: User type with its preference
Table 5: Different user pool
User type(0)
Preferences(class(e..)))
User Pool
User type (0) with (0)
A
cat (0.6), dog (0.4)
Pool 1
(0.2), B (0.2), (0.2), F (0.2)
dog (0.7) cat (0.3)
Pool 2
A (0.3), C (0.3), E (0.4)
airplane (0.5), ship (0.3), auto (0.2)
Pool 3
C (0.2), D (0.2), E (0.35). F (0.33)
auto (0.6), truck (0.4)
Pool 4
A (0.6), (0.4)
bird (0.4), deer (0.3),
A (0.1), (0.09), (0.16).
Pool 5
frog (0.2). horse (0.1)
D (0.32), E (0.17), (0.16) C (0.33), D (0.35), E (0.2), F (0.12)
(0.2), dog (0.2), airplane (0.15), auto (0.15), ship (0.1), truck (0.2)
Pool 6
F
Table 7: Outcomes of a 3-player setting under different user pools.
User Pool
D Supp
Dum
Weg
Pool
(M5, MM5, M5)
0.3333
0,2110
Pool 2
(M5, 55. M1)
0.3349
0.2117
Pool
(M5, M33, M3.
0.3335
0.1615
Pool 4
(M5, M5, M5)
0.3333
0.2366
Pool 5
(M5, M1, M1)
2
0,3856
0.1932
Pool
(M1, MI,
0.3333
0.1715
from the model pool. The user group configurations are provided in Table. 5 and the corresponding equilibrium outcomes are summarized in Table.
E.2
ALGORITHMIC BEST-RESPONSE ENTRY
E.2.1 ALGORITHM DETAILS
We provide the implementation details and hyperparameters used in our experiments for evaluating algorithmic performance. We first describe the specific procedures of the Resampling and Direct- Gradient methods, followed by the hyperparameters employed in training and evaluation. Unless otherwise specified, the same base diffusion backbone and optimization settings are applied across methods for fair comparison.
Resampling.
The algorithm for resampling method is shown Algorithm,
Algorithm Training Data Resampling
Require: Dataset D;; user types O; fixed opponents G {91.".. gad with scores {S.. (0));
parameters 8, y; outer rounds T; inner epochs E; evaluation budget b.
1: Compute S(0) maxjeM S)) for all 0.
2: for 1,..., do
3: Estimate (0) = r
4: Compute AS (0) S(0).
5: Compute
(3
6:
Type weights 0g = T(() (ao)² 50))
7:
Data weights: w(u)= x Eo 00 (1) or w(x) & Eo (z))
8:
Sample D with á as D
9: for e = 1,..,, E...E' do
10: Update o by minimizing the original loss in lb.
end for
12: end for
13:
return d.
The specific parameter details for algorithm-level:
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Published as a conference paper at ICLR 2026
Outer round, the time of resample T = 5.
Inner epcohs E' 50.
B
1.
Use adaptive scaling for og-
LoRA rank LoRA scaling coefficient 16, LoRA runtime multiplier 1.0.
Direct-Gradient. The algorithm for direct method is shown as Algorithm, 2
Algorithm Direct- Gradiett Optimization
Require: Dataset D; user types θ; fixed opponents {91, "M .9M with scores {S.m (0));
parameters X; Epochs E;
1: for == ...... E do
2: Estimate (0) E. og ro(x).
3: Compute 4g (0) S(0).
4: Compute
(3
5:
L (()) x Loee (0))g5((0)
6: o+ Φ VL.
7:
end for
8:
return
The specific parameter details for algorithm-level:
Epcohs F 20.
À 0.4.
Use adaptive scaling for ag-
Shared parameter.
The specific parameter details for shared training parameters:
The backbone network is trained on the full CIFAR-10 dataset for 200 epochs.
Batch size 256.
Learning rate 2 10 (for AdamW optimizer).
Diffusion steps 1000.
Data Distribution. As supplement to Fig. 6 A, provide the label distributions of 2, 000 sam- ples generated by three models, as shown in the Fig. 9
airplane dog
automobile
bird
cat
deer truck
frog
horse
ship
10%
38.2%
31.3%
27.4%
30.8%
70.3%
11.1%
75%
MS in original model pool
M samp
grad
Figure 9: Label distributions of 2, 000 generated samples from three models: (a) left: M2 from the original model pool (b) middle: Maamp by redampling method. (c) right: Mgrad by direct-gradient method.
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Published as a conference paper at ICLR 2026
0.4
0.3
0.3
B F
B F
C
C E
1-
p y-1
B-8 y-1
D
0.4-A
0,3
0.3
B
F
C E
E
|= 0
D
D
Figure 10: Different performance of model scores across user groups under different 3 and y com- pared to each user group's 's best score S.
0.4
0.4
0,3
0,3
2 6
2
2
0.1
0,1
5
5
3
3
+025
=055
=1
4
04
0,6
0.5
0.6
0.3
2 6
2
0.4
2
0.1
5
5
3
3
-02
-0.3
=044
Figure 11: Different performance of model scores across user groups under different À compared to each user group's 's best score S.
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Published as a conference paper at ICLR 2026
Table 8: Performance of four models on coding and reasoning benchmarks.
Model
HEval
Multi-language
Overall
Math
IFEval
MO: CodeLlama-3444 3a4B
0,5079
0.4297
0.1721
0.0413
0,4604
MI: Qwen2.5-CCode-332B
0.5710
0.6497
0.3326
0.3089
0,4363
M2: Nxcode-CQ-7B-orgg
0.8723
0.6688
0.1237
0,4007
0,4007
M3: Swen2.5-Coder. 3a---332--IInnccc
0.8320
0.7723
0,3989
0.4955
0.7265
Table 10: User type with its preference
Table 9: Different user pool
User type(0)
User Pool
User type (0) with T0)
Preferencee(class(((.
Pool
(0.2). B (0.2), C(0.2), D(0.2), (0.2)
A
HEval (0.6), Overall (0.2), (O.)
Pool 2
A (0.1), B (0.1), C(0.2). D(0.5). (0.1)
B
Overall (0.8), IFEval (0.2)
A (0.35), B (0.2). C(O.2),
C
H[Eal (0.8), IFEval (0.2)
Pool 3
D(0.35), E (0.2))
D
HEval (0.6), Muti-language (0.5)
Math (1.0)
E.2.2 ALGORITHM PARAMETER SENSITIVITY ANALYSIS
For resampling, we investigate the sensitivity of 3 and The result is shown Fig. 10 For direct- gradient optimaiztion, We investigate the sensitivity of which controls the trade-off between utility and the diffusion. The result is shown as Fig.
F EXPERIMENTS ON LANGUAGE MODELS
In this section, study the three-layer game in language setting using real large language models.
round
1.0
1.0
0.9
2 model
HHl diversity
0.8
player players
players player
diversity
0.9
Pool
models
players
players
Pool
models
0.6
0.7
HHI
0.6
0.6
0.4
0.5
0.5
0.2
0.4
0.3
step
round
step
(a) DHHI model pool increases.
(b) DHHI as player increases.
(c) DHHI as dæ increases.
round
2
round
0.65
0.65
0.60
0.60
0.7
0.55
0.6
0..50
0.45
0.40
0.35
model
models
models
models
0.35
0.40
--1 player - players
players player
0.4
Poot
players players
0.30
0.3
0
step
round
step
3
(d) V()) as model pool increases.
(c) V(f) as player increases.
(f) (f) as dæ increases.
Figure 12: Best-response simulations under different settings of language tasks.
Model Pool. We consider pool of four publicly available models that appear both the Big- Code models leaderboard (BigCode 2023) and the Open LLM Leaderboard (Fourrier et al.) 2024): Codel Jama-34B (Rozière al. 2024), (wwen.5-Coderr (Hui et al., 2024), Excode-CCQ771
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Table 11: Outcomes of 3-player setting under different user pools.
User Pool
Daupp
Dum
Weg
Pool
(M3, M3, M3)
0.3333
0.65945
Pool 2
(M3, M2, M2)
2
0.375
0.75949
Pool 3
(M3, M3, M2)
2
0.33375
0.73080
orpo (Hong & Thorne 2024) and
(Hui et al. 2024). For each model, we collect its HumanEEvllPyython score (HEval) and multi-lange maulti-annguge coding score (Multi-language, average over Java, JavaScript, and C++) from the BigCode leaderboard, well its overall, math- related scores and eestructionfolllwiiin evaluation (IFEval) from the Open LLM Leaderboard. Ta- ble 8 summarizes these benchmark results, which we treat pre-computed performance statistics.
User Group and reward function. We partition the user population into five groups, each char- acterized by heterogeneous preferences over metric preferences the details are given in Table 10 Then the reward for user type 0 is calculated by ro(x) Eeec de performance.
Simulation and Results. We conduct three sets experiments:
Expanding the model pool. With three players fixed, we gradually enlarge the model pool size from to 4 face the user pool 1. The resulting diversity DHHI and coverage value V(f) are shown in Fig. 12a and Fig. 12a respectively.
Increasing the number of players. With the full model pool available, we increase the number of players from to face the user pool 1. The resulting diversity DHHI and coverage value (())
are shown in Fig. and Fig. [2e] respectively.
Shifting user groups With the full model pool and three players, we vary the user pool as shown
in Table. 9]. The diversity DHHI and coverage value V (f) are shown in Fig. 12c and Fig. 121
respectively. The corresponding equilibrium outcomes are summarized in Table.
THE USE OF LARGE LANGUAGE MODELS
We used a large language model to aid in polishing grammar and phrasing. Consistent with ICLR policy, authors remain fully responsible for all content, including parts assisted by an LLM.
34


Reviews:
[{"review_id": "Ej7urJZiY2", "reviewer": "Reviewer_vtm9", "summary": "This paper formalizes GenAI ecosystems as a three-layer market game consisting of model providers, platforms, and user populations. Platforms strategically select genAI models from providers to serve user groups, while users choose platforms based on preferences. The work discusses the structure of equilibria and analyzes conditions for PNE, market differentiation, and user welfare, and provide algorithm for model provider for effective market entry.", "questions": "Please refer to my questions raised in the weaknesses section.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The topic is timely and relevant, theoretical analysis is nice and solid. Presentation is easy to follow. Despite the weaknesses I pointed out below, I really like the angle from which the paper formulates the problem and the style of presentation.", "weaknesses": "As a theoretical-oriented paper with the claimed strength lying on the proposed game-theoretic model, my main concern is that the model structure and theoretical finds are not sufficiently interesting for providing new insights. Here is some of my thoughts:\n1. The model seem to me is too stylistic. The hardmax user choice model is overly simplified, it would be more interesting to consider softmax or other alternative stochastic choice model and see if similar observation holds. If hardmax is a necessary simplification to derive theoretical results, I believe some discussion or simulations on alternative user behavior models will strengthen the paper. \n\n2. I do not see a main theoretical claim (there are lots of proposition, lemmas and corollaries, but no theorems). And those results seems to be straightforward and lacks insights. For example, \n - non-existence of PNE is standard as in my understanding these types of games can only be shown to have PNEs if it has a concave or potential structure. I would expect a deeper understanding like if the PNE does not exist, what would a typical best-response loop look like and how it reflects some phenomenon in reality. \n- Corollary 3.5 conveys a very intuitive message but if unfortunately it rely on a seemingly too restrictive set of assumption. I'm wondering how rare the situation happens when a model has a clear advantage over all other models for the majority user group, while performs similar to other models for all remaining user groups. \n- Lemma 4.2 does not contain any quantitative result. Isn't it trivial to simply saying the welfare at equilibrium can be less than the globally optimal welfare? It is just the definition of the price of Anarchy.\n3. The three-layer model does not seem to be tied to the nature of GenAI. The exact same model can be used to study the content creation market as well, if we view the model providers as content creator and candidate models as potential topics or genres of content creation strategies. That said, the proposed model is claimed to capture the market of genAI models but does not actually capture any nuance of the power of GenAI nor how it adds anything special to the market. I might have missed something, if so, I'd appreciate it if the author reemphasize how the competition model is uniquely relevant to genAI.  \n4. Section 5 provides little conceptual contribution. I'm not sure how practical the optimization procedure can be actually adopted by a GenAI model provider to determine the new market entry strategy, as it rely on transparent information of all the factors in the market. And one thing it fails to consider is the cost, which should be a very important factor for new players.\n5. Limitations in experiments. CIFAR-10 setting is somewhat toy-level and I believe results on language or multimodal models would better reflect generative model ecosystems. Results are lack of statistical significance as well, no error bar seen in figures. And also investigation on the sensitivity of results to various user distributions would be highly appreciated.\n\nMinor issues:\n1. typo in Definition 2.1 (Nash Equilibirum) -> Nash Equilibrium", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "vG2mfCVJYP", "reviewer": "Reviewer_1m6H", "summary": "The paper presents a three-layer framework consisting of the model, platform, and users to study competition in the generative model ecosystem. In this setting, foundation model providers such as Anthropic and OpenAI license their models to platforms like Microsoft Azure and Amazon Bedrock, which then compete to attract users. The authors derive conditions under which a pure Nash equilibrium exists and describe its structure, examining whether platforms use same models (a homogeneous equilibrium) or different ones (a heterogeneous equilibrium). They then analyze user welfare and show that greater competition, either through more models or more platforms, does not always lead to higher social welfare Finally, the paper considers the perspective of model providers and explores training strategies that can improve their overall utility", "questions": "- Please address the question raised in the weaknesses, particularly regarding platform model choice. For example, how could the framework be extended to allow platforms to deploy multiple models, and how would this impact the analysis and equilibrium outcomes?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The main strengths are:\n1) Originality in model formulation: The paper introduces a three-layer model, platform, user framework to study competition in generative AI markets. Prior work, typically focuses on two-layer settings involving only users and platforms/models. Their framework captures the distinct incentives at each layer and the competitive interactions (among platforms and among model providers).\n\n2) Section 3 is a highlight of the paper. It provides a clear and rigorous characterization of equilibrium conditions and market structure. The decomposition of platform utility into attraction and deviation components show when platforms converge on a single model versus when they diversify across different ones.\n\n3) Section 5 is also notably original, introducing a strategic perspective on model training. It shows how model providers can adjust their training objectives to improve adoption by competing platforms, with the direct-gradient optimization approach standing out as an innovative method.", "weaknesses": "1. Platforms Limited to a Single Model:\n\nThe modelling assumes that each platform selects one model provider. This does not reflect the papers motivation where platforms like Azure and Bedrock host multiple foundation models simultaneously. As a result, the framework cannot capture strategies such as model bundling, which are important for platform differentiation and for covering diverse user needs. This is not a major weakness, but it would be useful if the authors could discuss how multiple-model selection could be incorporated into the current framework.\n\n2. Inconclusive Welfare Insights (Section 4);\n\nThe findings in Section 4 are somewhat unclear. Figure 3 illustrates that adding a new model can reduce user welfare, highlighting the “paradox of competition.” At the same time, the authors provide sufficient conditions under which welfare increases. However, it is not clear how frequently these conditions are met in practice, and in the experiments (Section 6) user welfare appears to increase as more models are added. This makes it difficult to draw a consistent, general conclusion about the effect of competition on welfare.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "QpmFmLcTdH", "reviewer": "Reviewer_ZFbA", "summary": "This paper introduces a formal game-theoretic model of a three-layer generative AI market, comprising model providers, platforms, and heterogeneous users. It analyzes how platform competition for market share through model selection shapes equilibrium market structures, user welfare, and diversity. The key findings include necessary and sufficient conditions for the existence of pure Nash equilibria (both differentiated and homogeneous), a demonstration that increasing competition (via more models or platforms) does not necessarily improve welfare or diversity, and the proposal of two training methods for model providers to enter such a market strategically. Theoretical results are supported by experiments on synthetic and CIFAR-10 data.", "questions": "\tHow would the central conclusions regarding equilibrium existence and market structure change if the deterministic user choice model were replaced with a stochastic one (e.g., a softmax function based on scores)? Could this alleviate the non-existence of PNE in some cases?\n\n\tThe model relies on the strong assumption of complete information about user preferences. How do you expect the strategic dynamics and your conclusions to change in a more realistic setting with partial or asymmetric information, where platforms must learn user preferences over time?\n\n\tIn the current framework, when platforms and entrants formulate their best responses, they treat competitors' strategies as static. If we adopt an equilibrium concept where all platforms \"act simultaneously\" and anticipate each other's reactions (such as Nash equilibrium, which relies on static assumptions in its analysis), do you believe the model entry strategy proposed in Section 5 would remain effective? How should a \"strategic entrant,\" aware that its entry would trigger a recalibration of platform strategies, design its model?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "\tOriginality: The three-layer model formulation is a timely and non-trivial extension of prior work, better capturing the structure of modern generative AI ecosystems.\n\tTheoretical Rigor: The analysis of equilibrium conditions, linking market structure to the balance between average performance and deviation advantage, is a solid theoretical contribution.\n\tHolistic Approach: The paper cohesively analyzes the market from platform, user, and (to a limited extent) model provider perspectives, providing a relatively comprehensive initial view.", "weaknesses": "\tStrong and Potentially Unrealistic Assumptions: The model relies on several strong assumptions that limit the practical applicability of its conclusions.\n\nComplete Information: The model assumes that platforms and model providers have perfect knowledge of the user type distribution π_θ and their reward functions r_θ(x). This ignores the significant challenge of preference learning and the strategic implications of information asymmetry in real-world markets.\n\nDeterministic User Choice: The hard, winner-take-all user selection rule (Eq. 1) is a simplification. User behavior in practice is often stochastic, influenced by factors beyond a single quality score (e.g., UI, habit, discovery). A softer choice model (e.g., probabilistic) might lead to different equilibrium dynamics and welfare implications.\n\n\tStatic Nature of Competition:\n\nNon-Strategic Model Providers: The set of models G is treated as exogenous. Model providers are not modeled as strategic agents who dynamically develop or fine-tune models in response to market outcomes, which is a key feature of real-world competition.\n\nMyopic Best-Response: The analysis of platform and entrant strategies primarily considers myopic best-response to static competitors. It does not fully capture the simultaneous, forward-looking strategic interactions where all agents anticipate and react to each other's moves in a multi-stage game, potentially altering equilibrium outcomes.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "d6G9jcBZOF", "reviewer": "Reviewer_c5YV", "summary": "This paper studies the game-theoretical formulation and its Nash equilibrium (NE) of the \"model-platform-user\" market exemplified by LLM applications. Specifically, the paper considers the following three-layered market: (1) LLL company releases models, (2) each service platform chooses which LLM to use in their service, and (3) users choose which LLMs to use for their purpose. The paper finds that NE has to conditions that each service uses a different model or all the services use the same model and highlights the finding that increasing the choice of model does not necessarily lead to a social optimum.", "questions": "- I didn't really understand why score Z can be the negative of S when f is not in A (Def 3.2). I understand that this is needed to show the utility decomposition in Prop 3.3., however, it is not quite intuitive why Z can be anti-proportional when f is not in A.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The motivation for studying the \"model-platform-user\" market is well-explained, and I agree that this is a new and interesting market game to think about.\n\n- The paper provides a solid game-theoretic analysis of the given problem, and the formulation seems reasonable for the most part.\n\n- The paper also presents an algorithm to steer systems to achieve an equilibrium with diverse models, and shows that it works in the experiments.", "weaknesses": "- One discussion I have is whether the best-response dynamics are practical or not. In my understanding, the user and service greedily choose the best model or service for them. However, this may result in a quick change of behavior and does not seem realistic. Recent work on performative prediction considers a gradual change of participants (e.g., Brown et al., 22), and this looks like a more reasonable formulation.\n\n- Another discussion is whether it is reasonable to consider a single reward. This is because services may focus on different tasks such as coding or translation, and depending on the application, what kind of aspects of LLM, may be different from each other. In such cases, I wonder if homogenization or \"a-winner-takes-all\" dynamics can really happen.\n\nBrown et al., 22. Performative Prediction in a Stateful World. https://arxiv.org/abs/2011.03885", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)