Heterogeneous Data Game: Characterizing the Model Competition Across Multiple Data Sources
Authors: Renzhe Xu
Kang Wang
Bo Li
Venue: ICML 2025 poster (2025)
Keywords: Heterogeneous Data Game
Model Competition
Nash Equilibrium
Primary area: theory->game_theory

Abstract: Data heterogeneity across multiple sources is common in real-world machine learning (ML) settings. Although many methods focus on enabling a single model to handle diverse data, real-world markets often comprise multiple competing ML providers. In this paper, we propose a game-theoretic framework—the Heterogeneous Data Game—to analyze how such providers compete across heterogeneous data sources. We investigate the resulting pure Nash equilibria (PNE), showing that they can be non-existent, homogeneous (all providers converge on the same model), or heterogeneous (providers specialize in distinct data sources). Our analysis spans monopolistic, duopolistic, and more general markets, illustrating how factors such as the ``temperature'' of data-source choice models and the dominance of certain data sources shape equilibrium outcomes. We offer theoretical insights into both homogeneous and heterogeneous PNEs, guiding regulatory policies and practical strategies for competitive ML marketplaces.

Heterogeneous Data Game: Characterizing the Model Competition Across Multiple Data Sources
Renzhe Xu 12 Kang Wang? Bo Li
Abstract
demographics and discase prevalences. Such heterogeneous settings arise across diverse fields, including the digital economy and scientific research.
Data heterogeneity across multiple sources is com- mon in real-world machine learning (ML) set- tings. Although many methods focus enabling single model to handle diverse data, real-world markets often comprise multiple competing ML providers. In this paper, we propose a game- theoretic framework- Heterogeneous Data Game-to analyze how such providers compete across heterogeneous data sources. We investi- gate the resulting pure Nash equilibria (PNE), showing that they be non-existent, homoge- neous (all providers converge on the same model), or heterogeneous (providers specialize in distinct data sources). Our analysis spans monopolistic, duopolistic, and more general markets, illustrat- ing how factors such as the "temperature" of data- source choice models and the dominance of cer- tain data sources shape equilibrium outcomes. We offer theoretical insights into both homogeneous and heterogeneous PNEs, guiding regulatory poli- cies and practical strategies for competitive ML marketplaces.
Much of the existing literature on heterogeneous data fo- cuses on devising single ML method that performs ro- bustly across all data sources (Arjovsky al., 2019; Kuang et al., 2020; Liu al., 2021b; Duchi & Namkoong, 2021). However, real-world markets typically have multiple ML providers (Black et al., 2022: Jagadeesan et al., 2023a), each aiming to optimize its performance relative to others. For instance, competing diagnostic tool providers offer models to hospitals, which then choose a provider based on local performance criteria. This competitive interplay differs sig- nificantly from single-provider frameworks (Nisan al., 2007) and can lead to market dynamics unaddressed by previous approaches.
Several works (Ben-Porat & Tennenholtz, 2017; 2019; Feng et al., 2022; Jagadeesan et al., 2023a; lyer & Ke, 2024; Einav & Rosenfeld, 2025) have analyzed competition among mul- tiple ML model providers, examining Nash equilibria, social welfare, and agents' strategies under competition. However, these studies mainly focus on single data distribution and do not account for heterogeneity across multiple sources.
In this paper, we develop a game- -eeeettecoeetic framework to study multiple providers competing over heterogeneous data sources. We then analyze the resulting pure Nash equilibria to uncover how data heterogeneity and competitive forces shape providers' strategics.
1. Introduction
Data heterogeneity is commonplace in real-world machine learning (MI.) applications, where data often originate from multiple sources with distinct distributions (Li al., 2017: Hendrycks al., 2020; Gulrajani & Lopez-Paz, 2021; Liu et al., 2023). For example, in health care, patient data may be gathered from different hospitals, each serving varied
1.1. Overview of the Heterogeneous Data Game
We introduce the Heterogeneous Data Game model the competition among multiple ML model providers across diverse data sources. Consider K distinct data sources, each associated with weight wi representing its proportion, and joint distributions Px (z., y) over features x and labels In this market, cach of the ML model providers selects a model parameterized by 9,. Following previous works on model and platform competition (Jagadeesan et al., 2023a; Drezner & Hiselt, 2024), the utility of each model provider is determined by its market share across the different data sources. Specifically, each data source i selects an ML
'MoF Key Laboratory of Interdisciplinary Research of Computation and Economics, Shanghai University of Fi- nance and Economics, China. "Institute for Thcoretical Computer Science, Shanghai University of Finance and Eco- nomics. China. "School of Economics and Management, Ts- inghua University, China. Emails: xurenzhe @ sufe.edu.en. wangkang03330 (alp:300@@mmmm..mmmmmmmmmm com, tibo@sem.tsin www.seeeggmuuaeeeee edu.en. Corre- spondence to: Bo Li <libo@ sem wem.tsinghua..du.crm
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Heterogeneous Data Game
Heterogeneous Data Game under the Proximity Choice Model
Heterogeneous Data Game under the Probubility Choice Model
Monopoly (N = 1)
The single model chooses the parameter given by Eq. (9),
Duopoly
Equivalent condition for PNF existence (Thm. 5.1) PNE must be heterogeneous, if it exists (Thm. 5.1)
Equivalent condition for PNE existence (Thm. 5.2)
(N 2)
PNE must be homogeneous, if it exists (Thm. 5.2)
Equivalent condition for homogeneous PNE existence (Thm. 5.6)
Sufficient condition for PNE existence (Thm. 5.4 and Cor, 5.5) PNE must be heterogencous, if it exists (Prop. 5.3)
N
Sufficient condition for heterogeneous PNE existence (Thm. 5.7)
Example when both types of PNE cxist simultancously (Ex. 5.2)
Table 1. Overview of the results.
model based on the observed losses of available models. provider's utility then the sum of wy from all data sources that adopt its model. Consequently, each provider strategically chooses 8, to maximize its utility..
the weighted loss across all data sources, denoted by BM.
Duopoly (N = 2). Under the proximity choice model, we specify conditions for the existence of a PNE and show that, if PNE exists, both providers choose the ground-truth parameter of the data source with the maximal weight. In contrast, under the probability choice model, any PNE must be homogencous, with both providers choosing ΦM, the parameter that minimizes the weighted loss across sources.
Motivated by linear models, we represent each data source with two statistics: a ground-truth parameter 0g for Ps (x]lz) and a covariance matrix Σg for (z)) From a distribution- shift perspective, variations in 0% and Σg across sources correspond to concept shift and covariate shift, respectively- two common types of distribution shifts in practice (Liu et al., 2021b). Additionally, the loss of model l,2 on data source k is calculated as the squared Mahalanobis distance, (0,. 0x)) Σn (6,, 0z), ) corresponding to the mean squared error (MSE) in linear model settings.
More than two providers (N > 2). Under the proxim- ity choice model, if a PNE exists, providers tend to pick different models, leading to heterogeneous PNE. More- over, when a few data sources have significantly larger weights (Kairouz et al., 2021: Li et al., 2020), PNE exists if N lies within a certain range, and providers fully spe- cialize in those dominant sources. In contrast, under the probability choice model, both homogeneous and hetero- geneous PNE may arise, depending on the temperature t. Specifically, when is small, indicating that data sources are highly unlikely to choose sub-optimal models, only heterogeneous PNE may exist. Conversely, when t is large, meaning data sources are more likely to uniformly choose among all available models, only homogeneous PNE may exist. We also present an example where both types of PNE exist simultancously.
For data sources' choice models, we adopt two standard frameworks (Drezner & Eiselt, 2024): the proximity choice model (Hotelling, 1929; Plastria, 2001; Ahn et al., 2004), where each data source selects the provider with the low- est loss (with ties broken uniformly). and the probability choice model (Wilson, 1975; Hodgson, 1981; Bell et al., 1998), where data sources may choose sub-optimal models based on a logit framework (Train, 2009), controlled by a temperature parameter t.
1.2. Overview of the Results
An overview of these results is presented in Tab. 1. We investigate the pure Nash equilibria (PNE) of the Hetero- geneous Data Game and identify three patterns of PNEs across different game setups: (1) Non-existence of PNE, In this case, no PNE exists, leading to unstable ML model market. (2) Homogeneous PNE. Here, all model providers independently train their MI. models to minimize the wyy- weighted loss across all data sources. As a result, this type of PNE leads to the homogeneity of models available in the market. (3) Heterogeneous PNE. In this scenario, model providers offer different MIL models. Most specialize in single data source, typically adopting the ground-truth parameter Or of specific data source k.
Our theoretical findings yield several insights for multi- provider MI markets. First, they illuminate how the inter- play of data heterogeneity, choice models, and competition can produce either homogeneous or heterogeneous equilib- ria, thereby influencing the variety of models offered. Sec- ond, they indicate that when a few data sources dominate, providers tend to specialize in those sources, potentially overlooking smaller ones; this outcome calls for appropriate incentive mechanisms. Finally, market parameters- as the temperature in the probability choice model can be adjusted by market regulators to foster either heteroge- neous model offerings or convergence toward homogencous solutions. Taken together, these insights can inform both regulatory policy and practical strategies for building com- petitive MI. marketplaces.
Monopoly (N = 1). In this setting, a single provider can achieve the same utility with any ML model parameter. However, typically chooses the parameter that minimizes
2

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Heterogeneous Data Game
2. Related Works
tions; and (2) many models incorporate additional variables such as price or quantity, which reduce tractability and re- strict attention to small-scale settings. While few studies investigate high-dimensionas competition, they primarily address quantity competition (Anderson & Neven, 1990) or pricing (Bester, 1989), rather than spatial or parameter- based competition. By contrast, our setting considers source- specific distance metries arising from distributional shifts, along with high-dimensional strategy spaces driven by large number of data sources and model parameters. These distinctions introduce substantial challenges for theoretical analysis.
Data heterogeneity. In real-world scenarios, data often exhibit significant heterogeneity due variations in time, space, and population during the data collection process (Liu et al., 2023). The concept of data heterogeneity has been extensively studied across multiple disciplines, including ecology (Li & Reynolds, 1995), economies (Rosenbaum, 2005), and computer science (Wang et al., 2019). This work focuses on the implications of data heterogeneity in machine learning settings. In this context, considerable re- search has aimed to ensure that a single model performs robustly across diverse test environments (Liu et al., 2021b), leading to a range of effective methodological frameworks, including causal learning (Bühlmann, 2020; Peters et al., 2016), invariant learning (Arjovsky et al., 2019; Liu et al., 202 Koyama& & Yamaguchi, 2020), stable learning (Xu et al., 2022; Kuang et al., 2020; Yu et al., 2023), and distri- butionally robust optimization (Sinha et al., 2018; Duchi & Namkoong, 2021; Liu al., 2022). However, these existing approaches largely overlook the presence of multiple com- peting model providers and the strategic interactions that arise in such settings.
Other competitive frameworks. Finally, our work con- nects to competition scenarios in targeted advertising (lyer & Ke, 2024; lyer et al., 2024), online marketplaces (Liu et al., 2020; Hron et al., 2023; Jagadeesan et al., 2023b; Yao et al., 2024a:b), platform competition (Jullien & Sand Sand-Zantman,
2021; Calvano & Polo, 2021), and broader game- theoretc analyses (Immorlica al., 2011). Unlike these studies, we highlight how heterogeneous data distributions shape market equilibria among multiple ML model providers.
Competition in machine learning. Our work extends prior research on competition among machine learning model providers under homogeneous data settings (Ben- Porat & Tennenholtz, 2017; 2019; Feng et al., 2022: Ja- gadeesan et al., 2023a; Einav & Rosenfeld, 2025). Specifi- cally, Ben-Porat & Tennenholtz (2017: 2019) studied best- response dynamics and algorithmic methods for finding pure Nash equilibria (PNE) in regression tasks, while Einav & Rosenfeld (2025) extended these insights to classification. Feng et al. (2022) explored the bias-variance trade-off in competitive environments, showing that competing agents tend to favor variance-induced error over bias. Jagadeesan et al. (2023a) demonstrated that increasing model size does not necessarily improve social welfare. In contrast to these studies, we consider heterogeneous data sources with dis- tinct distributions, uncovering novel equilibrium structures and establishing new conditions for their existence.
3. Heterogeneous Data Game (HD-Game)
3.1. Notations
We begin by introducing several essential notations. For positive integer N, let [N] denote the set (1,,,...,NV The N N-dimensional simplex, denoted by AN. is defined as AN = ((x1,,2,... .N) Σ 1 Z; = 1 and Z 0, Vi E [N]]. For any square matrix A, we use A > 0 to indicate that A is positive definite, and A > 0 to indicate that is positive semi-definite. Furthermore, given a positive definite square matrix Σ O, the Ma- halanobis distance between two vectors x and y is defined as das (z, y: E) v(x Σ- (x v).
3.2. Game Setup
Heterogeneous data. Consider a setting with K 2 2 data sources. Each source l has true model parameter O2 E RDD and a covariance matrix Σx. These two terms capture concept shift (via 02) and covariate shift (via Ez), respectively (Liu et al., 2021b), as detailed in Sec. 3.3. We further assume 0g + 0y for all k + k', since any two sources with identical parameters can merged into one.
Competitive location models. Our framework is tech- nically related to competitive location models (Hotelling, 1929; Shaked, 1975; d' Aspremont et al., 1979; Eiselt et al., 1993; Plastria, 2001; Ahn et al., 2004), as comprehensively surveyed by Drezner& & Eiselt (2024). However, most exist- ing models focus on low-dimensional spaces or networks with uniform distance metrics, largely due to two factors: (1) applications in urban planning naturally align with one- dimensional (Hotelling, 1929; d' Aspremont et al., 1979), two-dimensional (Tsai& & Lai, 2005; Shaked, 1975; Led- erer& & Hurter Jr, 1986), or network- based (Eiselt & La- porte, 1991; 1993; Dorta-Gonzáles et al., 2005) formula-
For a model parameterized by 0 E RD, the loss associated with data source l is defined as the squared Mahalanobis distance between 0 and 0g with Σ i.e., d2, (0, 0ss "T²) As shown in Sec. 3.3, the Mahalanobis distance could corre- spond to the mean square error (MSE) of 8 on data source k in linear model settings, and it can measure the error caused by both concept shift and covariate shift.
3

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Heterogeneous Data Game
Additionally, each data source k is assigned weight wk. representing its proportion within the total data. Without loss of generality,
The Heterogeneous Data Game Given a strategy profile 0 (01, 02, ..., ",,, the utility of player n is
assume the weights are ordered and WK > O, with Ei=1 wy 1. Let
K
W1 > W2 > w = (Wi, 022 wx) denote the vector of weights.
Un (n) = Wegn
(4)
We note that for each n E [N], the term ln, is implicitly a function of On. as defined in Eq. (1). For any E IRP, we use (0, 0 O_n) to denote the strategy profile in which player n. deviates from their original strategy O,, to a new strategy E RD. We focus on the propertics of the Pure Nash Equilibrium (PNE), formally defined as follows.
Model providers. There are model providers (players)' that need to compete the models in these K data sources. Each player [AV] needs choose one model 0x e IR,, and the loss of player n for data source k, denoted as Cn is
Definition 3.1 (Pure Nash Equilibrium (PNE)), A strategy profile 0 = (01,..., ON) is pure Nash equilibrium if, for all [N] and R., un (0, O_.) (0).
Ln,k = ddu (O,, O2: ) = (0n. -6x) 0xx) Ex
0k). (1)
Data sources' choice model. The data sources will choose which model to deploy based on the losses k, Formally, let g RV + AN be the choice model. For data source k, given N losses fi .2. e, the func- tion (( (x,k) will output N-dimensional vector, and its r-th element, denoted as yn (fi,,..... &x,,), is the probability of choosing the n-th model. Following previous works (Jagadeesan et al., 2023a: Drezner & Eiselt, 2024), we consider two types of choice models for estimating the market share of different participants:
In practice model providers incur significant costs when re- training multiple models and typically deploy a single model rather than adopting a mixed strategy. Consequenty,, is more realistic to analyze the pure Nash equilibrium (PNE), where each provider commits to a specific model, rather than the mixed Nash equilibrium (MNE), which assumes randomized selection among multiple models.
Note that the utility function depends on whether the choice model (.(-)) in Eq. (4) is set as gh PROX or gn as defined in Eqs. (2) and (3). Consequently, different choice models yield different games and, therefore, different PNEs. For clarity, we refer to the heterogeneous data game under the proximity choice model as HD--Gam-Proximity and under the probability choice model as HD-Gaam-Prrobbbilit in the remainder of this paper.
Proximity choice model. Here, each data source chooses the model with the least loss. When several models exhibit the same loss, the data source will ran- domly choose one model with equal probabilities. For- mally,
PROX (41, CN..) In
3.3. Motivating Example - Linear Model
if jj E [N], lin < ba. otherwise.
Consider a linear model setting with K data sources, each associated with a distribution Ps(z, y) for k E [K]. where x E R denotes the feature vector and y E IR is the cor- responding label. Assume that is normalized such that Er [z]] = 0. The covariance matrix under PE is then given by Σg = P2 [xx] > O. Furthermore, assume that the con- ditional distribution P& xx) follows a linear model with parameter 3x. perturbed by Gaussian noise & ~ N of); that is, y x ~ ( x, o.).
(UE(N):
Probability choice model. Following (Jagadeesan et al., 2023a), we assume that data sources may nois- ily choose the models based on the following logit model (Train, 2009),
PROP PROP((1,,,,, (f1,,,..., CNNN) exp(-6m.k/1) In
(3)
Exprexp(-
Consider N players, each selecting parameter 8n. The MSE of player n on data source i is given by
with a temperature parameter t > 0. Intuitively, the pa- rameter /. controls the willingness for each data source to choose sub-optimal models. When - O, this model will become the proximity model as shown in Eq. (2). By contrast, when t + oo, all models become indifferent and the data source tends to choose models randomly.
Epc (3T
w)( (ST==v)) (30o-v)²
(B. -Be) (Bn-3a) En (Bn - -Bk) of
= dhir (Bnr Bki Ix oz
(5)
We use the terms "model provider" and "player" interchange- ably.
It is easy to verify that the noise term in Eq. (5) does not affect the choice models in Eqs. (2) and (3). Consequently,

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Heterogeneous Data Game
confirm that using the squared Mahalanobis distance as loss function aligns with the mean squared error (MSE), validating that the game effectively characterizes model provider competition in linear settings.
Assumption 4.2. For all i,j.k E [K] with + J.
das (0,, 2x ) + d (0j, O2: 2
Remark 4.3. This assumption ensures that two ground- truth models 0; and 0g from distinct data sources have iden- tical losses on a given data source k. Since different data sources typically have distinct ground-truth models, this condition is generally satisfied in practice.
Moreover, linear probing-where only the final linear layer is updated-is a widely used technique for adapting pre- trained models to downstream tasks, particularly when fine- tuning the full model is computationally expensive or prone to overfitting (Kumar et al., 2022). Since features can represent either raw inputs or embeddings from pretrained models, our framework also extends to scenarios where model providers employ the linear probing technique.
Problem-dependent constants. We introduce the follow- ing constant based on the game's parameters:
Lmax
max [E
(0:00x:
(8)
which represents the maximum possible loss for any strategy in 9. Intuitively, max quantifies the degree of heterogeneity among data sources. A small fmx indicates that any model incurs relatively low loss across all data sources, sug- gesting minimal variation among them. Conversely, large max implies greater difficulty in finding a single model that performs well across all sources, representing higher data heterogeneity.
4. Basic Properties and Assumptions
4.1. Basic Properties of HD-Game
We first characterize the possible strategy set in equilibria for each player.
Proposition 4.1. Denote the set 1 as follows:
I 2 (0(g) q = (g.,92....,94 qk) E Ak}.
(6)
5. Pure Nash Equilibria Analysis
where
In this section, we formally characterize the pure Nash equilibria (PNE) of our Heterogeneous Data Game under the different data-source choice models introduced earlier. As previewed in the introduction, we establish three possible outcomes, each governed by distinct sufficient conditions:
8(q) _ arg min
Σ
ahr (0, Ex
0
k=1
k
Ix
Σ k=1
qk En
1. Non-existence of PNE. In certain settings, no PNE arises, indicating that the model market remains fun- damentally unstable. In other words, providers contin- ually adjust their strategies in response to each other, preventing any long-term equilibrium.
Then, the following holds:
In
if a PNE exists, there must be one where every player's 's strategy belongs to O.
Homogeneous PNE. Here, a stable equilibrium exists in which all model providers converge on the same parameter (e.g., the one minimizing the we-weighted loss across sources). This outcome yields market dominated by essentially one "universal" model.
In 11--aa--rrbbbbllly any PNE necessarily re- quires all players' strategies to lie within 0.
Remark 4.1. Prop. 4.1 suggests that players will generally choose strategies from the set 0, which corresponds to mini- mizing a weighted loss over data sources, with each player determining their respective weights.
3. Heterogeneous PNE. Here, model providers differen- tiate themselves by specializing in distinct data sources. Typically, each provider adopts the ground-trun param- eter 0g of one source, resulting in a diverse array of models.
4.2. Assumptions
We introduce the following regularity assumption.
Assumption 4.1. For any 0 E RDD, there is at most one E AK such that 0(q)= = 0.
We observe that the concepts of "homogeneous PNE" and "heterogeneous PNE" are analogous to the ideas of "minimal" and "maximal" differentiation in location theory (Drezner & Eiselt, 2024). However, we maintain the terms "homo- geneous" and "heterogeneous"" because our distance metric differs across data sources.
Remark 4.2. When Σ₁ Σg reduces to requiring that 01, 0z.00.,,0 0x be affinely indepen- dent. For general settings, the number of parameters D is typically large, whereas the number of data sources K is rel- atively small, often satisfying K < D. Consequently, this assumption is generally reasonable real-worle settings.
Ex, this assumption
Below, we present the theoretical results for each outcome in the contexts of monopoly (Sec. 5.1,, duopoly (Sec. 5.2),
5

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Heterogeneous Data Game
and general multi-providee markets (Sec. 5.3).
PNE requires both players choose ÔM, established in the first part of this theorem. Notably, this inconsistency may not hold for N 2, which we demonstrate in Thm. 5.7.
5.1. Monopoly Setting
When N 1, the model choice is arbitrary, as there is no competition. However, the model provider typically selects strategy that minimizes the overall loss across all data sources. Consequently, the chosen strategy, denoted as 6M, is given by:
Deriving closed- form expression for the threshold is generally intractable. Empirically, we observe that & m Co (20mxx with 0 < Co < 1, where Co depends on the game's parameters. Experiments (see Sec. 6) consistently show that greater data-source heterogeneity- max in Eq. p0pphesss the threshold t upward. Hence, as heterogeneity grows, a homogeneous PNE can arise only when data sources exhibit an even stronger tendency to select sub-optimal models.
by
K wsdhu (0, 0x: 2x
jM A ((w)- = arg min
(9)
9
5.2. Duopoly Setting
5.3. General Cases with More than Two Model Providers
In this subsection, we consider the duopoly setting where there are only? 2 model providers in the market.
In this subsection, we analyze ML model markets with more than two model providers.
5.2.1. HD-GAME- PROMIIPROX
Theorem 5.1. Consider and suppose Assump. 4.1 holds. Then:
with N = 2,
5.3.1. HD-GAME- PRAXIIIPRX
Heterogeneity in PNE. We first show that in HD-Game- Proximity, any existing PNE tends to be heterogeneous.
1. wf < 0.5, a PNE does not exist.
2. If 2 0.5, a PNE exists, and 0 = (01,01) is a PNE. Moreover, if w/ > 0.5, (0,, 01) is the unique PNE.
and sup- pose Assump. 4.1 holds. Let Ô (01,.. ........ 0N} be PNE. For any E RD such that O & (0...... 0x }. let
Proposition 5.3. Consider HID-Game-Proximeg
Remark 5.1. Thm. 5.1 shows that in
m = Ki 03 = =0} Then, m S 1.
with N = 2, a PNE exists if and only if wh 2 0.5, indicating the presence of a dominant data source. Moreover, when W1 > 0.5, both model providers specialize in the dominant data source by selecting f1.
Remark 5.3. Prop. 5.3 shows that in HD-Game- Proomm-Po if a PNE exists, no two players will adopt the same model unless it corresponds to a ground- truth model of a data source. This suggests that model providers tend to offer distinct models, leading to heterogeneous PNE.
Although both providers adopt the same strategy, this PNE is still classified as heterogeneous, consistent with the general HD-Game-Proxinn Proximity results in Sec. 5.3. Notably, unlike the homogeneous PNE in HD-Game--robeen Probability, players in this PNE specialize in single dominant data source rather than optimizing across all sources.
Moreover, in some cases, achieving a PNE requires certain players to select strategies outside the set {0,, .2...., ......,, (k}. A detailed example provided in App. A.
Sufficient conditions for the existence of heterogeneous
5.2.2.
PNE. We derive sufficient conditions under which a het- erogeneous PNE exists and can be explicitly characterized.
Theorem 5.2. Consider HD-
with N = 2, and suppose Assump. 4.1 holds. If a PNE exists, then the only possible PNE is that both players choose 0M (desined in Eq. (9)).
Theorem 5.4. Consider HD-Game- CD-GGam--Poxiimity Proximity and suppose that Assumps. 4.1 and 4.2 hold. Assume there exists a con- stant ko E [K] such that Wko > 3 2K = Wz. Then PNE exists if N satisfies
Furthermore, there exists a constant < 24 depending on all game parameters, such that @M is PNE if and only ift 2
ko 3w'
Rg
-1) (10)
Remark 5.2. Compared to Thm. 5.1, in HD-Game- Probability, PNE may fail to exist even if w; 2 0.5 when I is smaller than the threshold L. This may seem counterin- tuitive, as one might expect the probability choice model to converge to the proximity choice model as + 0. However, the properties of PNEs are not consistent in this limit. This inconsistency arises because, for N- 2, the only possible
k=1
wk
k=1
k0+1 Wi
where for any lo such that 1 < k < kn
K V/ = wk+ Σ
l
k
arg min dm (0,, O2: E,5) 1ki'
j j=ka
6

---

AK such that Heterogeneous Data Game
Moreover, for any PNE 0- (Ō1,..., ON). it must hold .2.002.................. ........... :ko). In addition, let vi = Ki E [M] 0g = 0z be the number of players choosing strategy 0% in the PNE. Then,
ers choosing 02 and four choosing 0z. approximately pro- portional to (ui,, w(), where w/ = W1 + wa 0.63 and w22 w2 + w44 0.37.
that Vn E [N,, On E {01,02,...
In addition, Thm. 5.4 implies the following corollary.
Corollary 5.5. Consider HD- Came- and sup- pose that Assumps. 4.1 and 4.2 hold. When N Z ER= /w//WKk a PNE exists. Moreover, for any PNE 0 (01,...,6 NN). it holds that Yn E [N],6,
Vk E [ko],
m.
S1
(12)
where: 2" =sup 0: h(2) -
Ek=1 ko
2 N
{0,, 02....,00r 0x). In addition, let my = Ki E [M] 03 = 0x be the number of players that choose strategy 0g in the PNE. We have Vk: E [K], |m |wx/2"|| I1 s 1 where
Remark 5.4. Thm. 5.4 suggests that when a few data sources carry dominant weights- a phenomenon commonly ob- served in practice (Kairouz et al., 2021; Li et al., 2020) and the number of model providers N lies within a certain range, PNE exists in which all providers specialize in serving single data source. Moreover, in such equilibria, the num- ber of providers allocated to each source is approximately proportional to its weight.
=* = sup { >>>00 0: > k= [w]/=] 2 [ }
Remark 5.5. This result follows directly from Thm. 5,4 with kn set K. It implies that PNE always exists when N is sufficiently large.
Concretely, the choice of ko indicates that the top-ko data sources hold significantly higher weights, each at least three times the total weight of the remaining data sources. The constraints on N in Eq. (10) ensure that (1) providers con- sider the ko-th data source and (2) data sources with small weights are overlooked. Under these conditions, the ex- act form of the PNE can be derived. In a PNE, all model providers select a ground-truth model from the top-ko data sources. Consequently, the utility of non-dominant data sources is assigned to the nearest dominant data source, as characterized by Eq. (11). Furthermore, since 2" is fixed in Eq. (12), mg is proportional to w'. This aligns with intu- ition, as data sources with higher weights typically attract more model providers optimizing for them.
5.3.2. HD-GAME- HDDGAMM--RRBBBBLLTT
Unlike HD-Game- Groximtr, we show that both homoge- neous and heterogeneous PNEs can exist in HD-Game- Probability.
Homogeneous PNE. We first derive equivalent conditions for the existence of a homogeneous PNE, as well as suffi- cient condition for its uniqueness.
Theorem 5.6. Consider Proehbbbllu and sup- pose that Assump. 4.1 holds. Let 0 Homoo (ÔM QM ..... (M), where @M defined in Eq. (9). Then there exist two con- stants: L S 20 20 depending on all game parameters, and C > 0. depending only on {Ex, 0k, wx) Wk the following results hold:
This insight implies that policymakers can mitigate imbal- anced attention among different data sources by either intro- ducing more providers or incentivizing focus on underrepre- sented sources. Thm. 5.4 provides quantitative foundation for both interventions.
1. 0 Homo is a PNE if and only if 2 L.
2. 18t Z max/6C/N, 20max,, then 0 Homo PNE.
is the unique
Remark 5.6. Thm. 5.6 implies that a homogeneous PNE does not exist when t is small. As t increases, homo- geneous PNE may emerge, and for sufficiently large t, it becomes the unique PNE. The closed-form expressions of L and C are difficult derive. However, similar to Thm. 5.2, synthetic experiments in Sec. 6 suggest that the threshold t required for PNE existence is approximately Co x m26 where 0 < Co < is a game-specific constant. Moreover, as N increases, a homogencous PNE is more likely to be unique, as indicated by the second part of Thm. 5.6. This trend is further verified by our synthetic experiments in Sec. 6.
We further provide an example to explain Thm. 5.4.
Example 5.1. Consider a setting with K- data sources and weights (0.6, 0.35, 0.03, 0.02). The grouhd-truth models are given by 0, = (1, 0, 0). 02 (-1,0,0) 0g- = (1,0.1,)) and 02 = 0,0.1, with covariance matrices Σ1 = Σn = Σ3- = Σ₁ = I. In this case. the first two data sources have dominant weights.
Setting Ko- = 2, a heterogeneous PNE is guaranteed to exist when N lies within specific range (8, 19] in this case). In this PNE, model providers will only select strategies from {Oi, 02). Since data source 3 has similar ground-truth model to data source 1, and data source 4 to data source 2, providers selecting 0, benefit from data source 3, while those selecting fg benefit from data source 4.
Heterogeneous PNE. We next demonstrate that for suffi- ciently small i, heterogeneous PNE can exist.
OLDD-Gam--Proabbee Probability and sup- Theorem 5.7. Consider HD-Game-
For instance, when N 10, the PNE consists of six play-
pose that Assumps. 4.1 and 4.2 hold, with N z
7

---

Heterogeneous Data Game
E [3w// /w Additionally, assume that for all n, n' E [N] and distinct k. kf E [K) it holds that wk wx/n=² + we /nn.... Let Prox (0pms....,0
Homogencous PNE
Hetcrogcncous PNE
0.1250
0.125
......N'' ( be PNE in the correspond-
Uniiity
0.1225
0.120
ing Pa--Git--ro Then, there exists constant 0 such that for any t I i, a PNE Hele = (ōfter,... ( exists in HID--ame-PPrl Probability and satisfies
0.1200
0.115
Type Maddl Previdar
X
Type Madd Proviter Palky
0.1175
0.110-
Type Madd Providrr
x
x
Type Modd Provider Palky
0.0
0.5 Deviated Policy
1,0
0.0
0.5 Deviated Policy
1.0
Vn E [N],
| -0rm0 S S
Remark 5.7. For simplicity, we build on the conditions of Cor. 5.5. Thm. 5.7 establishes that when t is sufficiently small, heterogeneous PNE exists and closely approximates Game--roximing This is
Figure I. Utility of a single model provider with deviated policy for hoth homogeneous and heterogemeous PNE in the configuration of Ex. 5.2.
the PNE in the corresponding expected, as DeD-Gamm-Probbbiliig approaches HD-Game- Proximity for large .N as + O. However, proving this result is significantly more challenging due to the continuous nature of the potential strategy set for each model provider.
choice model (Eq. (3)) influences the existence of homoge- neous and heterogeneous PNEs in De---amm-PProbbbii
-attaggnn processes. Because our theoretical re- sults do not depend on the number of data sources K, we set K for simplicity. We also choose D 2 to fulfill Assump. 4.1. We randomly generate 10 game configurations with different {Me, 3k, Wk [k] Each covariance matrix is constructed so that its largest eigenvalue does not exceed 1. In addition, we set w2 2 0.1 to avoid a scenario where the first data source fully dominates the market. The num- ber of model providers N is varied from {2, 3, 4, ... to explore the effect of N on the critical values tf
Additionally, while the specified range for N is sufficient, it is not necessary. Our experiments in Sec. 6 suggest that a heterogeneous PNE may also exist for smaller values of N.
Thms. 5.6 and 5.7 show that, to promote model diversity (i.e., the emergence of heterogeneous PNE) and prevent convergence a single dominant model (i.e., homoge- neous PNE), policymakers can increase the rationality of data thhrc is, enhance their ability to select higher- performing which, in turn, encourages the forma- tion of heterogeneous PNE.
Calculating critical temperature parameters. Follow- ing Prop. 4.1, each model provider n's strategy 0, must take the form ((g,) with qn A2 and q, (a,, 1 1 for 0 I On S 1. To verify whether candidate strat- egy profile 0 is a PNE, we enumerate all possible de- viations: for each provider n, we check every a, e {0, 0.002, 0.004,.... 0.998, 1} to see if a profitable devi- ation exists. Using this enumeration, we identify:
Existence of both PNE at the same time. We now present an example to illustrate that both types of PNE can coexist in single game.
Example 5.2. Let N = 8 and K = 2 with 0 = (1, and 02- = (0, 1). The covariance matrices are Σ₁ = Ee = I, and the weights are w = (0.53, 0.47). The temperature param- eter is set to t = 0.4. Since = 2, Prop. 4.1 implies that each model provider selects a weight ax E [0, 1]. yielding
Homogeneous PNE, We seek the threshold given in Thms. 5.2 and 5.6. Specifically, we search over 2 (0.001,0.002, .... 0.999, 1} x (26 to find the
the final model 0,- = 0, + (1 - a2 = (an, 1).
In this setting, we identify two types of PNE: (1) a homoge- neous PNE, where all model providers choose a,, 0.53, and (2) a heterogeneous PNE, where four providers choose On RJ 0.76 (type 1) and the other four choose on, x 0.30 (type 2). In the heterogeneous PNE, model providers spe- cialize in different data sources, forming two distinct groups
minimal t for which the strategy profile 0 Homo (from
Thm. 5.6) is indeed a PNE
Heterogeneous PNE. We seek the maximal / for which a heterogeneous PNE exists. Since determining the ex- act maximum can be complex in game theory (Gottlob et al., 2003; Fabrikant et al., 2004), we adopt an em- pirical procedure inspired by the proof of Thm. 5.7. For each candidate t, we use Alg. 2 (discussed in App. B) to obtain a heterogeneous PNE candidate, then apply the same enumeration technique to ver- ify whether it is a PNE. We thus find the largest in {0.001, 0.002,... 0.999, 1} x (26max) for which our approach can produce a heterogeneous PNE. Although this does not guarantee the true maximum, it provides a useful lower bound.
As shown in Fig. 1, we plot each player's utility they deviate to a different policy. From the figure, it evident that no player benefits from deviating, thereby verifying the correctness of the PNEs.
6. Synthetic Experiments
In this section, we conduct synthetic experiments to inves- tigate how the temperature parameter in the probability
8

---

Heterogencous Data Game
0.2
0.15
0.2
0.10
WM
0.10-
0.10
0.1
W
0.05
0.05
0.05
r 2/2mmx
0.00
0.00
0.0
0.00
20
10
20
30
20
10
0.2
0.2
0.15
0.10
M
0.10
0.1
0.1
0.1
0.05
0.05
0.0
0.0
0.00
10
20
30
10
20
30
10
20
30
10
20
30
10
20
30
Number Model Providers
Homogeneous PNE: Threshold
1
Heterogeneous PNE: Maximal
The Lower Bound of N in Theorem 5.7
Figure 2. In the probability choice model, this figure reports, across 10 randomly generated games, the threshold / that guarantees the existence homogeneous PNE and the approximate largest value of that guarantees the existence of heterogeneous PNE, N varies.
Results and analysis. Fig. 2 presents our experimental results. We make the following observations:
of both PNE types becomes increasingly unlikely for large N.
1. Homogeneous PNE. The threshold temperature given Thms. 5.2 and 5.6 generally increases with N, but the growth curve flattens for larger N. This is consistent with Thms. 5.2 and 5.6, which guaran- tee that & S max and thus ensure the existence of homogeneous PNE once t 2 2fmax independent of N. Moreover, in our setups, the minimal is often significantly less than ma (roughly 0.1 x (26max) to 0.2 (26max)).
7. Conclusions
We propose the Heterogeneous Data Game to analyze com- petition among ML models in heterogeneous data markets. By studying PNE under proximity and probability choice models, we derive conditions for the existence of different PNE types, showing key factors that shape competitive ML marketplaces.
2. Heterogeneous PNE. Our empirical approach effec- tively finds hetcrogencous PNEs once N exceeds the lower bound given in Thm. 5.7. Nonetheless, because the conditions in Thm. 5.7 are sufficient but not neces- sary, we observe that a heterogeneous PNE can exist even when is smaller than that bound. The curve for the heterogeneous PNE is not smooth and exhibits peri- odic fluctuations. This is due the fact that the hetero- geneous PNE in PNE in the corresponding itself has non smothh dependence on N (Thm. 5.4 and Cor. 5.5).
A Acknowledgements
Renzhe Xu's research was supported by the National Key R&D Program of China (No. 2023YFA 1095000) the Na- tional Natural Science Foundation of China (No. 72442024), and the Shanghai Sailing Program (No. 24YF27 11600). Bo Li's research was supported by the National Natural Science Foundation of China (Nos. 72171 131 and 72133002).
depends the which
Impact Statement
This work presents a game-theoretic framework to study competition among ML providers across heterogencous data sources. By analyzing market equilibrium, it offers insights for designing policies that promote fair and diverse model offerings. Without such safeguards, smaller or less profitable data sources may be neglected, exacerbating in- equalities. We aim for our findings to guide policymakers and stakeholders in shaping responsible and equitable MI. marketplaces.
3. Coexistence of homogeneous and heterogeneous PNE. In some games, the heterogeneous PNE curve ap- pears above the homogeneous PNE curve, suggesting that both types may coexist. However, in other cases (e.g., the second row and second column of Fig. 2), no such coexistence is observed. In addition. as N increases, the maximal required for a heterogeneous PNE tends to be lower than the threshold t required for homogeneous PNE, indicating that the coexistence

---

Heterogencous Data Game
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gence, volume 37, pp.
12

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Heterogeneous Data Game
A. Omitted Examples
with 0, (0,0,1), 02 - 12, and 03 (0, 1)). The covariance matrices are Σ Σg Σg I, and the weights are given by w (w), W2, W3) (0.6, 0.25, 0.15). graphical explanation is provided in Fig. 3, where 1, 02. and 03 correspond to the vertices A, and of the triangle, respectively. In this case, the
Example A.1. Let
A (0.0, w2 0.6)
B(0 (2,0, 1), mg 0.25)
C(Os 10,1,1).s 0.15)
Figure 3. The graphical explanation of Ex. A.1.
Mahalanobis distance reduces to the standard Euclidean distance. It is straightforward verify that, at PNE, two players will choose strategy 01, while the remaining player will adopt strategy along the segment DE within the triangle (D and satisfy that BA = BE and CA CD), excluding the vertices D and E.
B. An Approach to Find a Potential Heterogeneous PNE in HD-Game-Probabillty
B.1. Approach Design
We first define the following mapping M from A A x... x AK AN. For (q), ........... .N.) E A
N
M(q 92.......................... .......9 9x) is calculated through Alg. 1.
Algorithm 1 The M mapping from A AN
1: Input: q1 ......, AN E Ax
2: 0n t ((qn) for all n E [N]
3: ln,k (On Oz; 21 1)
(0n - 0x) T (Ō, 0₂) for all [N] and k e [K]
4: Pn.k + exp(pp (onl-n/x/)/CCC0 Ei=1 (E exp(-1s,t (/,,x//)) for all [N] and k E [K]
5: an.k + WkPrik P ) (E WyPn, (1- pm,)) for all n E [N] and k E [K]
6: On + -ia,,,.... "o......,Kkkkk Kn.k) for all n E [N]
7: Output: (Q1, -2. .p.....,NN))
We also need the following definition.
Definition B.1 (k,). Given PNE 0 Hete
in HID-Game-Proxiiity define k, follows.
kn - (he index ie such that 0g 6
Based on the mapping M and constant k,, the pseudocode of our approach provided Alg. 2. The approach consists of several steps. First, compute the PNE 0 Prox of the corresponding HID-Game-Proximie using Thm. 5.4 and Cor. 5.5. Then, starting from this strategy profile, find fixed point of the mapping M defined in Alg. 1. Once fixed point is identified, we output the corresponding strategy profile 0 = (01....,0s)) where each 0,, 0(g,))
B.2. The Idea of the Approach
The rationality of the approach is based on the following results.
Lemma B.1. Let (2) (2) E Ag. Suppose (2) - 120 S e. Then, there exists constant C > 0, depending only K (E,, Og. wxlk k=1 such that
|( - s .c
13

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Heterogencous Data Game
Algorithm 2 An Approach Find Potential Heterogeneous PNE in HD-Game--robability
1: Input: Game parameters {En, 0k: WK]ke|k] and N
2: Calculate the PNE Prox based Thm. 5.4 and Cor. 5.5 in HD-Game-Proximity
3: Calculate ht for all [M] based on Def. B.1
4: In < (0,0.... 1,0, .,),....0)
the k. dh element is 5: while Not convergent do
6:
(q). .2. .N...... +- M(q qNx) given in Alg.
7: end while
HHor (())) for n E [N]
Hore (fflee, ..... (
9: 0
0 Hete
Output:
Let O be the constant that only depends on {Ex, 82, wk | k= in Lem. B.1. For any t >> O and B 2 1, define the space
(1.00)
(t,9)
(1.99)
(t.99
(13)
where Vn [N]
((4,3) A
q E Ax:
90,0, 1, 0, 0)
S 1
the k., th element is
20
Based on the above definition, we could provide two important properties of the mapping M.
Lemma B.2. For any (q1...., 9Nx) E A ler Ô = (01, . êN) where 0, 8(q,), Vn E [N]. If (g....., is fixed point of the mapping M, then for all E [N].
oun (0, θ_.))
09
1==6n
Lemma B.3. When B > 1, there exists constant t, depending only on {Ex, Os walk any (q1, 12.....90.................. 4N) E Q(..3) defined in Eq. (13), then M(qi M(qi-92...... .2...., E (
k= and B, such that when t S t, for
Based on the Brouwer fixed-point theorem, Lem. B.3 guarantees the existence a fixed point (q).... 4N) E (t..0) for the mapping M. Consequently, by Lem. B.2, the corresponding strategy profile 0 0 (fftete Na ). where each
0, Hete " = ((g,)) satisfies the zero partial gradient condition at @elete for un 0 Hete, for all n E [N], which a necessary
condition for PNE. Therefore, 0 Hete
serves as a candidate for a heterogeneous PNE in HD-Game-Probabiliyy
The complete proof is available in the full version of the paper at https: http://axxvvvvgg/aaa/// httpapps///aeaa...oo///////22/.0
C. Omitted Proofs
Additional notations. For any two vectors (21...., ZM) RM and dominated by y if for all i E [M], y: I z; and there exists a E [M], Uj < Zz.
R, say æ is
C.1. Proof of Prop. 4.1
We first prove Eq. (7).
Proof of Eq. (7). According to the definition of the Mahalanobis distance, have
K
0(q) = arg min
d (0, 0z: ET') = arg min
qe 0x) Tx) 0%)
h
14

---

Heterogencous Data Game
Denote the part in arg min in the right-hand side of the above equation L(0). And since o AK and Ex > 0, have
K
k (2) = Σ D > O.
( = 2 q -x)),
k=1
As result, L(0) is strictly convex and has the unique minimizer ((q). In addition, 0(q) must satisfy that VL(O(g)) = which means
K
Σ
(0(g)
02) 0.
k=
As result,
K 9h
0(q)=
Σ 1=
9h
Ds
EsOk
k=1
Now the claim follows.
For the proof of Prop. 4.1, we first introduce the following lemma.
Lemma C.1. Let be defined in Eq. (6). Then for any 0 4 U, there exists a 0 such that for all & [K]. (0" 0z: 2i (0, 0g; 1.
Proof of Lem. C.I. Define the following function f RDD BB as
f(0) = (100000;1;100
02: 02
da (0,00 1)
For any k [K], define (0) Lig (0, 0x: -3 ). Note that for any l E [K]. since Σg > O, we have dz (0, 0₁: Ex
(0- 0x) T '5 0x) is convex w.r.t. 0. As result, according Boyd (2004), for every Pareto optimal point of f, there
is some q E Aκ such that
8 arg min
Σ yh dh (f', 0z: Ex 0(q).
Hence, the set defined Eq. (6) contains all Pareto optimal points. Furthermore, for any points there must exist O' such that f(0') dominates f(0) (Ehrgott, 2005), which proves the claim.
Now could prove Prop. 4.1.
proof of Prop. 4.1. (1) In the proximity model, let Ô - (01, 0z,, Ōx) be a PNE. If Os 9, Vk [K], then the claim is already satisfied. Now suppose there exists an index n such that 6,, & 0. According to Lem, C.I, there exists a policy 0' E such that Vk [K], ",,x di (0', Os; )= d' (Ōnr Ozs "x²1) kn Consider the new strategy profile 0 (0', 0 O)..
We now show that 0 is also a PNE.
PROX is decreasing on the n-th element and Ô is PNE, we have that Since PROX
K
K
Un (0)
when
PROX
(LI, (N,K) S
Σ
PROX ween
(L1,6.... 1n-1,k Cn Cn latie...... (MNR) = un(0) S (0).
As a result, (,) u, (0) and player n will not benefit by deviation. Furthermore, Vk E [K]. we have
gf PRO C..,,,NA (N g PROX
PROX(L1, (fI,k. ln 1,6, Cl Cn enti.e.......................... (N,.). (Otherwise, player n would benefit by devia-
tion.) Since Lr < luck, we have
PROX (l.,,,..,, In
PROX (li,k..... en En lat1.k........ CN.K) = In
15

---

Heterogeneous Data Game
and hence ln *, > minGEIN a. Ak E [ As result, this deviation will not affect any other players' utility. Specifically, consider any other player j E (N]](n) We have
PROX (l 1.k k
En+1 (N.N) PROX ( (6,,,,...,,,,,,,)
(14)
Furthermore, since k < kn we have that for any 0" RD and fGNn d3r (0", Oki 2) have Vj [N].
gj (eI,....., (. kj in- AA bnt (N
(15)
PROX (l1,6
As result,
uy ((8" ((8"0--)) 0-)))
k
Wk 9j
PROX PROx(E1
Ej 111k............................
latiN...... (N..)
K
9j
PROX
f (NCK)
(by Eq. (15))
k=1
Σ wk Jj
PROX
(la,k...., &j. 1 (N,K)
(Ø is a PNE and hence u,(0) 2 u, (8", 0 0-;))
K
Wk
Dj
PROX (4 .n. En lntiik..... (N..)
(by Eq. (14))
1=1
uy (0).
Therefore, any player will not benefit by deviation from 0 and 0 is PNE.
To prove the original claim, we can keep this procedure. After at most N steps, we can a new PNE 0 = (67,.... .8)) from and o E 0, Ak e [K]. Now the claim follows.
(2) In the probability model, suppose there exists a PNE 0 = (02,02,.... On) and an index n [N] such that 0, $ Then according to Lem. C. there exists 0' E 0 such that Ak: [K],dx((' , O2; Ex < (On, O2; ) Since SPOPP is strictly decreasing on the n-th element, we could conclude that player will benefit if deviating to the policy 0', which leads to contradiction. Now the claim follows.
C.2. Proof of Thm. 5.1
We first introduce the following lemma.
Lemma C.2. Suppose that Assump. 4.1 holds. Let q E AK and = 0(q). Then for any ko E [K] such that Iko > 0, there exists a strategy such that
Vk E [K]] dxr(0, Ozi Ex < dm 0z; bx
Proof. For any [K] define e2 = Ex 0x). Furthermore, define the following matrix
A = (11
Ukg-1
Uko +1
...U
Then for any y K- such that y 2 and y + O, we must have Ay + O. Otherwise, we can construct new q' such that
yk/ EEk= ver)
if < ko-
d =
if k = ko:
yh 1k-1/
if k: > kp-
16

---

Heterogeneous Data Game
As result,
K En 0k) =
K Σ k=1
k-1 dh Ay/
Σ
yk = 0
and 0 = 0(q') = (q)). which violates Assump. 4.1.
According to Gordan's theorem (Lem. D.I. (Mangasarian, 1994)), there must exist a vector x E RD such that (-A) > 0. which means for all E [K]](Ko), [ko], we have uy x < O. Now construct the following 0g + L a with any 2 0. We can get that, for any [K)]
dd 10, 2x
d + x -8h)
Σx (0 + x 10--8p) 0x)
= 2x T Ex + 1.I -0x)
dt
dt
t=
t=
2x Ex - 0x) 2x T Uk 0.
As result, there must exist a la 0 such that for any 0 < t < tk. we have (o 02; xx (0, (0:000;:000 xx Now choose t' = min(ti, f2.............................. 1 t1g xx// and let 0' = 0 + x. Then for all & E [K]](ko) we must have dx((0' 0z: E3 < dM (0, 0x: Σx
Now if 0' E 0, the claim has already follows. When 0' g 0, according to the proof of Lem. C.I (see App. C.I). 1 contains all Pareto optimal points, As a result, there must exist a strategy E & such that for all & [K]] dvr(0: O2: 2, < d 0g: " < dM (0, 0z: Σ Now the claim follows.
Then we could prove Thm. 5.1.
Proof of Thm. 5.1. (1) We first consider the case where W1 < 0.5. Suppose a PNE 0 -(1,00) 02) exists. According to Prop. 4.1, we can assume that 0,.02 E d. Since the sum of the utilities of two players is 1, there must exist one player that has utility not greater than 0.5. Without loss of generality, we assume that u1 (0) S 0.5 and u2 2 0.5. Now we construct a new policy 0' for player 1 such that un (0', 0²)) > 0.5.
According to Assump. 4.1, there exists a unique q E AK such that 02 = ((q). Since E Ak. there must exist one element kn such that Qha > 0. According to Lem. C.2, there must exist a 0' E 0 such that
Vk [K]] d0 Oz; Ex dm (0, 0w; ).
Now by the proximity model as shown in Eq. (2), we have that
#i (0',0s 02)
wk Wko
2 1- >1 0.5 0.5 2 u1 (0).
As result, O is not a PNE, which leads contradiction.
(2) We then consider the case when wi 2 0.5.
We first show that 0- = (0),0)) is a PNE. In this strategy profile, Since two players choose the same strategy 0,, we have 41 (0) uz 0) = 0.5. In addition, when any player deviate from the strategy 01, he could have higher loss on data source 1 and hence could have utility at most Σk=z Wk
w1 I 0.5. As result, θ (01, 01) is a PNE.
Furthermore, consider the case when wi > 0.5. Suppose there exists another PNE 0' (Ø1, 02) + Ô. We consider the following two cases.
1. Suppose ò,, 02 + 0,. Since un (0') uz((') 1, there exist one player such that his utility is not greater than 0.5. Without loss of generality, we assume 41 (Ø) < 0.5. Then if player 1 choose strategy 01. he will get utility at least
Wi > 0.5 2 in (0') which leads to a contradiction.
2. Suppose one player choose 01 and the other player does not. Without loss of generality, we assume 0; = 02 and
8, # 0,. In this case, up (0') < Σk k= Wi = 1 - w1 < 0.5. However, player 2 choose strategy 0,, he will have the same strategy with player and get utility 0.5 > ug 0') ). which leads to contraction.
17

---

Heterogencous Data Game
To conclude, 0 (01, 01) is the unique PNE when w1 0.5.
Now the claim follows.
C.3. Proof of Thm. 5.2
Proof. (1) We first show that, if PNE exists, the only possible PNE is that both players choose 0(wr).
Suppose that 0 (fI, 62) is PNE. According to the definition of PNE, we must have
K
01 E arg max u1 (0, 02) = arg max
wk PI,a(O)
9
where
exp (- (00- - 0x) TR)TDD EE (0 0kk
PI,A p1,x (0)
exp (- (0 -0,)² Σn (0 - -0s) 0s) /1) + exp -0x) T
2x
(02 - 0k) /8
Note that
00
(- - 0x) E1 (0 - 0x) /t) exp
(-(82- -02) 0z)
Ey (02 0s)
exp
(-7 E. (0 0k)
exp (- (0 0k) Ex (0 0x) /t) + exp
00 0x
E3 ( 0z. /t)
(1)( - P1.A (z - 0x)-
As result,
( 02)
2
(KPP),((0)(1 (1)EE On).
Wk
00
00
k=1
Hence,
Ou, (0, 02)
2 K
0n) O.
(16)
00
1
0=
k=1
Similarly, we can get that
Suz (,,, 0)
2
K
WkP (02) - -2,, 0x) = O.
00
t
0=
k=1
where
exp
(- (0 - -kk) (x)TTiC
De (0 0xx) 0t)
P2,1
T
exp
(0 - 0x)
Ex (0, - 0z) /1) + exp (-(0-0s) --0-0s)
En (0 - 0z) 0z) /1)
Note that PI,x (01) + (f2) = 1. As result,
@uz(0,00 0z)
K
W.PL. - - - 0x) = - ukPi. (.p.:((0)pp.. - 0s) = 0.
00
p=ú;
km
Ouz(0,, 0)
Σ
K
WPP2,x(02)( (Ox)) 0x) 2
Σ
K
wapp 0z) 0.
00
0=60
k=1
k=1
Hence,
K Σ
W&PL, (8))) 0x) = - Σ
uPPLA (8)).. 0x)
18

---

Heterogeneous Data Game
and therefore
wapp:.(000p2..00005E..... 02) =
Define matrix A = Σx= WAPI& (0))),,x(02222) and we have A(Oi 02) 0. Note that for all h: E [K], wki (k.P1, (0,) P2,x (02) 0 and Ex 0. As a result, 0 and we must have 0₂ 02. Note that when 0₃ 02.
pl,x(01) = (22) = 1/2. Now Eq. (16) becomes
2
1 E2 0x) O.
Wk
As result, 0, ((w) 8g. Hence, if PNE exists, the only possible PNE is that both players choose 0(w). The claim then follows from the proof of the general result given by Thm. 5.6 (see Lems. C.3 and C.4 in App. C.6 for details).
C.4. Proof of Prop. 5.3
Proof. Suppose there exists a PNE = 020,0,,...,0 such that there are two players choose the same strategy and the 0 strategy is outside the set (01, 02, ... 0x). Without loss of generality, we assume that 0₂ 02 (01,...,0p
Define K as the set of data sources that player I and 2 could get positive utility, i.e.,
K a {k: Vj e [NV, dm (01, Ous 2x ) S dm (0,, On: 1)
Note that IC cannot be an empty set, as player could otherwise deviate to 01 and achieve a positive and higher utility. For any A KC, let
kn = Li : dm (0,, Ozs E = ddr(0, O2s "}
be the number of players that achieve the minimal loss in data source k: in the PNE Ô Note that kin 2 2 since 0, = 02. Then we can get that
u1 (0) uz (0)
Wk
kn
KEK
We consider two cases about I.
1. Consider the case when |K] and I. = {ku}. player 1 deviates to policy 0ko he will become the only player that has the smallest loss data source i and hence have utility at least Wkk Wko/ /nkg which leads to contradiction.
2. Consider the case when |AC] > 2. We further consider two cases about 0,.
(a) Consider the case when 01 & O. Then according the proof of Lem. C.1 (see App. C.1), there must exist 0 such that for all & E [K], ds (8, 8n; Bx;Ex 1) < d (02, 0x; 1 As result, if player 1 deviates to the policy 9, he will get utility least ΣkEi Wk > ΣtEx wk, = in (0), which leads to contradiction.
(b) Consider the case when 0, 9. Let ki be the smallest element in K. According to Assump. 4.1, let q E Ar be the unique vector such that 0, 0(q). Since i + Oko by assumption, there must exist kj E [K]]((k)) such that qk, 0. Now according to Lem. C.2, there exists a strategy 0 E 9 such that for all & [K]](Aa] we have
d (0, O4: 2 ) < (02, O2: 2 ). Let hi be the second smallest element K. As a result,
un (0, 1) kE
Wk 2
Wk
Wko
Wkk
Wk 2 un (0).
2
2
kn
(E
ke
Therefore, player 1 will have a higher utility if deviating to the policy 8, which leads to a contradiction. To conclude, all cases lead to contradiction. As a result, the claim follows.
19

---

Heterogeneous Data Game
C.S. Proof of Thm. 5.4
Proof. (I) We first show the existence of PNE under the condition in Eq. (10). Note that 2" exists since h(z)) when & 0 and h(2) = 0 when 1. Moreover, since h(2) is right continuous, it must hold that h(z*) 2 N. Under the condition in Eq. (10), we have
w/o
Avo
300/
S N.
3
wko
In addition, for any c O, we have
Whe h
Ko
3wy k. ka-
Ko
3rlk
3w
+
c
+2
+
$ N.
k
2 + 3E
k=1
ko
+ 3e
W ko
wkk
Hence, it must hold that =² S /3. /.. Then define
w
Vk E [ko],
m4 =
As result, we have that m 2 m/oo = 110/o /2" enough, we have that
2 3 for all & E [No] In addition, due to Eq. (10), by making >> 0 small
h
Σ j=
K
Wi
+
ho =
W/zs
ko
w/
] -1) 2 N.
1=
Σ
E
+
k=1
Ei==o+1 w/)
We have that 2* >
Ex =1= Wj-
We construct the PNE based on two cases.
1. Consider the scenario when h(2²)= = Ek= m4 N. Then let ma = m/ for all k E [Ao]]
2. Consider the scenario when h(2²) Eko m4y > N. Note that by the choice of 2", h(2 + c) < N for all c > 0. Define the set IC {k [ko] w2//22 m As result, when e E 0, h(2- +) = h(2²) - |K] N. Hence,
must hold that IC| h(2")- N. Let K' be the set of the (h(2*) N) smallest elements in K. Define m follows.
mg
if & KY if K..
Vk [kl]
m
(17)
m -1
Now it holds that Eken mi N.
Construct a strategy profile 0" = (0i, 02,00... 0n) such that mi players choose strategy 0,, m² players choose strategy 0z...., and mł, players choose strategy Oz.- In this profile, for any player that chooses strategy 0x, by the construction of wa in Eq. (II), he will get utility at least w/// /f/m Moreover, by the construction of my and m/k we have that ///mm/ 2 w// ///mm, 2 2* Hence, all players have utility at least 2*. Then we show that for any player that chooses strategy 6g with k [ko]. he could only get utility at most =" by deviation. Consider the two cases of the deviated strategy 0'.
1. Consider the case when the deviated strategy 6' E {01..... Ok-1, 0841,.... 0k, }. Suppose the player deviates to strategy k * k. As a result, the player will utility w + 1). When k' E K', we have that w///(m². + 1) S w'y////. = =". When k' & K', we have that w + 1) = w/// + 1) < =*, Hence, he could get utility at most =" by deviation.
2. Consider the case when the deviated strategy 0' E (01, 02,00.,000....
0k2 ). Note that m 2 m/ - 2 2 by the construction of mi. As result, for any strategy 0g with & E [ku], at least one player chooses it even if player deviates to 0'. As result, player i could get utility at most KK +k= Wkk <
20

---

Heterogencous Data Game
To conclude, in the strategy profile 0" every player obtains utility of least and can achieve at most 2* by deviating. Therefore, 0 is PNE.
(2) We then show that for any PNE = (01....,(N) we must have Vn E [N], E {01, 02..0., k.. To prove this claim, we have several steps.
I: We show that for all & E [Ko]. there exists E [N] such that 0, On. We prove this by contradiction. Suppose that there exists E [Ro] such that for all i E [N], 0; + 2g. Since the sum of the utilities of all players is 1, there must exist a player j such that (0) - 1/N. Now let 6' = 4z. Since all other players do not choose 8k. player j could become the only player that achieves the minimal loss on data source k. As a result, u (0', 0_-)) wk- Note that
ko
3w B
Wkk
N 2 Wko
Wko
2//
k=1
ko
330/
Wko
wkk
k= 1
3wkka
kn wkke
(Because Σ we. = >E= Wk- = 1)
w
3wkk
Ko Wko
(By Eq. (11))
+ Σk kk== Wk'
Wko
310kg
ko . Wko
(By the assumption Wko 3 K = wg)
Wko
+ Wko//
ko
9
Wkk'
(Because w1 > W2 > ... > wwk)
4
==1
9
(Because Σx=1 W/: 1)
4
5
1.
4
Hence, w2 > 1/N, implying that player j would achieve a higher utility by deviating to strategy Oz. leading to a contradiction.
II: Suppose there exists at least two players i E [N] such that 0,,0; 01 & (01,.... Oxo J. According to the first step, for any k E [Ko]] there exists at least one player that choose strategy k. As result, the sum of the utilities of players i and is at most Σ kk= kn Wk- Hence, at least one player has utility at most Σk Lk=ko+1 +1 wx/2. Without loss of generality, we assume this player player i. Since two players do not choose strategies in {61,... . OR. J. there must exist k [Ko] such that the number of players that choose Og is less than m (defined in Eq. (17)). Hence, if player deviates to strategy Ox, the utility
is at least
WK
w/y
Ek=ha+1 Wk
w/y
Ek= Wh
Σ - Wk
Σ
K k= kn+1 wkk
2
my
my
mK
mkk'
2
This leads to contradiction.
III: Suppose there is only one player i that chooses a strategy outside the set {01, O2...., Ono The utility of player i is at most Σ Wk 2" In addition, there must exist k E [ko] such that in the PNE, at most m players choose strategy 0g. Therefore, if player i deviates to strategy 0g. he will get utility at least w///m² 2 w/// o///m/, m// 2 2". This leads to contradiction.
(3) For any PNE 0, let my = Hi E [M] : 0, = =x2]l I be the number of players that choose strategy 02 in the PNE. We finally show that |m m// <
Suppose there exists k [ko) such that |m, -m²1 2 Consider cases.
1. Consider the case when my m/, 2 2. Suppose that player i chooses strategy Oz. The utility of player i is at most
w///m, S w///mm/, + 2) < z² Since Eke my Eie m N and my 2 m 2 m + 2, there must exist a
21

---

Heterogeneous Data Game
M E [Nol: // * k such that my mi.. As a result, if player deviates strategy 0g, he will obtain utility of least w/ge/ (m+ + 1) Z w 2 //m// 2 2" > (0), which leads to a contradiction.
2. Consider the case when mie me 2. Since Σt= ma Σ m N and m m/ - 1 my 1, there must cxist a k' E [Kol, k' # such that my my. Let i be any player that chooses strategy 0N in the PNE. Then (=(0) =wye/ w/y//mx S wy///mp, 1) S 2². However, if player deviates strategy 8k, he will obtain utility of
least (m/ + 1) 2 / (m/ 1) z", which leads to contradiction.
Now the claim follows.
C.6. Proof Sketch of Thm. 5.6
We prove Thm. 5.6 by dividing it into three parts.
Lemma C.3. Under the same assumption as Thm. 5.6, then O Homo (@M, (M.....
...... ....,.M) is a PNE if mm
Lemma C.A. Under the same assumption as Thm. 5.6, then there exists constant & such that O Homo
-60 6M ...., GM)
PNE if and only ift 2 t.
Lemma C.S. Under the same assumption Thm. 5.6, then there exists constant C such that if t N
max[6C/N N226max). then 0 HHomo is the unique PNE.
Now Thm. 5.6 follows from Lems. C.3 to C.5. The complete proof is available in the full version of the paper at petps://arxxor otg./iivrrrreee
07688.
C.7. Proof Sketch of Thm. 5.7
We first need the following propositions. Proposition C.6. Under the same conditions as in Thm. 5.7, there is a constant & > 0, depending only on (Ex, 0x wes )
such that whenever i S L a stralegy profile 0 (0), .2, E ON exists with
8n.
graex n
$ for all n E [N].
and
Oun (0, (0,0 0_n)
00
We introduce the following constants.
Definition C.1 (Ln). Since 0 * 0j for any distinct i,j E [K]. we can find constant t > 0, depending only on (Ex, 0n, we JK 1= such that
AO E 0, { [K] di (0, 0z: E s ED 1
Definition C.2 (m²). Let my = Hi E [N] 0RTO = 0 H be the number of players that choose strategy 0g in the PNE 0 Prex
Based on Def'. B.1 and C.1, for any n E [N], we could partition the space 0 into four parts.
(11) 0 0: (0, OR.; Σm. - 83/2
((2)
(DE E 0: 3/2 dy (0, Oz. Em {
0 (3) {0 E 0: Vk E [K] di (0, O2i Dx {n}
044 {0 E 0: 3:3k [K] dh (0, 0z: Ex <
It holds that & = ( U ! ( U ((4))
n.t- Denote the constant t to and the strategy profile 0 Ø Hete in Prop. C.6. Proposition C.7. Under the same conditions as in Thm. 5.7, there is constant >> O with to such that when t S the Hote
following holds: for all n E [N] and 0 E (.)). have un (0 Hete
2 un (0,0
22

---

Heterogencous Data Game
Proposition C.8. Under the same conditions as in Thm. 5.7, there is constant
0 with & Lo such that when & S the
jollowing holds: for all E [N] and 0 o have (0
Hete
(0,0 Hete
un
Proposition C.9. Under the same conditions as in Thm. 5.7, there is constant & > 0 with bo such that when t S t, the Hete
following holds: for all E [N] and 0 E D have un
Here
(0, (0,0-..
Un
Proposition C.10. Under the same conditions as in Thm. 5.7, there is a constant t > 0 with t S to. such that when t S t. g.a)) the following holds: for all n E [N] and 0 E n,u we have Un Hote 2 Un 0, 0n
Hete
Now Thm. 5.7 follows directly by combining Props. C.6 to C.10.
The complete proof available in the full version of the paper at httph::t
D. Important Lemmas
We need the following variants of Farkas's Lemma (Perng, 2017).
Lemma D.1 (Gordan's Theorem (Mangasarian, 1994)). For each given matrix A, exactly one of the following is true.
1. There exists vector x such that Az O.
2. There exists vector y 2 0 and y + 0 such that A ATy 0.
23


Reviews:
[{"review_id": "kx5r6YxFR2", "reviewer": "Reviewer_C4o6", "summary": "This work considers a game between model providers who choose which data sources to include in their models training. The approach is one like facility location games: the data sources are also the customers, and so e.g. a model provider could prioritize one data source to ensure that they win business from that customer. Or, in the presence of no competition, the model provider could equally weight all data sources, or weight them according to their sizes within the market. The example up front given is hospitals: they are both providing the data for training the algorithms, and they are the consumers of medical models that are trained on data from their and other hospitals.\n\nThe paper considers pure Nash equilibria, and whether or not equilibria: a) do not exist, b) exist where all model providers converge to same outcome, or c) where model providers specialize on different sources or mixtures of sources. These are considered under two regimes: in which the facilities / data providers choose exactly the best model provider, or noisily choose providers.\n\nWith two competing firms, the firms converge to the dominant data source if there is one; otherwise, there is no pure Nash equilibrium. However, this is called Heterogenous for some strange reason. I do not understand that. \nWith more than two competing firms, the paper gives conditions on PNE existence that translate to usually finding heterogeneous strategies when data source provider is chosen optimally, and can be heterogenous or homogeneous under noisily chosen parameters. Examples and characterizations of the spaces that lead to either are included.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your reviewing efforts and constructive comments.\n\n> ***Comparison with Competitive Location Models***\n\n**1. Differences in Setting**  \nOur model captures two key features of ML markets often missing in prior work: (1) **Source-specific distance metrics** from distribution shifts, and (2) **High-dimensional strategy spaces** due to many sources and model parameters.\n\nIn contrast, most competitive location models focus on **low-dimensional spaces** or **networks** with **uniform distance metrics**, driven by two factors: (1) applications in urban planning naturally fit 1D, 2D, or network settings; and (2) many models involve additional variables like price or quantity, limiting tractability and leading to smaller-scale formulations.\n\nWhile a few papers explore high-dimensional competition, they do so in the context of quantity competition [1] or pricing [2], rather than spatial or parameter competition.\n\n**2. Technical Distinctions and Contributions**  \nWhile prior models also observe homogeneous and heterogeneous equilibria, our key contribution lies in characterizing *when* they arise under ML-specific conditions.\n\n- **Heterogeneous distance metrics**: When $\\Sigma_k$ differ across sources, Proposition 4.1 shows that the PNE strategy set becomes a non-linear manifold in $\\mathbb{R}^D$, making equilibrium analysis substantially more complex.\n- **High-dimensionality**: To our knowledge, no prior work studies PNE in high-dimensional continuous spaces, even under Euclidean distance. As prior results rely heavily on geometric properties [3, 4], existing theories do not extend to our setting.\n\nWe acknowledge connections to low-dimensional results. For example, [3] analyzes a 2D probabilistic model with conditions for homogeneous PNE, while [5] studies location–quality competition, showing that proximity choice leads to homogeneous PNE, and probabilistic choice allows both types of PNE. However, these insights do not extend to **high-dimensional settings with heterogeneous metrics**. Classical economic models (e.g., Bertrand, Cournot) focus on price or quantity competition under simpler assumptions and do not capture the geometric complexity introduced by heterogeneous loss functions in our setting.\n\nTechnically, analyzing PNE requires new tools. To prove Theorem 5.7, we apply a fixed-point theorem to find a local equilibrium and show, via a partition of the strategy space (Section C.7), that it satisfies PNE conditions. The analysis is non-trivial due to the geometric complexity involved.\n\n> ***Policy Implications***\n\n**Our model captures key aspects of ML markets overlooked in prior work, as previously noted.** As discussed in Section 1.2, our theoretical results offer the following novel policy insights:\n\n1. When certain data sources (e.g., hospitals) have dominant weights, providers tend to focus on them (Theorem 5.4). Policymakers can counter this by adding more providers or incentivizing attention to smaller sources; our results help quantify both.\n2. To promote model diversity and avoid convergence to a single dominant model (e.g., in medical ML), Theorems 5.6 and 5.7 show that increasing the rationality of data sources—i.e., improving their ability to choose better models—encourages heterogeneous PNE.\n\n> ***Writing Issues***\n\nWe thank the reviewer for the helpful suggestions and will revise the paper accordingly. In addition, in HD-Game-Proximity with $N=2$, we label the PNE as \"heterogeneous\" since the strategy is $(\\theta_1,\\theta_1)$, not the homogeneous PNE given in the paper, consistent with the heterogeneous structure observed for $N>2$.\n\n> ***Justification of Experiments***\n\nOur synthetic experiments verify theoretical results and explore scenarios beyond theories. Ten representative simulations align well with theory. We further ran 100 random instances and report the average and standard deviation of temperature thresholds, confirming, for example, that the minimal temperature for a homogeneous PNE is typically well below $2\\ell_{\\max}$.\n\n|N|5|10|15|20|25|30|\n|:--|:--|:--|:--|:--|:--|:--|\n|Homogeneous PNE: Minimal $t/(2\\ell_{\\max})$|0.11 (0.07)|0.15 (0.08)|0.17 (0.08)|0.17 (0.08)|0.18 (0.09)|0.18 (0.09)|\n|Hetegeneous PNE: Maximal $t/(2\\ell_{\\max})$|0.09 (0.07)|0.12 (0.07)|0.12 (0.07)|0.11 (0.07)|0.11 (0.07)|0.11 (0.07)|\n\n---\n\n[1] Simon P. Anderson et al. Spatial Competition a la Cournot: Price Discrimination by Quantity‐Setting Oligopolists. Journal of Regional Science. 1990.\n\n[2] Helmut Bester. Noncooperative bargaining and spatial competition. Econometrica. 1989.\n\n[3] Dodge Cahan et al. Spatial competition on 2-dimensional markets and networks when consumers don't always go to the closest firm. International Journal of Game Theory. 2021.\n\n[4] Zvi Drezner et al. Competitive location models: A review. European Journal of Operational Research. 2024.\n\n[5] M. Elena Sáiz et al. On Nash equilibria of a competitive location-design problem. European Journal of Operational Research. 2011.", "claims_and_evidence": "I did not review the proofs in detail. The claims do not seem to be problematic.", "methods_and_evaluation_criteria": "It is helpful that the methods include both theoretical proofs and examples.", "theoretical_claims": "I did not verify the correctness of the proofs in the supplemental appendix.", "experimental_designs_or_analyses": "The paper simulates these games experimentally, but the results are just aggregated by showing 10 sample runs, not any statistics aggregated over the outputs or treatment of whether these are representative.", "relation_to_broader_scientific_literature": "This paper proposes a model based on facility location, which has an extensive literature. That relationship is an interesting one to draw on.  Those connections should be further discussed - what are the closest results? Is the intuition similar or different when moving from euclidean space into ML model competition?", "essential_references_not_discussed": "Not to my knowledge.", "other_strengths_and_weaknesses": "The topic is interesting, and facility location makes for a good literature to draw on. I don't know that the results and model formulation are of the utmost interest. It seems clear that sometimes firms would consolidate, and sometimes differentiate. For a full treatment, this should be compared to other results in the facility location literature and general economics competition literature - what is different about the existence, homogeneous vs heterogeneous results in this setting vs other settings?  The paper speaks to a desire to have policy impacts, but the results are not discussed at the level that would result in policy discussions. What are the policy implications here? How should a policy maker evaluate this in the example scenario of training on hospital data? \n \nThe paper is poorly written, which makes it hard to follow. Here are some examples.\n- Proposition 4.1 is described as characterizing the \"strategy set\" for each player, but it is really talking about best response strategies.\n- In the definition of The Heterogeneous Data Game, the type parameter \\theta on the left hand side of the equation does not show up on the right side, it is implicit in l_{n,k}. This is confusing. Adjust your notation so that you have enough space to include the type explicitly.\n- g_n(l_{1,k}, ..., l_{n,k}) is the weight for facility k on data provider n, which must satisfy \\sum_{n\\ in N} g_n(l_{.., k})=1. That seems strange, for it to be a choice of k and not n. Perhaps put a k superscript on g. Then you could say: g_n^k(  l(theta hat) ). That notation makes much more sense.", "other_comments_or_suggestions": "Nit to pick: In 3.2, notationally, you should be defining g, not g^{prop,prox}. You could though say g = g^prop = ....", "questions_for_authors": "How do your heterogeneous and homogeneous equilibrium results compare to the findings in the facility location? If a policy maker takes their intuition from general facility location games, what do they lose out on vs understanding the model you present?", "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": "Hu4URAO2xD", "reviewer": "Reviewer_1w5x", "summary": "This work analyze competition through Nash equilibria between multiple ML model providers across heterogeneous data sources. The game is characterized under two different data source choice models and provides conditions for each type of equilibrium. Synthetic experiments are conducted.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your reviewing efforts and constructive comments.\n\n> ***Focus on Linear Models, IID Assumptions, and Lack of Empirical Validation***\n\n1. **Linear Models**: Despite the rise of deep learning, linear models remain widely used for their interpretability, particularly in high-stakes domains [1], and can outperform deep models when data is limited [2]. Our framework also aligns with the linear probing setup, where only the final layer of a foundation model is updated—common when full fine-tuning is costly or prone to overfitting [3]. This makes our model relevant for analyzing competition in foundation model markets.\n2. **IID Assumption**: Our setting is inherently non-IID, as data sources follow different distributions, reflected by distinct covariance matrices $\\Sigma_k$. This heterogeneity is central to our analysis. Extending to more complex or dynamic distributions is a promising direction for future work.\n3. **Empirical Validation**: Our synthetic experiments are designed to rigorously verify our theoretical results and explore cases beyond analytical coverage. Synthetic data allows us to exhaustively check whether a strategy profile constitutes a PNE, which is difficult to do reliably in real-world high-dimensional settings. While experimental validation in real applications is indeed valuable, we emphasize that our theoretical results apply to such settings and remain relevant even when empirical verification is challenging.\n\n> ***In line 128 \"There are N model providers (players) that need to compete the models in these K data sources.\"***\n\nThank you for pointing this out. If the concern is about clarity, we are happy to revise the sentence to: \"There are $N$ model providers (players), each deploying a single model to compete across $K$ data sources.\" Please let us know if a different clarification was intended.\n\n> ***Explain how a data source queries each model provider for $\\ell_{n,k}$. Is this validation loss of each data source?***\n\nYes, $\\ell_{n,k}$ can be interpreted as the validation loss of model provider $n$ on data source $k$. The availability of such loss information is a standard modeling assumption in prior work on ML model competition [4].\n\nIn practice, this value can be obtained through interactions between model providers and data sources. A common scenario is: the data source shares a private validation set with model providers, who return predictions. The data source then evaluates the loss and may report it back to the provider.\n\n> ***How do the model providers optimize $\\hat{\\theta}$ in practice?***\n\n1. Our focus is on analyzing the properties of equilibrium among model providers, representing the stable market states. Nash equilibrium is a standard and widely used solution concept in both ML model markets [4] and broader competitive settings [5].\n2. The process by which providers optimize $\\hat{\\theta}$ is an interesting direction for future work. While not the focus of our paper, a common approach—also used in prior work on ML model markets without heterogeneous data [6]—is **best-response dynamics**, where each provider updates $\\hat{\\theta}_n$ to maximize their utility in Equation (3), given others' strategies. A formal analysis of convergence in our setting remains open.\n\n> ***How do the model providers learn their loss on each data source?***\n\nAs noted above, model providers can obtain loss information through interactions with data sources. For example, a data source may provide a validation set, and after receiving predictions from the provider, compute and share the resulting loss. This feedback allows providers to estimate $\\ell_{n,k}$, which in turn guides how they weight different data sources during model training.\n\n> ***Stationarity Assumption and Robustness to Distribution Shifts***\n\nOur current analysis assumes fixed (stationary) data distributions. However, the equilibrium is robust to moderate distributional shifts, as key parameters (e.g., $\\ell_{\\max}$, $\\hat{\\theta}^M$) vary smoothly with the distribution. Thus, small changes lead to small adjustments in PNE. Thus, our theoretical insights continue to hold under mild distributional shifts. Studying dynamic or non-stationary settings remains an important direction for future work.\n\n---\n\n[1] Cynthia Rudin. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature machine intelligence. 2019. \n\n[2] Muhammad Arbab Arshad et al. The Power Of Simplicity: Why Simple Linear Models Outperform Complex Machine Learning Techniques--Case Of Breast Cancer Diagnosis. 2023.\n\n[3] Ananya Kumar et al. Fine-Tuning can Distort Pretrained Features and Underperform Out-of-Distribution. ICLR. 2022.\n\n[4] Omer Ben-Porat et al. Regression equilibrium. ACM Conference on Economics and Computation. 2019.\n\n[5] Noam Nisan et al. Algorithmic game theory. Cambridge Univ. Press. 2007.\n\n[6] Omer Ben-Porat et al. Best response regression. NeurIPS. 2017.", "claims_and_evidence": "Characterization of conditions for PNE existence is supported with theoretical analysis. \nStronger evidence is the practical generalizability beyond linear models.", "methods_and_evaluation_criteria": "Theoretical findings are evaluated with synthetic experiments for various parameters such as model choice temperature and number of model providers. Experiments are rather basic and additional real-world examples would have strengthened the evaluation.", "theoretical_claims": "Statements in the main text look right. Proofs in the appendix were not checked.", "experimental_designs_or_analyses": "Only experiments with synthetic data are conducted, which validate theoretical analysis.", "relation_to_broader_scientific_literature": "The related section seems comprehensive and their work is well-situated in the literature. The main novelty seems to be extending the study of equilibrium from homogeneous data source to heterogeneous data source.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "A limitation of this analysis is focused on linear models and IID distributions. Limited empirical validation with real data.", "other_comments_or_suggestions": "* In line 128 \"There are N model providers (players) that need to compete the models in these K data sources.\"\n* Explain how a data source queries each model provider for $\\ell_{n,k}$. Is this validation loss of each data source? Mention why assuming this is available is reasonable for a data market setting.", "questions_for_authors": "* How do the model providers optimize $\\hat{\\theta}$ in practice? \n* How do the model providers learn their loss on each data source? This is not discussed. \n* Does this assume stationarity of data source distribution? How would distribution shifts affect equilibria. How robust are your equilibrium results to non-stationarity in data distributions?", "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": "UxJni5nswk", "reviewer": "Reviewer_GPAn", "summary": "The paper explores the HDG problem, aiming to identify the Pure Nash Equilibrium (PNE) in distributing data resources among machine learning (ML) model producers. Each producer provides parameters, influencing resource allocation based on both their own and the ML parameters. The analysis covers conditions for PNE existence in scenarios involving one producer, two producers, and multiple producers.\n\n## update after rebuttal\nI keep the score because I think the presentation can be improved.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your reviewing efforts and constructive comments.\n\n> ***It is unclear if Nash equilibrium is good solution concept here. What is the role of pure Nash equilibria in this scenario? How to use it in the real world?***\n\n1. **Nash equilibrium is a well-established and widely used solution concept in game theory for analyzing strategic interactions among competing agents [1]**. In the context of ML model markets, recent works have also focused on characterizing equilibrium outcomes among model providers [2, 3]. Beyond ML model markets, the use of Nash equilibrium extends to other competitive domains such as recommender systems [4] and targeted advertising [5], where it is employed to model stable market outcomes. Therefore, applying the concept of Nash equilibrium in our setting is both standard and well-supported in the literature.\n2. **In our model, a pure Nash equilibrium represents a stable state of the market**, where each model provider (player) chooses a strategy (i.e., model parameters) such that no one can unilaterally deviate to improve their utility. This notion of stability is central to real-world markets, where firms adapt until no player can gain from further adjustments, given the choices of others. Thus, analyzing PNE offers insight into the long-term strategic behavior and positioning of providers in a competitive ML ecosystem.\n3. **Beyond theoretical interest, studying the structure of Nash equilibria can offer practical guidance for market design and policy-making**. As discussed in the last paragraph of Section 1.2, our results provide actionable insights:\n   -  When certain data sources have dominant weights, providers tend to focus on them exclusively (Theorem 5.4). Our analysis quantifies how adding more providers or incentivizing attention to underrepresented sources can shift the equilibrium toward more balanced coverage.\n   - To avoid the risks of model monoculture (i.e., all data sources converging to the same model), Theorems 5.6 and 5.7 show that increasing the rationality of data sources—i.e., improving their ability to select high-performing models—encourages heterogeneous equilibria, leading to more diverse model deployment across the ecosystem.\n\n> ***I think the written is not satisfying. Maybe dividing each theorem into two or three will make it better. And also the proof is tedious.***\n\n1. We thank the reviewer for the valuable suggestion. **We will revise the paper to improve clarity, organization, and overall readability.** In particular, we will consider breaking down complex theorems—such as Theorems 5.4 and 5.6—into smaller components to make them easier to follow. If there are specific theorems or sections you found especially difficult to read, we would greatly appreciate further feedback, which would help us target our revisions more effectively.\n2. **Regarding the proofs, we acknowledge that some arguments are lengthy, but this reflects the non-trivial nature of the results.** The main technical challenges arise from the high-dimensional continuous strategy space and heterogeneous covariance matrices $\\Sigma_k$, which together form a non-linear manifold of PNE strategies (Proposition 4.1). This complexity is particularly evident in HD-Game-Probability (Section 5.3.1), where we first use a fixed-point theorem to establish a local equilibrium, and then, via a structured partition of the strategy space (Section C.7), show that it satisfies full PNE conditions. We will revise the proofs to improve clarity and provide more intuitive explanations in the future version of the paper.\n\n> ***How you get that PNE? Enumerating all possible profile in $(\\theta_1, ..., \\theta_K)^N$?***\n\n**The initial PNE for HD-Game-Proximity is derived using Theorem 5.4 and Corollary 5.5.** These results show that, at equilibrium, the number of model providers focusing on each data source is approximately proportional to that source’s weight. The specific allocation is computed using Equation (11), which determines how many providers should select each $\\theta_k$.\n\n**This method provides an efficient way to construct a candidate PNE and does not rely on exhaustive enumeration over all possible strategy profiles in $(\\theta_1, ..., \\theta_K)^N$.** As a result, it is computationally efficient and scalable in our experimental setup.\n\n---\n\n[1] Noam Nisan et al. Algorithmic game theory. Cambridge Univ. Press. 2007.\n\n[4] Omer Ben-Porat et al. Regression equilibrium. ACM Conference on Economics and Computation. 2019.\n\n[3] Meena Jagadeesan et al. Improved bayes risk can yield reduced social welfare under competition. NeurIPS. 2023.\n\n[4] Meena Jagadeesan et al. Supply-side equilibria in recommender systems. NeurIPS 2023.\n\n[5] Ganesh Iyer et al. Competitive model selection in algorithmic targeting. Marketing Science. 2024.", "claims_and_evidence": "Yes, all claim has its clear and convincing evidence.", "methods_and_evaluation_criteria": "The evaluation criteria are Nash Equilibrium, a well-known notion. For method, the paper focus on proving the existence. They also shown an efficient method to find the lower bound of t in experiment. \n\nIt is unclear if Nash equilibrium is good solution concept here. What is the role of pure Nash equilibria in this scenario? How to use it in the real world?", "theoretical_claims": "Yes. I have checked all proofs and I think they are all correct.", "experimental_designs_or_analyses": "Yes. I have checked the experiment designs and analysis, and I think they are sound and valid.", "relation_to_broader_scientific_literature": "Unsure", "essential_references_not_discussed": "Unsure", "other_strengths_and_weaknesses": "The paper is technically sound and the result is significant. However, I think the written is not satisfying.", "other_comments_or_suggestions": "Maybe dividing each theorem into two or three will make it better. And also the proof is tedious.", "questions_for_authors": "In your approach to find heterogeneous PNE in HD-Game-Probability, you start with a PNE for HD-Game-Proximity. However, how you get that PNE? Enumerating all possible profile in (\\theta_1, …, \\theta_K)^N?", "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": "mmXmfpmat2", "reviewer": "Reviewer_3G31", "summary": "This paper investigates the phenomenon of model competition across multiple data sources. The authors propose heterogeneous data game, where each of the model providers decide to deploy a single model, aiming to win the choose from data sources as much as possible. The model is characterized by its parameters and covariance matrix, and the loss incurred when a data source decides to choose a model is measured via Mahalanobis distance. The authors study the cases where the data sources pick models by proximity choice and by probability choice, and prove equivalent condition for pure Nash equilibrium existence when the number of providers $N=2$ in proximity choice case, and give sufficient conditions as well as some basic properties if PNE exists in other cases.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your reviewing efforts and constructive comments.\n\n> ***Linear Model Assumption***\n\n1. Despite the prevalence of deep learning, **linear models remain important due to their interpretability—an essential requirement in high-stakes domains** such as healthcare and the judicial system [1]. Additionally, when training data is limited, linear models have been shown to outperform deep learning methods in various settings [2, 3], and thus are still widely used in practice.\n2. As discussed in the last paragraph of Section 3.3, **our model also aligns with the linear probing paradigm**, where only the final linear layer of a pretrained foundation model is updated. This approach is common when fine-tuning full models is costly or risks overfitting [4]. In this context, our framework can be interpreted as modeling competition over linear heads on fixed representations, offering insights into the emerging market dynamics around foundation model adaptation.\n\n> ***On the Sufficiency, Necessity, and Practical Value of Theoretical Results***\n\n1. While prior work discusses the existence of homogeneous and heterogeneous PNE, **our main contribution lies in rigorously characterizing *when* they arise under ML-specific conditions**—namely, distribution shifts and high-dimensional strategy spaces. As noted in our response to Reviewer C4o6, most existing models assume low-dimensional spaces with uniform distance metrics, limiting relevance to ML markets.\n2. **We also further characterize the exact necessary and sufficient condition for homogeneous PNE in HD-Game-Probability** (see our response below on the interval in Theorem 5.6), addressing the gap between sufficiency and necessity.\n3. **Although the existence of PNE types may seem intuitive, the technical challenge lies in identifying the exact conditions under which they emerge**. The difficulty stems from the high-dimensional continuous strategy space and heterogeneous $\\Sigma_k$, which, per Proposition 4.1, yield a non-linear manifold. Analyzing such cases—particularly in HD-Game-Probability (Section 5.3.1)—requires non-standard tools: we first establish a local equilibrium via a fixed-point theorem and then show it is a true PNE using a careful partition of the strategy space (Section C.7).\n4. From a policy perspective (see Section 1.2), our results offer actionable insights: (1) Theorem 5.4 shows that dominant data sources attract provider focus; our results help quantify how many additional providers or how much incentive is needed to support underrepresented sources. (2) Theorems 5.6 and 5.7 demonstrate that increasing the rationality of data sources can foster heterogeneous equilibria, thus helping to mitigate the risks associated with the dominance of a single model in the market.\n\n> ***Justification of Assumption 4.1***\n\nFor any fixed $\\theta \\in \\mathbb{R}^D$, the condition $\\bar{\\theta}(\\mathbf{q}) = \\theta$ is equivalent to solving $\\sum_{k=1}^K q_k \\Sigma_k (\\theta_k - \\theta) = 0$ under the constraint $\\sum_{k=1}^K q_k = 1$. When the covariance matrices $\\Sigma_k$ differ, the vectors $\\Sigma_k(\\theta_k - \\theta)$ are typically in general position—especially in high-dimensional settings where $D \\gg K$. In such cases, the corresponding overdetermined linear system is injective, and the only solution (except for a measure-zero set with degenerate alignment) is the one uniquely determined by the normalization constraint.\n\n> ***Interval between case 1 and 2 in Theorem 5.6***\n\nWe have strengthened our theoretical result to address this point: **In HD-Game-Probability, there exists a threshold $t_0 > 0$ such that the homogeneous PNE exists if and only if $t \\ge t_0$.**\n\nThe proof uses a constructive argument: assume the homogeneous PNE exists at $t = t'$. For any $t'' > t'$ and deviation $\\theta$, the convexity of the Mahalanobis loss implies that the utility of a player deviating to $\\theta$ under temperature $t''$ is no greater than the utility of deviating to $c\\theta + (1-c)\\hat{\\theta}^M$ with $c = t'/t''$ under $t'$, which is at most $1/N$. Thus, the deviation is not profitable, and the homogeneous PNE remains valid for all $t'' > t'$, confirming the existence region is an interval $[t_0, \\infty)$.\n\nAlthough the exact value of $t_0$ is hard to compute analytically, our synthetic experiments suggest it is approximately a constant multiple of $2\\ell_{\\max}$.\n\n---\n\n[1] Cynthia Rudin. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature machine intelligence. 2019. \n\n[2] Shuming Jiao et al. Does deep learning always outperform simple linear regression in optical imaging?. Optics express. 2020.\n\n[3] Muhammad Arbab Arshad et al. The Power Of Simplicity: Why Simple Linear Models Outperform Complex Machine Learning Techniques--Case Of Breast Cancer Diagnosis. 2023.\n\n[4] Ananya Kumar et al. Fine-Tuning can Distort Pretrained Features and Underperform Out-of-Distribution. ICLR. 2022.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I've only checked the correctness roughly, while I do not discover any technical flaw.", "experimental_designs_or_analyses": "Yes. The experiment focuses on synthetically studying the effect of temperature parameter on existence of Nash equilibrium in the probability choice model of data sources, and is generally intuitive.", "relation_to_broader_scientific_literature": "No.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "- Strength: This paper is self-contained. As a theoretical paper, the mathematical modeling of the considered real-world problem is well-defined. Most steps of the abstractions as well as the assumptions in proving theorems are justified. All different cases inside the proposed mathematical model is studied. Generally speaking, this is an interesting paper, and I do not find any serious technical flaw or vulnerability in this research that leading to a straightforward rejection. \n- Weakness: I maintain a reserved stance regarding the significance of this paper:\n    - The game-theoretical modeling ensures the change of model parameters linearly influence the outcomes, which is a strong simplification of the initial real-world problem. \n    - The sufficient conditions are overly intuitive, giving readers a sense of apparent correctness, while the position of necessary conditions, as well as the area between necessity and sufficiency is less explored. And because of this, the practical guide of theoretical results seems not enough, as claimed in abstract.", "other_comments_or_suggestions": "- I suggest that Assumption 4.1 could be justified more detailedly. The justification in Remark 4.2 gives an intuitive explanation of its satisfiability, while personally I think an elaboration on the case when the covariances are not equal to each other is also needed, since Assumption 4.1 is a key assumption in the rest of the paper.\n- In Theorem 5.6, there is an interval between case 1 and case 2. Is the existence of PNE not sured in that interval? The authors would better point out what happened in that interval to make the discussion complete.", "questions_for_authors": "No.", "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:

Meta-review: This paper considers a data market where multiple ML providers compete for heterogeneous data sources, heterogeneity being the key novelty. Under various conditions, the paper establishes what equilibria the ML providers will converge to, if any (e.g., all the same model, or each a specialized model). Reviewers had a favorable view of the paper. The main feedback is that the writing and exposition can be improved. This includes clearly delineating the setting of the paper from others, improving the mathematical language, and clearly articulating the motivation/description of experiments (possibly adding non-synthetic examples). Some of the limitations of the setting should also be better discussed (e.g., linear models). We urge the authors to closely follow all the suggestions of the reviewers and improve the clarity of the paper.
Decision: Accept (poster)