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Heterogeneous Data Game: Characterizing the Model Competition Across Multiple Data Sources

Renzhe Xu, Kang Wang, Bo Li

TL;DR

The paper introduces the Heterogeneous Data Game (HD-Game), a game-theoretic framework for analyzing competition among multiple ML model providers across heterogeneous data sources. It defines a Mahalanobis-distance-based loss and two data-source choice models (Proximity and Probability with temperature $t$) to study Pure Nash Equilibria under monopoly, duopoly, and multi-provider settings. The authors derive conditions for the existence of homogeneous and heterogeneous PNE, showing that heterogeneity is common under proximity when dominant data sources exist, while temperature in the probability model can toggle between homogeneous and heterogeneous equilibria and even allow coexistence. Synthetic experiments elucidate how data-heterogeneity magnitude and $t$ influence equilibrium structure, providing insights for policy design aimed at fostering diverse, fair model offerings in competitive ML marketplaces. Overall, the HD-Game framework links distribution shifts, market structure, and strategic model deployment to offer principled guidance for regulators and practitioners.

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

TL;DR

The paper introduces the Heterogeneous Data Game (HD-Game), a game-theoretic framework for analyzing competition among multiple ML model providers across heterogeneous data sources. It defines a Mahalanobis-distance-based loss and two data-source choice models (Proximity and Probability with temperature ) to study Pure Nash Equilibria under monopoly, duopoly, and multi-provider settings. The authors derive conditions for the existence of homogeneous and heterogeneous PNE, showing that heterogeneity is common under proximity when dominant data sources exist, while temperature in the probability model can toggle between homogeneous and heterogeneous equilibria and even allow coexistence. Synthetic experiments elucidate how data-heterogeneity magnitude and influence equilibrium structure, providing insights for policy design aimed at fostering diverse, fair model offerings in competitive ML marketplaces. Overall, the HD-Game framework links distribution shifts, market structure, and strategic model deployment to offer principled guidance for regulators and practitioners.

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.
Paper Structure (63 sections, 37 theorems, 151 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 63 sections, 37 theorems, 151 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Overview of the Heterogeneous Data Game
  3. Overview of the Results
  4. Monopoly ($N=1$).
  5. Duopoly ($N=2$).
  6. More than two providers ($N>2$).
  7. Related Works
  8. Data heterogeneity.
  9. Competition in machine learning.
  10. Competitive location models.
  11. Other competitive frameworks.
  12. Heterogeneous Data Game (HD-Game)
  13. Notations
  14. Game Setup
  15. Heterogeneous data.
  16. ...and 48 more sections

Key Result

Proposition 4.1

Denote the set $\vartheta$ as follows: where Then, the following holds:

Figures (3)

  • Figure 1: Utility of a single model provider with a deviated policy for both homogeneous and heterogeneous PNE in the configuration of \ref{['example:probability-both-existence']}.
  • Figure 2: In the probability choice model, this figure reports, across 10 randomly generated games, the threshold $\underline{t}$ that guarantees the existence of a homogeneous PNE and the approximate largest value of $t$ that guarantees the existence of a heterogeneous PNE, as $N$ varies.
  • Figure 3: The graphical explanation of \ref{['example:proximity-model']}.

Theorems & Definitions (88)

  • Definition 3.1: Pure Nash Equilibrium (PNE)
  • Proposition 4.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Theorem 5.1
  • Remark 5.1
  • Theorem 5.2
  • Remark 5.2
  • Proposition 5.3
  • ...and 78 more