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A Geometric Embedding Approach to Multiple Games and Multiple Populations

Bastian Boll, Jonas Cassel, Peter Albers, Stefania Petra, Christoph Schnörr

TL;DR

The paper introduces a geometric meta-simplex embedding that reduces multi-population replicator dynamics to single-population dynamics on a joint simplex, enabling unified analysis of multi-population and multi-game dynamics. It centers on two isometric embedding theorems implemented via linear maps $M$ and $Q$ and a nonlinear payoff transformation $\widehat{F}=Q\circ F\circ M$, with an isometric embedding $T:\mathcal{W}\to\mathcal{T}\subseteq\mathcal{S}_N$ and maximum-entropy marginalization. The framework extends to nonlinear payoffs, tangent-space parameterizations, and learning from data, showing how standard asymptotic results for single-population replicator dynamics transfer to the multi-population setting, and connects to Segre embeddings and potential quantum-state generalizations. Overall, the work provides a robust mathematical toolkit for analyzing complex population dynamics across biology and related fields, clarifying when multi-population and multi-game models align and how to study them within a unified geometric framework.

Abstract

This paper studies a meta-simplex concept and geometric embedding framework for multi-population replicator dynamics. Central results are two embedding theorems which constitute a formal reduction of multi-population replicator dynamics to single-population ones. In conjunction with a robust mathematical formalism, this provides a toolset for analyzing complex multi-population models. Our framework provides a unifying perspective on different population dynamics in the literature which in particular enables to establish a formal link between multi-population and multi-game dynamics.

A Geometric Embedding Approach to Multiple Games and Multiple Populations

TL;DR

The paper introduces a geometric meta-simplex embedding that reduces multi-population replicator dynamics to single-population dynamics on a joint simplex, enabling unified analysis of multi-population and multi-game dynamics. It centers on two isometric embedding theorems implemented via linear maps and and a nonlinear payoff transformation , with an isometric embedding and maximum-entropy marginalization. The framework extends to nonlinear payoffs, tangent-space parameterizations, and learning from data, showing how standard asymptotic results for single-population replicator dynamics transfer to the multi-population setting, and connects to Segre embeddings and potential quantum-state generalizations. Overall, the work provides a robust mathematical toolkit for analyzing complex population dynamics across biology and related fields, clarifying when multi-population and multi-game models align and how to study them within a unified geometric framework.

Abstract

This paper studies a meta-simplex concept and geometric embedding framework for multi-population replicator dynamics. Central results are two embedding theorems which constitute a formal reduction of multi-population replicator dynamics to single-population ones. In conjunction with a robust mathematical formalism, this provides a toolset for analyzing complex multi-population models. Our framework provides a unifying perspective on different population dynamics in the literature which in particular enables to establish a formal link between multi-population and multi-game dynamics.
Paper Structure (21 sections, 23 theorems, 95 equations, 3 figures, 1 table)

This paper contains 21 sections, 23 theorems, 95 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview, Contribution
  3. Organization
  4. Basic Notation.
  5. Preliminaries
  6. Fisher-Rao Geometry, Replicator Dynamics
  7. Data Labeling and Assignment Flows
  8. Embedding the Assignment Manifold
  9. Multiple Populations and Multiple Games
  10. Tangent Space Parameterization
  11. Learning Replicator Dynamics from Data
  12. Asymptotic Behavior
  13. Conclusion
  14. Funding
  15. Competing interests
  16. ...and 6 more sections

Key Result

Theorem 3.1

The map $T\colon {\mathcal{W}}\to{\mathcal{T}}\subseteq {\mathcal{S}_N}$ is an isometric embedding of ${\mathcal{W}}$ equipped with the product Fisher-Rao geometry, into ${\mathcal{S}_N}$ equipped with the Fisher-Rao geometry. On its image $T({\mathcal{W}}) =: {\mathcal{T}}\subseteq {\mathcal{S}_N}$

Figures (3)

  • Figure 1: Marginals distributions $S = (S_1, S_2)$ and two possible conforming joint distributions. Joint distribution values are scaled by a factor of $c$ for visual clarity.
  • Figure 2: The embedded submanifold ${\mathcal{T}}\subseteq {\mathcal{S}_N},\, N=4$. For two marginal distributions, this is known as the Wright manifoldHofbauer:1998, Chamberland:2000.
  • Figure 3: Left: Noisy input assignment of $c=47$ colors to the pixels of an image. Center: Limit of an EGN flow \ref{['eq:vec_egn_field']} with learned interaction in $3\times 3$ pixel neighborhoods. Right: Ground truth noise-free color assignment.

Theorems & Definitions (46)

  • Theorem 3.1: assignment manifold embedding
  • proof
  • Proposition 3.2: maximum entropy property
  • proof
  • Lemma 3.3: Lifting Map Lemma
  • proof
  • Lemma 3.4: $Q$ Adjoint Lemma
  • proof
  • Theorem 3.5: Multi-Population Embedding Theorem
  • proof
  • ...and 36 more