Disc Game Dynamics: A Latent Space Perspective on Selection and Learning in Games

2025-11-24

AI Summary

This paper addresses the analytical intractability of evolutionary dynamics in symmetric zero-sum games. We propose the *disk game embedding* method: by axiomatically constructing a latent variable space, we reduce each agent’s optimal strategy update direction to a function solely of the opponent’s coordinates, thereby decomposing the original game into a linear combination of disk games. The method establishes an exact equivalence—under a coordinate transformation—between continuous replicator dynamics and adaptive dynamics, and yields a constrained oscillation equation. Leveraging Hamiltonian system theory and finite-rank approximation, we derive a minimal-rank representation that substantially reduces simulation complexity in high dimensions. Our core contribution is a tractable oscillatory system characterization enabling either closed-form finite-dimensional solutions or optimal low-rank approximations, providing a unified, generalizable analytical framework for evolutionary game theory.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Machine Learning: Evolutionary Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

Evolutionary game theory studies populations that change in response to an underlying game. Often, the functional form relating outcome to player attributes or strategy is complex, preventing mathematical progress. In this work, we axiomatically derive a latent space representation for pairwise, symmetric, zero-sum games by seeking a coordinate space in which the optimal training direction for an agent responding to an opponent depends only on their opponent's coordinates. The associated embedding represents the original game as a linear combination of copies of a simple game, the disc game, in a new coordinate space. In this article, we show that disc-game embedding is useful for studying learning dynamics. We demonstrate that a series of classical evolutionary processes simplify to constrained oscillator equations in the latent space. In particular, the continuous replicator equation reduces to a Hamiltonian system of coupled oscillators that exhibit Poincaré recurrence. This reduction allows exact, finite-dimensional closure when the underlying game is finite-rank, and optimal approximation otherwise. It also establishes an exact equivalence between the continuous replicator equation and adaptive dynamics in the transformed coordinates. By identifying a minimal rank representation, the disc game embedding offers numerical methods that could decouple the cost of simulation from the number of attributes used to define agents. These results generalize to metapopulation models that mix inhomogeneously, and to any time-differentiable dynamic where the rate of growth of a type, relative to its expected payout, is a nonnegative function of its frequency. We recommend disc-game embedding as an organizing paradigm for learning and selection in response to symmetric two-player zero-sum games.

Problem

• Derives latent space representation for symmetric zero-sum games
• Simplifies evolutionary dynamics to constrained oscillator equations
• Provides efficient simulation methods for complex game interactions

Innovation

• Latent space embedding for symmetric zero-sum games
• Reduces replicator dynamics to Hamiltonian oscillator system
• Enables efficient simulation decoupled from agent attributes