Complexity as Advantage: A Regret-Based Perspective on Emergent Structure

TL;DR

This paper proposes a new framework called Complexity as Advantage (CAA) that measures system complexity based on observer-dependent predictive regret, unifying various notions of emergent behavior and explaining the functional value of complexity.

Contribution

The paper introduces CAA, a novel observer-relative complexity measure that unifies multiple concepts of emergence and explains the utility of complexity in systems.

Findings

CAA unifies multiscale entropy, predictive information, and observer-dependent structure.

Systems are complex when they create differentiated regret for different observers.

The framework has implications for learning, evolution, and artificial intelligence.

Abstract

We introduce Complexity as Advantage (CAA), a framework that defines the complexity of a system relative to a family of observers. Instead of measuring complexity as an intrinsic property, we evaluate how much predictive regret a system induces for different observers attempting to model it. A system is complex when it is easy for some observers and hard for others, creating an information advantage. We show that this formulation unifies several notions of emergent behavior, including multiscale entropy, predictive information, and observer-dependent structure. The framework suggests that "interesting" systems are those positioned to create differentiated regret across observers, providing a quantitative grounding for why complexity can be functionally valuable. We demonstrate the idea through simple dynamical models and discuss implications for learning, evolution, and artificial agents.