The algebra of predicting agents

TL;DR

This paper explores the algebraic structure of predicting agents within the category of open games, revealing how their properties approximate a compact closed category with complex algebraic features.

Contribution

It introduces a new algebraic perspective on agents with predictive goals in open game categories, connecting to bicategory structures and lax bialgebras.

Findings

Open games with predictive agents form a symmetric monoidal bicategory.

The structure approximates a compact closed category with lax commutative bialgebras.

The algebraic properties relate to earlier work on selection functions.

Abstract

The category of open games, which provides a strongly compositional foundation of economic game theory, is intermediate between symmetric monoidal and compact closed. More precisely it has counits with no corresponding units, and a partially defined duality. There exist open games with the same types as unit maps, given by agents with the strategic goal of predicting a future value. Such agents appear in earlier work on selection functions. We explore the algebraic properties of these agents via the symmetric monoidal bicategory whose 2-cells are morphisms between open games, and show how the resulting structure approximates a compact closed category with a family of lax commutative bialgebras.