Morphisms of open games

TL;DR

This paper introduces a formal framework for morphisms between open games, connecting lenses and game theory, enabling a structured approach to analyzing equilibria and strategic interactions.

Contribution

It defines a new notion of morphisms between open games using lenses, constructing a symmetric monoidal double category that unifies various equilibrium concepts.

Findings

Established a double category structure for open games and lenses.

Demonstrated the framework with a market entry game example.

Unified Nash and subgame perfect equilibria within the same formalism.

Abstract

We define a notion of morphisms between open games, exploiting a surprising connection between lenses in computer science and compositional game theory. This extends the more intuitively obvious definition of globular morphisms as mappings between strategy profiles that preserve best responses, and hence in particular preserve Nash equilibria. We construct a symmetric monoidal double category in which the horizontal 1-cells are open games, vertical 1-morphisms are lenses, and 2-cells are morphisms of open games. States (morphisms out of the monoidal unit) in the vertical category give a flexible solution concept that includes both Nash and subgame perfect equilibria. Products in the vertical category give an external choice operator that is reminiscent of products in game semantics, and is useful in practical examples. We illustrate the above two features with a simple worked example from microeconomics, the market entry game.