# The game semantics of game theory

## TL;DR

This paper bridges game theory and game semantics using a reformulation of compositional game theory, employing lenses and category theory to model open games and their interactions.

## Contribution

It introduces a novel categorical framework connecting game theory with game semantics through the use of lenses and the Int-construction.

## Key findings

- Reformulation of open games as systems and environments.
- Construction of a compact closed category of computable open games.
- Application of wave-style geometry of interaction to game semantics.

## Abstract

We use a reformulation of compositional game theory to reunite game theory with game semantics, by viewing an open game as the System and its choice of contexts as the Environment. Specifically, the system is jointly controlled by $n \geq 0$ noncooperative players, each independently optimising a real-valued payoff. The goal of the system is to play a Nash equilibrium, and the goal of the environment is to prevent it. The key to this is the realisation that lenses (from functional programming) form a dialectica category, which have an existing game-semantic interpretation. In the second half of this paper, we apply these ideas to build a compact closed category of `computable open games' by replacing the underlying dialectica category with a wave-style geometry of interaction category, specifically the Int-construction applied to the cartesian monoidal category of directed-complete partial orders.