Bayesian open games

TL;DR

The paper addresses the limitations of deterministic, complete-information open games in Ghani et al. by introducing stochastic environments, stochastic actions, and incomplete information handled via Bayesian reasoning. It develops a probabilistic open-game framework by using the probability monad $D$ and existential/coend lenses, embedding behavioural strategies into a monoidal categorical setting. Incomplete information is treated with a Harsanyi-type transformation to games of imperfect information, culminating in a Bayesian Nash equilibrium concept and a three-part context $(\Theta, D(\Theta \times X), \Theta \times Y \to D(R))$ related to double lenses. The resulting framework provides a unified, compositional approach to Bayesian open games, enabling modular reasoning about stochastic environments and information asymmetries with diagrammatic tooling.

Abstract

This paper generalises the treatment of compositional game theory as introduced by Ghani et al. in 2018, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the notion of a game in the work by Ghani et al. is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issues all at once.