From Lawvere to Brandenburger-Keisler: interactive forms of diagonalization and self-reference

TL;DR

This paper explores the Brandenburger-Keisler paradox in epistemic game theory, connecting it to Lawvere's diagonal argument, and develops a categorical and compositional framework for understanding self-reference and assumptions in multi-agent systems.

Contribution

It recasts the paradox as a fixpoint in regular categories, links it to Lawvere's diagonal argument, and provides a compositional, multi-agent generalization framework.

Findings

Reformulation of the paradox as a categorical fixpoint

Reduction to Lawvere's diagonal argument

Development of multi-agent generalizations

Abstract

We analyze the Brandenburger-Keisler paradox in epistemic game theory, which is a `two-person version of Russell's paradox'. Our aim is to understand how it relates to standard one-person arguments, and why the `believes-assumes' modality used in the argument arises. We recast it as a fixpoint result, which can be carried out in any regular category, and show how it can be reduced to a relational form of the one-person diagonal argument due to Lawvere. We give a compositional account, which leads to simple multi-agent generalizations. We also outline a general approach to the construction of assumption complete models.