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

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 coalgebraic approach to the construction of assumption-complete models.