Some Non-Classical Approaches to the Branderburger-Keisler Paradox

In this paper, we discuss a well-known self-referential paradox in foundational game theory, the Brandenburger - Keisler paradox. We approach the paradox from two different perspectives: non-well-founded set theory and paraconsistent logic. We show that the paradox persists in both frameworks for category theoretical reasons, but, with different properties.

💡 Research Summary

The paper revisits the Brandenburger‑Keisler (BK) paradox – a self‑referential inconsistency that arises in epistemic game theory when two players, Ann and Bob, each hold beliefs about the other’s beliefs and assumptions – using two non‑classical frameworks: non‑well‑founded set theory (NWF) and paraconsistent logic.

The authors begin by recalling the standard ZFC‑based formulation of the paradox. In the usual belief model ((U_a,U_b,R_a,R_b)) two modal operators are defined: a belief operator (\Box) (written (\bigcirc) in the original) and an assumption operator (\Diamond) (written (\heartsuit)). The BK sentence “Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong” leads to a contradiction, showing that no belief model can be complete for the first‑order language that contains the relevant predicates.

1. Non‑well‑founded set theory approach
The paper replaces the axiom of foundation with Aczel’s Anti‑Foundation Axiom, allowing sets that contain themselves. A model is now a pair (M=(W,V)) where (W) is a non‑well‑founded “hyper‑set” and (V) assigns propositional variables to elements of (W). The semantics of the basic modal operators are given by membership:

• (M,w\models^{+}\Diamond\varphi) iff there exists (v\in w) with (M,v\models^{+}\varphi).
• (M,w\models^{+}\Box\varphi) iff for all (v\in w), (M,v\models^{+}\varphi).

Using these, belief and assumption modalities are re‑defined as:
\