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Uncommon Belief in Rationality

Qi Shi, Pavel Naumov

TL;DR

This paper introduces RBR graphs, a graph-based language to model higher-order beliefs about rationality beyond common knowledge, capturing both real and doxastic agents and their belief hierarchies. It defines doxastic rationalisability as the natural extension of rationalisability to uncommon RBR systems and proves key equivalence results that tie node-level beliefs to global graph equivalence. A partition-refinement algorithm is developed to compute a canonical, minimal equivalent RBR graph, with time complexity $O(|\mathcal{A}|\cdot|N|^2\cdot\log|N|)$, ensuring uniqueness up to isomorphism. The framework provides a principled, scalable way to reason about bounded rationality and nested beliefs, with potential applications in epistemic game theory and multi-agent systems.

Abstract

Common knowledge/belief in rationality is the traditional standard assumption in analysing interaction among agents. This paper proposes a graph-based language for capturing significantly more complicated structures of higher-order beliefs that agents might have about the rationality of the other agents. The two main contributions are a solution concept that captures the reasoning process based on a given belief structure and an efficient algorithm for compressing any belief structure into a unique minimal form.

Uncommon Belief in Rationality

TL;DR

This paper introduces RBR graphs, a graph-based language to model higher-order beliefs about rationality beyond common knowledge, capturing both real and doxastic agents and their belief hierarchies. It defines doxastic rationalisability as the natural extension of rationalisability to uncommon RBR systems and proves key equivalence results that tie node-level beliefs to global graph equivalence. A partition-refinement algorithm is developed to compute a canonical, minimal equivalent RBR graph, with time complexity , ensuring uniqueness up to isomorphism. The framework provides a principled, scalable way to reason about bounded rationality and nested beliefs, with potential applications in epistemic game theory and multi-agent systems.

Abstract

Common knowledge/belief in rationality is the traditional standard assumption in analysing interaction among agents. This paper proposes a graph-based language for capturing significantly more complicated structures of higher-order beliefs that agents might have about the rationality of the other agents. The two main contributions are a solution concept that captures the reasoning process based on a given belief structure and an efficient algorithm for compressing any belief structure into a unique minimal form.

Paper Structure

This paper contains 17 sections, 53 theorems, 24 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Rationality and Rationalisation
  3. Terminologies and Solution Concept
  4. Equivalence in RBR Graphs
  5. Minimisation of RBR Graphs
  6. Concluding Discussion
  7. Additional Notations and Properties
  8. Rational Solution and Theorem \ref{['th:rational solution']}
  9. Doxastic Equivalence Between Nodes
  10. Necessity
  11. Sufficiency
  12. Equivalence Between RBR Graphs
  13. Minimisation of RBR Graphs
  14. Canonical RBR Graph
  15. Partition Sequence of the Node Set
  16. ...and 2 more sections

Key Result

Theorem 1

There is an integer $i\geq 0$ such that $\mathbb{S}=\mathbb{R}^j(S^{\Delta})$ for each integer $j\geq i$.

Figures (2)

  • Figure 1: Different RBR graphs among agents $a,b,c$.
  • Figure 2: The collection of individual beliefs in Figure \ref{['fig:3AgentUncommonBelief']}.

Theorems & Definitions (72)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 62 more