Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalising Aumann's Agreement Theorem

Matthew Leifer, Cristhiano Duarte

TL;DR

The paper demonstrates that Aumann's agreement theorem—rational agents with common priors cannot 'agree to disagree' when outcomes are common knowledge—extends beyond classical probability to hybrid quantum-classical settings and generalised probabilistic theories (GPTs). It achieves this by formalising a fixed knowledge model and using conditioning notions that generalise the law of total probability: in the quantum case with density-operator valued measures (DOVMs) and in GPTs with state-valued measures (SVMs). The main results show that, under common knowledge, agents' posteriors (or updated states) must coincide, i.e., $\sigma_1=\cdots=\sigma_N=\rho_{|C(E)}$ (quantum) and $\mu_1=\cdots=\mu_N=\mu_{|C(E)}$ (GPTs). The work clarifies that the impossibility is a mathematical property of probability theories over knowledge models, not a physical principle distinguishing theories, and it discusses limitations, comparisons with related work, and directions for dynamic epistemic extensions. These findings deepen our understanding of how common knowledge constrains epistemic updates across broad probabilistic frameworks with potential implications for quantum foundations and decision theory.

Abstract

According to Aumann's celebrated theorem, rational agents cannot agree to disagree. In other words, agents who once shared a common prior probability distribution and who have common knowledge about their posteriors cannot assign different probability distributions to a given proposition. Common knowledge imposes strong restrictions on assigned probabilities. In fact, Aumann's agreement theorem was one of the first attempts to formalise and explore the role played by common knowledge in decision theory. Recently, the debate over possible (quantum) extensions of Aumann's results has resurfaced. This paper contributes to this discussion. First, we argue that agreeing to disagree is impossible in quantum theory. Secondly, by building on the quantum argument, we show that agreeing to disagree is also forbidden in any generalised probability theory. The upshot is that in its probabilistic version, the agreement theorem is a direct consequence of how we choose to condition upon acquiring new information.

Generalising Aumann's Agreement Theorem

TL;DR

The paper demonstrates that Aumann's agreement theorem—rational agents with common priors cannot 'agree to disagree' when outcomes are common knowledge—extends beyond classical probability to hybrid quantum-classical settings and generalised probabilistic theories (GPTs). It achieves this by formalising a fixed knowledge model and using conditioning notions that generalise the law of total probability: in the quantum case with density-operator valued measures (DOVMs) and in GPTs with state-valued measures (SVMs). The main results show that, under common knowledge, agents' posteriors (or updated states) must coincide, i.e., (quantum) and (GPTs). The work clarifies that the impossibility is a mathematical property of probability theories over knowledge models, not a physical principle distinguishing theories, and it discusses limitations, comparisons with related work, and directions for dynamic epistemic extensions. These findings deepen our understanding of how common knowledge constrains epistemic updates across broad probabilistic frameworks with potential implications for quantum foundations and decision theory.

Abstract

According to Aumann's celebrated theorem, rational agents cannot agree to disagree. In other words, agents who once shared a common prior probability distribution and who have common knowledge about their posteriors cannot assign different probability distributions to a given proposition. Common knowledge imposes strong restrictions on assigned probabilities. In fact, Aumann's agreement theorem was one of the first attempts to formalise and explore the role played by common knowledge in decision theory. Recently, the debate over possible (quantum) extensions of Aumann's results has resurfaced. This paper contributes to this discussion. First, we argue that agreeing to disagree is impossible in quantum theory. Secondly, by building on the quantum argument, we show that agreeing to disagree is also forbidden in any generalised probability theory. The upshot is that in its probabilistic version, the agreement theorem is a direct consequence of how we choose to condition upon acquiring new information.
Paper Structure (7 sections, 7 theorems, 35 equations, 1 figure)

This paper contains 7 sections, 7 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Aumann's Original Argument
  3. Quantum Version of the Agreement Theorem
  4. Aumann's Theorem in GTPs
  5. Limitations
  6. Comparison to other works
  7. Conclusions

Key Result

Lemma 4

Let $(\Omega, Q_1, Q_2,...,Q_N, \Sigma)$ be a knowledge model and $E$ an event. If $C(E) \neq \emptyset$, then for each $i \in [N]$ there exists a finite family $\{D_{i}^{1},...,D_{i}^{k_i}\} \subseteq Q_i$ such that for every $i \in [N]$ and for every $l \neq l' \in [k_{i}]$

Figures (1)

  • Figure 1: Knowledge model for two agents: yellow and orange (colours online). Regardless of the state of the world $\omega \in \Omega$, neither agent can know the purple $E$ event. Similarly, only the orange agent can know the green $F$ event.

Theorems & Definitions (21)

  • definition 1: Knowledge Model
  • definition 2: Knowledge
  • definition 3: Mutual and Common Knowledge
  • Lemma 4
  • proof
  • Theorem 5: Aumann's Agreement Theorem
  • proof
  • Theorem 6: Aumann's Theorem - Second Version
  • definition 7: DOVM
  • definition 8: Conditional State
  • ...and 11 more