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arXiv:2202.02156 · Quantum

Generalising Aumann's Agreement Theorem

Matthew Leifer, Cristhiano Duarte

Published Sep 18, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.22331/q-2026-09-18-2210

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Agreement theorems are no-go results about rational disagreement: if two agents start from a common prior and their posterior beliefs are common knowledge, they cannot assign different probabilities to the same event. Standard treatments of the result have the agents reason about an underlying state of the world, which has lead some to ask whether the result can extend to quantum or postquantum phenomena, where such a description may no longer be appropriate. We derive an operational version of Aumann's agreement theorem without assuming an objective state of the world and instead focusing only on what is observed. This allows us to establish the theorem's validity in quantum theory and even in situations with indefinite causal order or involving hypothetical postquantum phenomena. We comment on seemingly contradictory results in the literature and point to the one place where the theorem might fail: Wigner's friend-type situations.
Intersubjective consistency asks when different observers of a physical system, reasoning from a shared theory, will agree on their probability assessments. Classical probability theory provides a benchmark in the form of the Agreement Theorem (Aumann, 1976). We study the quantum mechanics (QM) analog to the classical Agreement Theorem in a finite-dimensional tripartite setting in which two observers obtain information via local projective measurements. We represent their mutual awareness by common certainty, which is formally an infinite hierarchy of certainty operators concerning the probability of an event of interest. We first prove that if the observers' measurements commute with one another and with the property of interest, common certainty forces their probability assessments to coincide, thereby recovering a QM analog to the classical Agreement Theorem. We then construct a one-parameter family of qutrit-qubit-qubit models in which one observer's measurement does not commute with the property and common certainty of disagreement occurs -- a distinctively QM phenomenon. The mechanism is coherence between branches distinguished by one but not the other observer's measurement. When measurement outcomes are stored in a classical register, the resulting dephasing removes this coherence and restores agreement. Finally, we prove that QM does not permit the extreme 0-1 configuration in which Alice is certain of a property and is also certain that Bob is certain of its negation. These results aim to turn debate in QM about observer-dependent facts into precise mathematical conditions for when intersubjective agreement or disagreement in QM can be sustained.
, 1236-1239. (doi:10.1214/aos/1176343654)) states that if two Bayesian agents start with a common prior, then they cannot have common knowledge that they hold different posterior probabilities of some underlying event of interest. In short, the two agents cannot 'agree to disagree'. This result applies in the classical domain where classical probability theory applies. But in non-classical domains, such as the quantum world, classical probability theory does not apply. Inspired principally by their use in quantum mechanics, we employ signed probabilities to investigate the epistemics of the non-classical world. We find that here, too, it cannot be common knowledge that two agents assign different probabilities to an event of interest. However, in a non-classical domain, unlike the classical case, it can be common certainty that two agents assign different probabilities to an event of interest. Finally, in a non-classical domain, it cannot be common certainty that two agents assign different probabilities, if communication of their common certainty is possible-even if communication does not take place. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.
Is the world quantum? An active research line in quantum foundations is devoted to exploring what constraints can rule out the postquantum theories that are consistent with experimentally observed results. We explore this question in the context of epistemics, and ask whether agreement between observers can serve as a physical principle that must hold for any theory of the world. Aumann's seminal Agreement Theorem states that two observers (of classical systems) cannot agree to disagree. We propose an extension of this theorem to no-signaling settings. In particular, we establish an Agreement Theorem for observers of quantum systems, while we construct examples of (postquantum) no-signaling boxes where observers can agree to disagree. The PR box is an extremal instance of this phenomenon. These results make it plausible that agreement between observers might be a physical principle, while they also establish links between the fields of epistemics and quantum information that seem worthy of further exploration.