Generalising Aumann’s Agreement Theorem
1Schmid College of Science and Technology, Chapman University, One University Dr., Orange, CA 92866, USA
2Institute for Quantum Studies, Chapman University, One University Dr., Orange, CA 92866, USA
3Instituto de Física, Universidade Federal da Bahia, Campus de Ondina, Rua Barão do Geremoabo, s.n., Ondina, Salvador, BA 40210-340, Brazil
4Universidade Federal de Juiz de Fora, Departamento de Física, Juiz de Fora, MG, Brasil
| Published: | 2026-09-18, volume 10, page 2210 |
| Editor: | Alexander Wilce |
| Eprint: | arXiv:2202.02156v4 |
| Doi: | https://doi.org/10.22331/q-2026-09-18-2210 |
| Citation: | Quantum 10, 2210 (2026). |
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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.
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► References
[1] Scott Aaronson ``The complexity of agreement'' Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC ’05 634 (2005).
https://doi.org/10.1145/1060590.1060686
http://portal.acm.org/citation.cfm?doid=1060590.1060686
[2] Robert J. Aumann ``Agreeing to Disagree'' The Annals of Statistics 4 (1976).
https://doi.org/10.1214/aos/1176343654
[3] Maria Baghramianand J. Adam Carter ``Relativism'' Metaphysics Research Lab, Stanford University https://plato.stanford.edu/archives/spr2021/entries/relativism/ (2021).
https://plato.stanford.edu/archives/spr2021/entries/relativism/
[4] Giacomo Bonannoand Klaus Nehring ``Agreeing to Disagree: A Survey'' (1997).
http://hdl.handle.net/10419/189462
[5] Kok-Wei Bong, Aníbal Utreras-Alarcón, Farzad Ghafari, Yeong-Cherng Liang, Nora Tischler, Eric G. Cavalcanti, Geoff J. Pryde, and Howard M. Wiseman, ``A strong no-go theorem on the Wigner’s friend paradox'' Nature Physics 16, 1199–1205 (2020).
https://doi.org/10.1038/s41567-020-0990-x
[6] Adam Brandenburger, Patricia Contreras-Tejada, Pierfrancesco La Mura, Giannicola Scarpa, and Kai Steverson, ``Agreement and disagreement in a non-classical world'' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 382, 20230004 (2024).
https://doi.org/10.1098/rsta.2023.0004
[7] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner, ``Bell nonlocality'' Rev. Mod. Phys. 86, 419–478 (2014).
https://doi.org/10.1103/RevModPhys.86.419
[8] Dagmar Bruss ``Characterizing entanglement'' Journal of Mathematical Physics 43, 4237–4251 (2002).
https://doi.org/10.1063/1.1494474
[9] Eric G. Cavalcanti ``The View from a Wigner Bubble'' Foundations of Physics 51 (2021).
https://doi.org/10.1007/s10701-021-00417-0
[10] Carlo Cepollaroand Andrea Di Biagio ``Aumann's theorem beyond ontology: quantum, postquantum, and indefinite causal order'' (2026).
arXiv:2603.23595
[11] Patricia Contreras-Tejada, Giannicola Scarpa, Aleksander M. Kubicki, Adam Brandenburger, and Pierfrancesco La Mura, ``Observers of quantum systems cannot agree to disagree'' Nature Communications 12, 7021 (2021).
https://doi.org/10.1038/s41467-021-27134-6
https://www.nature.com/articles/s41467-021-27134-6
[12] L. Demey ``Believing in Logic and Philosophy. PhD thesis, KU Leuven.'' (2014).
[13] Lorenz Demey ``Agreeing to disagree in probabilistic dynamic epistemic logic'' Synthese 191, 409–438 (2014).
https://doi.org/10.1007/s11229-013-0280-6
[14] Hans van Ditmarsch, Wiebe van der Hoek, and Barteld Kooi, ``Dynamic epistemic logic'' Springer (2008).
[15] Cédric Dégremontand Oliver Roy ``Agreement Theorems in Dynamic-Epistemic Logic'' Journal of Philosophical Logic 41, 735–764 (2012).
https://doi.org/10.1007/s10992-012-9236-4
[16] María García Díaz, Adam Brandenburger, and Giannicola Scarpa, ``The Quantum Agreement Theorem'' (2026).
arXiv:2511.21258
[17] Christopher A. Fuchs ``Quantum Mechanics as Quantum Information (and only a little more)'' (2002).
https://arxiv.org/abs/quant-ph/0205039
[18] John D Geanakoplosand Heraklis M Polemarchakis ``We can’t disagree forever'' Journal of Economic Theory 28, 192–200 (1982).
https://doi.org/10.1016/0022-0531(82)90099-0
https://linkinghub.elsevier.com/retrieve/pii/0022053182900990
[19] Peter Janottaand Haye Hinrichsen ``Generalized probability theories: what determines the structure of quantum theory?'' Journal of Physics A: Mathematical and Theoretical 47, 323001 (2014).
https://doi.org/10.1088/1751-8113/47/32/323001
[20] Andrei Khrennikov ``Quantum version of Aumann’s approach to common knowledge: Sufficient conditions of impossibility to agree on disagree'' Journal of Mathematical Economics 60, 89–104 (2015).
https://doi.org/10.1016/j.jmateco.2015.06.018
[21] Andrei Khrennikovand Irina Basieva ``Possibility to agree on disagree from quantum information and decision making'' Journal of Mathematical Psychology 62–63, 1–15 (2014).
https://doi.org/10.1016/j.jmp.2014.09.003
[22] Harvey Lederman ``PEOPLE WITH COMMON PRIORS CAN AGREE TO DISAGREE'' The Review of Symbolic Logic 8, 11–45 (2015).
https://doi.org/10.1017/S1755020314000380
[23] M. S. Leiferand Robert W. Spekkens ``Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference'' Phys. Rev. A 88, 052130 (2013).
https://doi.org/10.1103/PhysRevA.88.052130
[24] Itamar Pitowsky ``Quantum probability - quantum logic'' Springer-Verlag (1989).
https://doi.org/10.1007/BFb0021186
[25] Martin Plávala ``General probabilistic theories: An introduction'' Physics Reports 1033, 1–64 (2023).
https://doi.org/10.1016/j.physrep.2023.09.001
https://linkinghub.elsevier.com/retrieve/pii/S0370157323002752
[26] Davide Poderini, Giovanni Rodari, George Moreno, Emanuele Polino, Ranieri Nery, Alessia Suprano, Cristhiano Duarte, Fabio Sciarrino, and Rafael Chaves, ``Device-independent witness for the nonobjectivity of quantum dynamics'' Phys. Rev. A 108, 032201 (2023).
https://doi.org/10.1103/PhysRevA.108.032201
[27] Duncan Pritchard ``Scepticism: a very short introduction'' Oxford University Press (2019).
[28] Duncan Pritchard ``What is this thing called knowledge?'' Routledge, Taylor Francis Group (2013).
[29] Carlo Rovelli ``‘Space is blue and birds fly through it’'' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20170312 (2018).
https://doi.org/10.1098/rsta.2017.0312
[30] Ariel Rubinsteinand Asher Wolinsky ``On the logic of “agreeing to disagree” type results'' Journal of Economic Theory 51, 184–193 (1990).
https://doi.org/10.1016/0022-0531(90)90057-Q
[31] Dov Samet ``Agreeing to disagree: The non-probabilistic case'' Games and Economic Behavior 69, 169–174 (2010) Special Issue In Honor of Robert Aumann.
https://doi.org/10.1016/j.geb.2008.09.032
https://www.sciencedirect.com/science/article/pii/S0899825608001759
[32] S. Samet ``The impossibility of agreeing to disagree: an extension of the sure-thing principle'' (2020).
https://www.tau.ac.il/~samet/papers/stp.pdf
[33] Julio C. F. Silva, B. F. Rizzuti, and Cristhiano Duarte, ``Agreement and Compatibility in Wigner's Friend Paradox'' (2026).
arXiv:2605.27424
[34] Susan Vineberg ``Dutch Book Arguments'' Metaphysics Research Lab, Stanford University https://plato.stanford.edu/archives/spr2016/entries/dutch-book/ (2016).
https://plato.stanford.edu/archives/spr2016/entries/dutch-book/
[35] Alexander Wilce ``Quantum Logic and Probability Theory'' Metaphysics Research Lab, Stanford University https://plato.stanford.edu/archives/sum2024/entries/qt-quantlog/ (2024).
https://plato.stanford.edu/archives/sum2024/entries/qt-quantlog/
Cited by
[1] John B. DeBrota and Peter J. Love, "Bayesian rational agents in iterated quantum games", Physical Review A 113 4, 042456 (2026).
[2] María García Díaz, Adam Brandenburger, and Giannicola Scarpa, "The Quantum Agreement Theorem", arXiv:2511.21258, (2025).
[3] Lucas L. Brugger, Cristhiano Duarte, and Bruno F. Rizzuti, "Recasting Schrödinger's Cat Thought Experiment as a Remote Measurement Problem", arXiv:2512.17991, (2025).
[4] Carlo Cepollaro and Andrea Di Biagio, "Aumann's theorem beyond ontology: quantum, postquantum, and indefinite causal order", arXiv:2603.23595, (2026).
[5] Adam Brandenburger, Patricia Contreras-Tejada, Pierfrancesco La Mura, Giannicola Scarpa, and Kai Steverson, "Agreement and disagreement in a non-classical world", Philosophical Transactions of the Royal Society of London Series A 382 2268, 20230004 (2024).
[6] Thales B. S. F. Rodrigues, Lucas L. Brugger, Vinicius G. Valle, Bruno F. Rizzuti, and Cristhiano Duarte, "Emergent Quantum Dynamics as a Bayesian Inference Problem: A Critical Analysis", arXiv:2605.04112, (2026).
[7] Julio C. F. Silva, B. F. Rizzuti, and Cristhiano Duarte, "Agreement and Compatibility in Wigner's Friend Paradox", arXiv:2605.27424, (2026).
The above citations are from SAO/NASA ADS (last updated successfully 2026-10-23 01:55:49). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref's cited-by service no data on citing works was found (last attempt 2026-10-23 01:55:47).
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