Agreement Theorem Holds Even Without Reality

A fundamental principle about rational disagreement, long thought to rely on an objective reality, has been shown to hold even when such a reality is absent. In a paper published on arXiv, researchers Carlo Cepollaro and Andrea Di Biagio extend Aumann's agreement theorem—a classic result stating that two rational agents with a common prior cannot disagree if their beliefs are common knowledge—to quantum mechanics, indefinite causal orders, and hypothetical postquantum theories. This operational version of the theorem removes the need for an underlying state of the world, focusing solely on observable measurement outcomes, and reveals that the core of agreement lies in probabilistic consistency rather than ontological assumptions. The finding has for understanding consensus in complex systems, from quantum computing to artificial intelligence, where traditional notions of reality may not apply.

The key finding is that rational agents must assign the same probabilities to an event if they share a common prior and their posterior beliefs are common knowledge, regardless of whether an objective reality exists. The researchers derived an operational agreement theorem that applies directly to measurement outcomes, without referencing a true state of the world. In classical scenarios, this aligns with the original Aumann's theorem, but the new formulation extends it to noncommuting quantum measurements, situations with indefinite causal order, and even postquantum frameworks. For example, in a quantum setup with noncommuting measurements on a four-dimensional system, the theorem ensures that whenever Alice and Bob have common knowledge of their posteriors—such as when Alice observes a specific outcome and infers Bob's—their assigned probabilities must coincide, as illustrated in the paper with specific probability values like qA and qB from equations (15).

Ology involves reformulating the agreement theorem in terms of joint probability distributions over measurement outcomes, rather than an underlying state space. Alice, Bob, and a third measurement (for event E) have outcomes i, j, and k, respectively, with a shared prior p(i, j, k). The researchers defined sets of outcomes where agents assign specific posteriors (e.g., A0 and B0 from equation (6)) and iteratively built common knowledge conditions (An and Bn from equation (7)). The proof, detailed in Theorem 2, shows that if posteriors are common knowledge—meaning outcomes satisfy the intersection condition in equation (8)—then the posteriors must be equal, leveraging Bayesian conditioning and the existence of a joint distribution. This approach is agnostic to how probabilities are derived, allowing application to quantum theory where p(i, j, k) might come from trace formulas like equation (12), or to indefinite causal order scenarios using process matrices as in equation (17).

Analysis from the paper demonstrates the theorem's validity across diverse settings. In quantum theory, even with noncommuting measurements, the joint probability distribution ensures agreement when common knowledge holds; for instance, in the example with specific parameters (θ, φ, q, r), posteriors qA and qB match under common knowledge conditions. The researchers also show that the theorem applies to measurements in any order, such as when event E occurs between Alice's and Bob's measurements, using probability assignments like equation (16). Beyond quantum mechanics, the theorem holds for indefinite causal order described by W-matrices (equation (17)) and postquantum theories like generalized probabilistic theories, provided a joint distribution exists. The paper contrasts this with previous literature, noting that apparent contradictions arise from different assumptions, such as Contreras–Tejada et al.'s focus on counterfactual events or Díaz et al.'s nonstandard update rules for noncommuting measurements.

Of this work are significant for fields relying on probabilistic reasoning and consensus-building. In quantum computing and cryptography, where agents may need to agree on outcomes without assuming a shared reality, the theorem provides a foundation for secure protocols and error correction. For artificial intelligence and multi-agent systems, it offers insights into how rational entities can reach agreement in environments with uncertain or non-classical dynamics, such as in robotics or network coordination. The research also touches on foundational debates in physics, suggesting that agreement among observers does not require an objective ontology, aligning with interpretations like QBism that emphasize information over reality. This could influence how scientists model complex systems in cosmology or particle physics, where traditional causal structures may break down.

Limitations of the study, as noted in the paper, arise in scenarios where no joint probability distribution can be meaningfully assigned. The researchers identify Wigner's friend-type situations—thought experiments involving nested observers—as a potential failure point for the agreement theorem. In such cases, outcomes from different observers may not be treated as jointly existing events, challenging the consistency assumptions underlying the theorem. The paper references recent discussions on Wigner's friend paradoxes, suggesting that these scenarios provide a natural setting to explore departures from standard agreement frameworks. Additionally, the theorem assumes finite measurement outcomes and a common prior, which may not hold in all practical applications, though extensions to infinite outcomes are mentioned in the context of prior literature. Future work could investigate these edge cases to further understand the boundaries of rational agreement in quantum and postquantum contexts.