AI Judge changed title after evaluation, original title: "Axiom Math and Scott Kominers use Lean to formally verify Aumann's Agreement Theorem"
The theorem proves that rational agents cannot agree to disagree
Carina Hong#1102
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
Many users praised Axiom Math AI's formalization of the Aumann Agreement Theorem because it makes assumptions clearer, supports theorem proving in economics, and could help popularize tools like Lean.
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Fabulous! I argued a couple years back that journals will require formal proofs very soon, as AI reduces the cost of doing so. Huge value! (Alternative: us MEDS PhD students back in the day had an Aumann shrine + many discussions of charges vs countable additivity. No joke!)
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
This next year will be major for economic theory.
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
For economics, in particular, there's a deeper takeaway:
Formalization isn't about replacing intuition. It's about assumption accounting – making transparent which modeling choices carry which logical consequences.
Theorists of the world, unite!
Super exciting. Can’t wait to see where this will take economic theory.
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
In short:
Formalization helps us agree about Agreement, QED. 🤝
Co-authors: ➕ @chen_ruize3962 ➕ Ben Eltschig ➕ @KenOno691 ➕ Jujian Zhang
📜 Paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6837298
💾 Code: https://github.com/AxiomMath/AgreeToDisagree/
This is cool!
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
We also formalized the Monderer–Samet (1989) p-belief version, with proving assistance from @axiommathai's AxiomProver.
This illustrates the power of composability: Once the primitives (beliefs, partitions, etc.) are formal, we can automate probing for and proving extensions.
This perspective leads to a modular version of Aumann's theorem.
Rather than starting from a full common prior assumption μ₁ = μ₂, we isolate the minimal requirement:
μ₁(E | C) = μ₂(E | C).
The common-prior theorem then becomes a corollary.
Formalization clarifies which assumptions are logically essential and which are structural shortcuts.
This is a fantastic thread
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
formal proofs are entering econ now
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
It’s certainly conceptually interesting to make implicit assumptions in economic models explicit, but I’m not sure how useful it is, since most economic modeling assumptions are violated in real life (e.g., rational agents, equilibrium, …).
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
One way might have been -- @axiommathai needed basically one pass to formalize the final proof -- and bulk of the authorship work was unpacking/interpreting the proof, noticing "assumption playing two roles" -- and other teachable moments for readers (e.g., your contrasts with how econs usually teach Aumann). 2/
(And with Lean formalization, we can be fully confident in the output of such automated proof searches – if there's an error anywhere, then the proof won't compile 😉.)
too cool
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
QED 🙌
Super excited to share joint work with @axiommathai that kicks off a broader project of formalization in economics.
Aumann's celebrated theorem says we can't "agree to disagree."
But what does that actually mean – formally? 👀
Thanks so much! For what it's worth, the way I think about it is that surfacing assumptions – and, in particular, the way Lean makes it possible to track which parts of the economic argument rests on which parts of which assumptions – is valuable precisely because it lets us reason about whether the key assumptions are "close enough" to accurate for a practical economic situation were trying to understand.
@skominers @axiommathai Alternatively, maybe it required a lot of back and forth with the @axiommathai software, and/or more automated guidance about what aspects were interesting to surface. /3
@skominers @axiommathai Wow, that is super neat!! How long did it take you to formalize the proof? How long to learn how to use lean? Do you think this is realistic to do for a paper before submitting it?
The classic result: Bayesian agents with a common prior can't have common knowledge of different posteriors.
I.e., if we start with the same background knowledge and fully learn each other's beliefs, then disagreements can't persist.
(A foundation of rational expectations!)
@skominers @axiommathai Finally, I'm curious how you settled on Aumann (1976). Did you have a hunch something was "there"? 4/4
@skominers @axiommathai @skominers this is awesome. I'm interested to hear more about the result & paper came about. For example --
