An application of Aumann's agreeing-to-disagree result: certain kinds of war between rational countries are puzzling.

Since war involves destruction, better for one to surrender and bargain. But maybe each believes it'll get more by fighting? Suppose a country fights iff

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it thinks it will win with prob > 0.5. In the notation below, let X = indicator that I win. Then war means it's common knowledge that Y>.5>Z.

That's impossible, but the proof below doesn't quite show it. Exercise: show there's no CK event E on which Y>Z. https://twitter.com/ben_golub/status/1318231515797594113?s=20

War here is like speculative trade (cf. no-trade theorem): it can't happen due to different information ALONE, because by Aumann both of us can't rationally expect to win.

Thus, if war is zero-sum or worse, it entails irrationality or different priors.

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(A darker possible explanation is that, taking only elites' interests into account, war actually generates "surplus" -- both would agree to it even if neither expected to win.)

And, to be clear, this is a teaser, suitable for putting on a problem set, for a big and deep literature -- see https://www.cambridge.org/core/journals/international-organization/article/rationalist-explanations-for-war/ for an introduction... this is an active literature -- see e.g., https://onlinelibrary.wiley.com/doi/full/10.1111/j.1540-5907.2007.00278.x and https://www.journals.uchicago.edu/doi/full/10.1086/707767