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pswoo comments on The Aumann's agreement theorem game (guess 2/3 of the average) - Less Wrong
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There is no "infinity" to be considered here.
We are given a single equation
P = (2/3)P
with the unique solution P=0.
P = (2/3)P
P-(2/3)P = 0
P(1-2/3) = 0
P(1/3) = 0
P=0
QED
As a general rule, you shouldn't even mention infinity except in very select circumstances. Especially not when the solution is so simple!
But the correct equation is Pwin = (2/3)Pavg.