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CannibalSmith comments on The Aumann's agreement theorem game (guess 2/3 of the average) - Less Wrong
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Naively appealing, but if the third step is "so you'll pick 44 instead", the first step claiming that "You will pick 100, and I know that" is incorrect.
Okay.
You will pick X - a number whose value I don't know. I know that, so I'll pick X*2/3. You know that I know that, so you'll pick X*4/9 instead. But I know that you know that I know that, so I'll pick X*8/27 instead. But you know that I know that you know that I know that, so you'll pick X*16/81 instead. But I know-
This continues to infinity until X is multiplied by 0. At this point the value of X doesn't matter.
Nope, it's the same argument. You can't know that I pick X and at the same time know that I pick X*4/9 instead. From the outset, you can't assume to know precisely what I pick, and considering all possible values that I pick and you know I pick (a set of situations indexed by X) doesn't fix that.