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Alicorn comments on The Aumann's agreement theorem game (guess 2/3 of the average) - Less Wrong
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Comments (149)
If everybody reasons as you describe then everyone will guess 1/∞ and everyone will tie. You can't get closer to 2/3 of an infinitesimal than an infinitesimal, so it's stable.
Disclaimer: I'm not mathy. Maybe you actually can get closer to 2/3 of an infinitesimal than an infinitesimal.
The question required us to provide real numbers, and infinitesimals are not real numbers. Even if you allowed infinitesimals, though, 0 would still be the Nash equilibrium. After all, if 1/∞ is a valid guess, so is (1/∞)*(2/3), etc., so the exact same logic applies: any number larger than 0 is too large. The only value where everyone could know everyone else's choice and still not want to change is 0.
No, everyone who prefers to win outright will do the logic you just did and decide that 1/∞ is too small, for the reason you state.
Also, those of us who prefer to win outright will note that someone has to take one for the team and vote high to give the possibility of an outright winner, unless we believe cousin_it...
Reminds me of yet another semi-interesting game: everyone chooses a positive integer in base-10 notation, and the second highest number wins.
If the game as I've understood it is played over and over, I think it'd be much like rock, paper, scissors: http://www.cs.ualberta.ca/~darse/rsbpc.html
Really? I'd expect the numbers to spiral higher and higher.
After the first round, I wouldn't think there's much reason to guess higher than the previous highest number (or lower than the previous third number), which suggests convergence. If everyone updates based just on what other people did last time, then won't they cycle progressively closer around the initial second number? (Actually, I'm pretty sure the same holds even if people anticipate other's responses, provided they all reason forward the same number of steps.)
ETA: What happens If there is a tie for the highest number? Does the third highest guess win, or the two highest together? What if everyone guesses the same thing?
Eh? Why would anyone take one for the team? If it's bad to tie, surely it's worse to lose for the purpose of helping someone else win.
I should have explicitly stated this (you've caused me to realize that it's necessary for my whole line of reasoning), but it's also the case that losing is not a worse outcome for me than tying.
Ah. OK. I think that's generally supposed to be excluded by the set up (though in this case it was admittedly ambiguous). But given that you have those preferences, I agree that your reasoning makes sense.
There's no prize offered, but in theory, these people could collaborate to share whatever prize makes winning better than tying. Since there's no prize except winning, losing to help someone else does seem like it would be bad.