A generous patron makes an offer to you. You are to pick out a positive integer n and you will get 2n units of value. You have the ability to pick out any positive integer at no cost to yourself (maybe you can engage in a supertask and name long numbers really fast).
You think about naming a million, but then a billion would pay so much better, and a billion and two is four times better! You agonize. And then you have a brilliant idea. You will randomize by choosing positive integer n with probability 2−n (say, by flipping a coin until you get heads and counting how many flips that took). Your expected payoff will be
- (1/2)(2) + (1/4)(4) + (1/8)(8) + ... = ∞.
That beats any specific number you could choose. So you go for it.
And, poof, you get 4. Regrets! You don’t want to stick to what the random choice gave you, as you’ll “only” get 24 = 16 units of value. Disappointing! So you try again. You choose another positive integer. Now it is, mirabile dictu, a billion and two. But you think: 21000000002 may be a lot, but infinity is more, and if you randomly choose another number, your expected payoff is ∞. So you randomly choose again. And whatever you get, you are dissatisfied.
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