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Counterfactual Mugging and Logical Uncertainty
Followup to: Counterfactual Mugging.
Let's see what happens with Counterfactual Mugging, if we replace the uncertainty about an external fact of how a coin lands, with logical uncertainty, for example about what is the n-th place in the decimal expansion of pi.
The original thought experiment is as follows:
Omega appears and says that it has just tossed a fair coin, and given that the coin came up tails, it decided to ask you to give it $100. Whatever you do in this situation, nothing else will happen differently in reality as a result. Naturally you don't want to give up your $100. But Omega also tells you that if the coin came up heads instead of tails, it'd give you $10000, but only if you'd agree to give it $100 if the coin came up tails.
Let's change "coin came up tails" to "10000-th digit of pi is even", and correspondingly for heads. This gives Logical Counterfactual Mugging:
Omega appears and says that it has just found out what that 10000th decimal digit of pi is 8, and given that it is even, it decided to ask you to give it $100. Whatever you do in this situation, nothing else will happen differently in reality as a result. Naturally you don't want to give up your $100. But Omega also tells you that if the 10000th digit of pi turned out to be odd instead, it'd give you $10000, but only if you'd agree to give it $100 given that the 10000th digit is even.
This form of Counterfactual Mugging may be instructive, as it slaughters the following false intuition, or equivalently conceptualization of "could": "the coin could land either way, but a logical truth couldn't be either way".
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Counterfactual Mugging
Related to: Can Counterfactuals Be True?, Newcomb's Problem and Regret of Rationality.
Imagine that one day, Omega comes to you and says that it has just tossed a fair coin, and given that the coin came up tails, it decided to ask you to give it $100. Whatever you do in this situation, nothing else will happen differently in reality as a result. Naturally you don't want to give up your $100. But see, Omega tells you that if the coin came up heads instead of tails, it'd give you $10000, but only if you'd agree to give it $100 if the coin came up tails.
Omega can predict your decision in case it asked you to give it $100, even if that hasn't actually happened, it can compute the counterfactual truth. Omega is also known to be absolutely honest and trustworthy, no word-twisting, so the facts are really as it says, it really tossed a coin and really would've given you $10000.
From your current position, it seems absurd to give up your $100. Nothing good happens if you do that, the coin has already landed tails up, you'll never see the counterfactual $10000. But look at this situation from your point of view before Omega tossed the coin. There, you have two possible branches ahead of you, of equal probability. On one branch, you are asked to part with $100, and on the other branch, you are conditionally given $10000. If you decide to keep $100, the expected gain from this decision is $0: there is no exchange of money, you don't give Omega anything on the first branch, and as a result Omega doesn't give you anything on the second branch. If you decide to give $100 on the first branch, then Omega gives you $10000 on the second branch, so the expected gain from this decision is
-$100 * 0.5 + $10000 * 0.5 = $4950
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