Counterfactual mugging is a thought experiment for testing and differentiating decision theories, stated as follows:

Omega, a perfect predictor, flips a coin. If it comes up tails Omega asks you for $100. If it comes up heads, Omega pays you $10,000 if it predicts that you would have paid if it had come up tails.

Depending on how the problem is phrased, intuition calls for different answers. For example, Eliezer Yudkowsky has argued that framing the problem in a way Omega is a regular aspect of the environment which regularly asks such types of questions makes most people answer 'Yes'. However, Vladimir Nesov points out that Rationalists Should Win could be interpreted as suggesting that we should not pay. After all, even though paying in the tails case would cause you to do worse in the counterfactual where the coin came up heads, you already know the counterfactual didn't happen, so it's not obvious that you should pay. This issue has been discussed in this question.

Formal decision theories also diverge. For Causal Decision Theory, you can only affect those probabilities that you are causally linked to. Hence, the answer should be 'No'. In Evidential Decision Theory any kind of connection is accounted, then the answer should be 'No'. Timeless Decision Theory answer seems undefined, however Yudkowsky has argued that if the problem is recurrently presented, one should answer 'Yes' on the basis of enhancing its probability of gaining $10000 in the next round. This seems to be Causal Decision Theory prescription as well. Updateless decision theory 1 prescribes giving the $100, on the basis your decision can influence both the 'heads branch' and 'tails branch' of the universe.

Regardless of the particular decision theory, it is generally agreed that if you can pre-commit in advance that you should do so. The dispute is purely over what you should do if you didn't pre-commit.

Eliezer listed this in his 2009 post Timeless Decision Theory Problems I can't Solve, although that was written before Updateless Decision Theory.

Variants

The Counterfactual Prisoner's Dilemma is a symmetric variant of he original independently suggested by Chris Leong and Cousin_it:

Omega, a perfect predictor, flips a coin. If if comes up heads, Omega asks you for $100, then pays you $10,000 if it predict you would have paid if it had come up tails and you were told it was tails. If it comes up t

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