Counterfactual Mugging

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

Omega, a perfect predictor, flips a coin. If if 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 in 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....

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