Yes, defining what “winning” means is indeed part of the problem of evaluating decision theories, and is largely the difference between many decision theories.
One way of defining “winning” is getting the most you can (on expectation) given the information you have at a given instant of time. In the Counterfactual Mugging situation where you are asked for $100, winning obviously means not paying by this definition. In the other situation you have no choices to make and so automatically “win” because in that local situation there is no better action you can take.
Another way of defining “winning” is getting the most you can (on expectation) across all scenarios weighted by their probabilities. In Counterfactual Mugging, this means paying up because the $100 loss in half the scenarios by probability is greatly outweighed by the $1000 gain in the other half.
Note that the scenario is always presented as “Omega asks you to pay up”. While this is the only scenario in which you get to make a decision, it also biases perception of the problem by directing attention away from the equally prevalent scenarios in which Omega just turns up and (if you would have paid) gives you $1000 or (otherwise) tells you that you get nothing.
Yes, defining what “winning” means is indeed part of the problem of evaluating decision theories, and is largely the difference between many decision theories.
One way of defining “winning” is getting the most you can (on expectation) given the information you have at a given instant of time. In the Counterfactual Mugging situation where you are asked for $100, winning obviously means not paying by this definition. In the other situation you have no choices to make and so automatically “win” because in that local situation there is no better action you can take.
Another way of defining “winning” is getting the most you can (on expectation) across all scenarios weighted by their probabilities. In Counterfactual Mugging, this means paying up because the $100 loss in half the scenarios by probability is greatly outweighed by the $1000 gain in the other half.
Note that the scenario is always presented as “Omega asks you to pay up”. While this is the only scenario in which you get to make a decision, it also biases perception of the problem by directing attention away from the equally prevalent scenarios in which Omega just turns up and (if you would have paid) gives you $1000 or (otherwise) tells you that you get nothing.