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
So, this straightforward calculation tells that you ought to give up your $100. It looks like a good idea before the coin toss, but it starts to look like a bad idea after the coin came up tails. Had you known about the deal in advance, one possible course of action would be to set up a precommitment. You contract a third party, agreeing that you'll lose $1000 if you don't give $100 to Omega, in case it asks for that. In this case, you leave yourself no other choice.
But in this game, explicit precommitment is not an option: you didn't know about Omega's little game until the coin was already tossed and the outcome of the toss was given to you. The only thing that stands between Omega and your 100$ is your ritual of cognition. And so I ask you all: is the decision to give up $100 when you have no real benefit from it, only counterfactual benefit, an example of winning?
P.S. Let's assume that the coin is deterministic, that in the overwhelming measure of the MWI worlds it gives the same outcome. You don't care about a fraction that sees a different result, in all reality the result is that Omega won't even consider giving you $10000, it only asks for your $100. Also, the deal is unique, you won't see Omega ever again.
Well, if we're designing an AI now, then we have the capability to make a binding precommitment, simply by writing code. And we are still in a position where we can hope for the coin to come down heads. So yes, in that privileged position, we should bind the AI to pay up.
However, to the question as stated, "is the decision to give up $100 when you have no real benefit from it, only counterfactual benefit, an example of winning?" I would still answer, "No, you don't achieve your goals/utility by paying up." We're specifically told that the coin has already been flipped. Losing $100 has negative utility, and positive utility isn't on the table.
Alternatively, since it's asking specifically about the decision, I would answer, If you haven't made the decision until after the coin comes down tails, then paying is the wrong decision. Only if you're deciding in advance (when you still hope for heads) can a decision to pay have the best expected value.
Even if deciding in advance, though, it's still not a guaranteed win, but rather a gamble. So I don't see any inconsistency in saying, on the one hand, "You should make a binding precommitment to pay", and on the other hand, "If the coin has already come down tails without a precommitment, you shouldn't pay."
If there were a lottery where the expected value of a ticket was actually positive, and someone comes to you offering to sell you their ticket (at cost price), then it would make sense in advance to buy it, but if you didn't, and then the winners were announced and that ticket didn't win, then buying it no longer makes sense.
You're fundamentally failing to address the problem.
For one, your examples just plain omit the "Omega is a predictor" part, which is key to the situation. Since Omega is a predictor, there is no distinction between making the decision ahead of time or not.
For another, unless you can prove that your proposed alternative doesn't have pathologies just as bad as the Counterfactual Mugging, you're at best back to square one.
It's very easy to say "look, just don't do the pathological thing". It's very hard to formalize that into an actual dec... (read more)