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thrawnca comments on Counterfactual Mugging - Less Wrong
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I don't think so; I think the element of repetition substantially alters it - but in a good way, one that makes it more useful in designing a real-world agent. Because in reality, we want to design decision theories that will solve problems multiple times.
At the point of meeting a beggar, although my prospects of obtaining a gold coin this time around are gone, nonetheless my overall commitment is not meaningless. I can still think, "I want to be the kind of person who gives pennies to beggars, because overall I will come out ahead", and this thought remains applicable. I know that I can average out my losses with greater wins, and so I still want to stick to the algorithm.
In the single-shot scenario, however, my commitment becomes worthless once the coin comes down tails. There will never be any more 10K; there is no motivation any more to give 100. Following my precommitment, unless it is externally enforced, no longer makes any sense.
So the scenarios are significantly different.
So say it's repeated. Since our observable universe will end someday, there will come a time when the probability of future flips is too low to justify paying if the coin lands tails. Your argument suggests you won't pay, and by assumption Omega knows you won't pay. But then on the previous trial you have no incentive to pay, since you can't fool Omega about your future behavior. This makes it seem like non-payment propagates backward, and you miss out on the whole sequence.
I wouldn't trust myself to accurately predict the odds of another repetition, so I don't think it would unravel for me. But this comes back to my earlier point that you really need some external motivation, some precommitment, because "I want the 10K" loses its power as soon as the coin comes down tails.
This is the point of the thought experiment.
Omega is a predictor. His actions aren't just based on what you decide, but on what he predicts that you will decide.
If your decision theory says "nah, I'm not paying you" when you aren't given advance warning or repeated trials, then that is a fact about your decision theory even before Omega flips his coin. He flips his coin, gets heads, examines your decision theory, and gives you no money.
But if your decision theory pays up, then if he flips tails, you pay $100 for no possible benefit.
Neither of these seems entirely satisfactory. Is this a reasonable feature for a decision theory to have? Or is it pathological? If it's pathological, how do we fix it without creating other pathologies?
But in the single-shot scenario, after it comes down tails, what motivation does an ideal game theorist have to stick to the decision theory?
Like Parfit's hitchhiker, although in advance you might agree that it's a worthwhile deal, when it comes to the point of actually paying up, your motivation is gone, unless you have bound yourself in some other way.
That's what the problem is asking!
This is a decision-theoretical problem. Nobody cares about it for immediate practical purpose. "Stick to your decision theory, except when you non-rigorously decide not to" isn't a resolution to the problem, any more than "ignore the calculations since they're wrong" was a resolution to the ultraviolet catastrophe.
Again, the point of this experiment is that we want a rigorous, formal explanation of exactly how, when, and why you should or should not stick to your precommitment. The original motivation is almost certainly in the context of AI design, where you don't HAVE a human homunculus implementing a decision theory, the agent just is its decision theory.
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 decision theory, without creating new pathologies. I feel obnoxious to keep repeating this, but that is the entire problem in the first place.