"Willpower is not exhaustible" is not necessarily the same claim as "willpower is infallible". If, for example, you have a flat 75% chance of turning down sweets, then avoiding sweets still makes you more likely to not eat them. You're not spending willpower, it's just inherently unreliable.

I'm pretty sure that decision theories are not designed on that basis.

You are wrong. In fact, this is a totally standard thing to consider, and "avoid back-chaining defection in games of fixed length" is a known problem, with various known strategies.

Yes, that is the problem in question!

If you want the payoff, you have to be the kind of person who will pay the counterfactual mugger, even once you no longer can benefit from doing so. Is that a reasonable feature for a decision theory to have? It's not clear that it is; it seems strange to pay out, even though the expected value of becoming that kind of person is clearly positive before you see the coin. That's what the counterfactual mugging is about.

If you're asking "why care" rhetorically, and you believe the answer is "you shouldn't be that kind of person", then your decision theory prefers lower expected values, which is also pathological. How do you resolve that tension? This is, once again, literally the entire problem.

Your decision is a result of your decision theory, and your decision theory is a fact about you, not just something that happens in that moment.

You can say - I'm not making the decision ahead of time, I'm waiting until after I see that Omega has flipped tails. In which case, when Omega predicts your behavior ahead of time, he predicts that you won't decide until after the coin flip, resulting in hypothetically refusing to pay given tails, so - although the coin flip hasn't happened yet and could still come up heads - your yet-unmade decision has the same effect as if you had loudly precommitted to it.

You're trying to reason in temporal order, but that doesn't work in the presence of predictors.

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.

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?

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.

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.

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?

Decision theory is an attempt to formalize the human decision process. The point isn't that we really are unsure whether you should leave people to die of thirst, but how we can encode that in an actual decision theory. Like so many discussions on Less Wrong, this implicitly comes back to AI design: an AI needs a decision theory, and that decision theory needs to not have major failure modes, or at least the failure modes should be well-understood.

If your AI somehow assigns a nonzero probability to "I will face a massive penalty unless I do this really weird action", that ideally shouldn't derail its entire decision process.

The beggars-and-gods formulation is the same problem. "Omega" is just a handy abstraction for "don't focus on how you got into this decision-theoretic situation". Admittedly, this abstraction sometimes obscures the issue.

Precommitments are used in decision-theoretic problems. Some people have proposed that a good decision theory should take the action that it would have precommitted to, if it had known in advance to do such a thing. This is an attempt to examine the consequences of that.

I'm not sure you've described a different mistake than Eliezer has?

Certainly, a student with a sufficiently incomplete understanding of heat conduction is going to have lots of lines of thought that terminate in question marks. The thesis of the post, as I read it, is that we want to be able to recognize when our thoughts terminate in question marks, rather than assuming we're doing something valid because our words sound like things the professor might say.