An alternative to bounded utility is to suppose that probabilities go to zero faster than utility. In fact, the latter is a generalisation of the former, since the former is equivalent to supposing that the probability is zero for large enough utility.
However, neither “utility is bounded” nor “probabilities go to zero faster than utility” amount to solutions to Pascal’s Mugging. They only indicate directions in which a solution might be sought. An actual solution would provide a way to calculate, respectively, the bound or the limiting form of probabilities for large utility. Otherwise, for any proposed instance of Pascal’s Mugging, there is a bound large enough, or a rate of diminution of P*U low enough, that you still have to take the bet.
Set the bound too low, or the diminution too fast (“scope insensitivity”), and you pass up some gains that some actual people think extremely worth while, such as humanity expanding across the universe instead of being limited to the Earth. Telling people they shouldn’t believe in such value, while being unable to tell them how much value they should believe in isn’t very persuasive.
This alternative only works because it asserts that such and such a bet is impossible, e.g. there is a reward that you would pay $100 for if the odds were one in a googolplex of getting the reward, but in fact the odds for that particular reward are always less than one in a googolplex.
That still requires you to bite the bullet of saying that yes, if the odds were definitely one in a googolplex, I would pay $100 for that bet.
But for me at least, there is no reward that I would pay $100 for at that odds. This means that I cannot accept your alternative. And I don’t think that there are any other real people who would consistently accept it either, not in real life, regardless of what they say in theory.
That still requires you to bite the bullet of saying that yes, if the odds were definitely one in a googolplex, I would pay $100 for that bet.
The idea of P*U tending to zero axiomatically rules out the possibility of being offered that bet, so there is no need to answer the hypothetical. No probabilities at the meta-level.
Or, if you object to the principle of no probabilities at the meta-level, the same objection can be made to bounded utility. This requires you to bite the bullet of saying that yes, if the utility really were that enormous etc.
The same applies to any axiomatic foundation for utility theory that avoids Pascal’s Mugging. You can always say, “But what if [circumstance contrary to those axioms]? Then [result those axioms rule out].”
The two responses are not equivalent. The utility in a utility function is subjective in the sense that it represents how much I care about something; and I am saying that there is literally nothing that I care enough about to pay $100 for a probability of one in a googolplex of accomplishing it. So for example if I knew for an absolute fact that for $100 I could get that probability of saving 3^^^^^^^^^3 lives, I would not do it. Saying the utility can’t be that enormous does not rule out any objective facts: it just says I don’t care that much. The only way it could turn out that “if the utility really were that enormous” would be if I started to care that much. And yes, I would pay $100 if it turned out that I was willing to pay $100. But I’m not.
Attempting to rule out a probability by axioms, on the other hand, is ruling out objective possibilities, since objective facts in the world cause probabilities. The whole purpose of your axiom is that you are unwilling to pay that $100, even if the probability really were one in a googolplex. Your probability axiom is simply not your true rejection.
Saying the utility can’t be that enormous does not rule out any objective facts: it just says I don’t care that much.
To say you don’t care that much is a claim of objective fact. People sometimes discover that they do very much care (or, if you like, change to begin to very much care) about something they did not before. For example, conversion to ethical veganism. You may say that you will never entertain enormous utility, and this claim may be true, but it is still an objective claim.
And how do you even know? No-one can exhibit their utility function, supposing they have one, nor can they choose it.
As I said, I concede that I would pay $100 for that probability of that result, if I cared enough about that result, but my best estimate of how much I care about that probability of that result is “too little to consider.” And I think that is currently the same for every other human being.
(Also, you consistently seem to be implying that “entertaining enormous utility” is something different from being willing to pay a meaningful price for small probability of something: but these are simply identical—asking whether I might objectively accept an enormous utility assignment is just the same thing as asking whether there might be some principles which would cause me to pay the price for the small probability.)
An alternative to bounded utility is to suppose that probabilities go to zero faster than utility. In fact, the latter is a generalisation of the former, since the former is equivalent to supposing that the probability is zero for large enough utility.
However, neither “utility is bounded” nor “probabilities go to zero faster than utility” amount to solutions to Pascal’s Mugging. They only indicate directions in which a solution might be sought. An actual solution would provide a way to calculate, respectively, the bound or the limiting form of probabilities for large utility. Otherwise, for any proposed instance of Pascal’s Mugging, there is a bound large enough, or a rate of diminution of P*U low enough, that you still have to take the bet.
Set the bound too low, or the diminution too fast (“scope insensitivity”), and you pass up some gains that some actual people think extremely worth while, such as humanity expanding across the universe instead of being limited to the Earth. Telling people they shouldn’t believe in such value, while being unable to tell them how much value they should believe in isn’t very persuasive.
This alternative only works because it asserts that such and such a bet is impossible, e.g. there is a reward that you would pay $100 for if the odds were one in a googolplex of getting the reward, but in fact the odds for that particular reward are always less than one in a googolplex.
That still requires you to bite the bullet of saying that yes, if the odds were definitely one in a googolplex, I would pay $100 for that bet.
But for me at least, there is no reward that I would pay $100 for at that odds. This means that I cannot accept your alternative. And I don’t think that there are any other real people who would consistently accept it either, not in real life, regardless of what they say in theory.
The idea of P*U tending to zero axiomatically rules out the possibility of being offered that bet, so there is no need to answer the hypothetical. No probabilities at the meta-level.
Or, if you object to the principle of no probabilities at the meta-level, the same objection can be made to bounded utility. This requires you to bite the bullet of saying that yes, if the utility really were that enormous etc.
The same applies to any axiomatic foundation for utility theory that avoids Pascal’s Mugging. You can always say, “But what if [circumstance contrary to those axioms]? Then [result those axioms rule out].”
The two responses are not equivalent. The utility in a utility function is subjective in the sense that it represents how much I care about something; and I am saying that there is literally nothing that I care enough about to pay $100 for a probability of one in a googolplex of accomplishing it. So for example if I knew for an absolute fact that for $100 I could get that probability of saving 3^^^^^^^^^3 lives, I would not do it. Saying the utility can’t be that enormous does not rule out any objective facts: it just says I don’t care that much. The only way it could turn out that “if the utility really were that enormous” would be if I started to care that much. And yes, I would pay $100 if it turned out that I was willing to pay $100. But I’m not.
Attempting to rule out a probability by axioms, on the other hand, is ruling out objective possibilities, since objective facts in the world cause probabilities. The whole purpose of your axiom is that you are unwilling to pay that $100, even if the probability really were one in a googolplex. Your probability axiom is simply not your true rejection.
To say you don’t care that much is a claim of objective fact. People sometimes discover that they do very much care (or, if you like, change to begin to very much care) about something they did not before. For example, conversion to ethical veganism. You may say that you will never entertain enormous utility, and this claim may be true, but it is still an objective claim.
And how do you even know? No-one can exhibit their utility function, supposing they have one, nor can they choose it.
As I said, I concede that I would pay $100 for that probability of that result, if I cared enough about that result, but my best estimate of how much I care about that probability of that result is “too little to consider.” And I think that is currently the same for every other human being.
(Also, you consistently seem to be implying that “entertaining enormous utility” is something different from being willing to pay a meaningful price for small probability of something: but these are simply identical—asking whether I might objectively accept an enormous utility assignment is just the same thing as asking whether there might be some principles which would cause me to pay the price for the small probability.)