There is a problem with completeness that requires studying the actual theorem and its construction of utility from preference. The preference function does not range over just the possible “outcomes” (which we suppose are configurations of the world or of some local part of it). It ranges over lotteries among these outcomes, as explicitly stated on the VNM page linked above. This implies that the idea of being indifferent between a sure gain of 1 util and a 0.1% chance of 1000 utils is already baked into the setup of these theorems, even before the proof constructs the utility function. A theorem cannot be used to support its assumptions.
The question, “Why should I maximise expected utility (or if not, what should I do instead)?” is a deep one which I don’t think has ever been fully answered.
The argument from iterated gambles requires that the utility of the gambles can be combined linearly. The OP points out that this is not the case if utility is nonlinear in the currency of the gambles.
Even when the utilities do combine linearly, there may be no long run. An example now well-known is one constructed by Ole Peters, in which the longer you play, the greater the linearly expected value, but the less likely you are to win anything. Ever more enormous payoffs are squeezed into an ever tinier part of probability space.
Even gambles apparently fair in expectation can be problematic if their variances are infinite and so the Central Limit Theorem does not apply.
So iterated gambles do not answer the question.
Logarithmic utility does not help: just imagine games with the payouts correspondingly scaled up.
Bounded utility does not help, because there is no principled way to locate the bound, and because if the bound is set large enough finite versions of the problems with unbounded utility still show up. See SBF/FTX. Is that debacle offset by the alternate worlds in which the collapse never happened?
Very interesting observations. I woudln’t say the theorem is used to support his assumption because the assumptions don’t speak about utils but only about preference over possible outcomes and lotteries, but I see your point.
Actually the assumptions are implicitly saying that you are not rational if you don’t want to risk to get a 1′000′000′000′000$ debt with a small enough probability rather than losing 1 cent (this is strightforward from the archimedean property).
There is a problem with completeness that requires studying the actual theorem and its construction of utility from preference. The preference function does not range over just the possible “outcomes” (which we suppose are configurations of the world or of some local part of it). It ranges over lotteries among these outcomes, as explicitly stated on the VNM page linked above. This implies that the idea of being indifferent between a sure gain of 1 util and a 0.1% chance of 1000 utils is already baked into the setup of these theorems, even before the proof constructs the utility function. A theorem cannot be used to support its assumptions.
The question, “Why should I maximise expected utility (or if not, what should I do instead)?” is a deep one which I don’t think has ever been fully answered.
The argument from iterated gambles requires that the utility of the gambles can be combined linearly. The OP points out that this is not the case if utility is nonlinear in the currency of the gambles.
Even when the utilities do combine linearly, there may be no long run. An example now well-known is one constructed by Ole Peters, in which the longer you play, the greater the linearly expected value, but the less likely you are to win anything. Ever more enormous payoffs are squeezed into an ever tinier part of probability space.
Even gambles apparently fair in expectation can be problematic if their variances are infinite and so the Central Limit Theorem does not apply.
So iterated gambles do not answer the question.
Logarithmic utility does not help: just imagine games with the payouts correspondingly scaled up.
Bounded utility does not help, because there is no principled way to locate the bound, and because if the bound is set large enough finite versions of the problems with unbounded utility still show up. See SBF/FTX. Is that debacle offset by the alternate worlds in which the collapse never happened?
Very interesting observations. I woudln’t say the theorem is used to support his assumption because the assumptions don’t speak about utils but only about preference over possible outcomes and lotteries, but I see your point.
Actually the assumptions are implicitly saying that you are not rational if you don’t want to risk to get a 1′000′000′000′000$ debt with a small enough probability rather than losing 1 cent (this is strightforward from the archimedean property).