Replacing single utilities with time-indexed sequences of utilities doesn’t help for representing preferences. If you have to make a decision between two options, each of which will result in a different sequence of time-indexed utilities, you still need to decide which option is better overall, which means you’ll need a one-dimensional scale to compare these utility-sequences on. The VNM theorem tells you that, under certain fairly weak assumptions, a single real-valued utility is the appropriate measure to use for this.
Reals might not be “continuous enough” to do the job. A hard limit case.
Option A: 1 utility on day 1, 0 utility for the rest of days
Option B: 2 utility on day 1, 0 utility for the rest of days
Option C: 3 utility on day 1, 0 utility for the rest of days
Option D: 1 utility on day 1, 1 utiltity for the rest of days
Option E: 2 utility on day 1, 1 utility for the rest of days
Continuity means when L<M<N then there should a p such that pL+(1−p)N∼M
So there are values p1A+(1−p1)C∼B and p2A+(1−p2)D∼B and p3A+(1−p3)E. If p1, p2 adn p3 are from the reals and different then they should be finite multiples of each other. So while one can do with one real to differentiate between A,B,C and D,E to me it seems the jump between the types of cases is not finite and the reals can’t provide that at the same time as keeping the resolution on differentiating betwen day 1 utilities.
With surreals the probabilities could be infinidesimal and the missing probabilities exist.
If you have surreal-valued utilities, you can just round infinitesimals to 0 to get real-valued utilities, and then continuity can be satisfied with real-valued probabilities again. The resulting real-valued utility function is correct about your preferences whenever it assigns higher utility to one option than the other, and is deficient only in the case where it assigns the same utility to two different options that you value differently. But it is very unlikely for two arbitrary reals to be exactly the same, and even when this does happen, the difference is infinitesimally unimportant compared to other preferences, so this isn’t a big loss.
Replacing single utilities with time-indexed sequences of utilities doesn’t help for representing preferences. If you have to make a decision between two options, each of which will result in a different sequence of time-indexed utilities, you still need to decide which option is better overall, which means you’ll need a one-dimensional scale to compare these utility-sequences on. The VNM theorem tells you that, under certain fairly weak assumptions, a single real-valued utility is the appropriate measure to use for this.
Reals might not be “continuous enough” to do the job. A hard limit case.
Option A: 1 utility on day 1, 0 utility for the rest of days
Option B: 2 utility on day 1, 0 utility for the rest of days
Option C: 3 utility on day 1, 0 utility for the rest of days
Option D: 1 utility on day 1, 1 utiltity for the rest of days
Option E: 2 utility on day 1, 1 utility for the rest of days
Continuity means when L<M<N then there should a p such that pL+(1−p)N∼M
So there are values p1A+(1−p1)C∼B and p2A+(1−p2)D∼B and p3A+(1−p3)E. If p1, p2 adn p3 are from the reals and different then they should be finite multiples of each other. So while one can do with one real to differentiate between A,B,C and D,E to me it seems the jump between the types of cases is not finite and the reals can’t provide that at the same time as keeping the resolution on differentiating betwen day 1 utilities.
With surreals the probabilities could be infinidesimal and the missing probabilities exist.
If you have surreal-valued utilities, you can just round infinitesimals to 0 to get real-valued utilities, and then continuity can be satisfied with real-valued probabilities again. The resulting real-valued utility function is correct about your preferences whenever it assigns higher utility to one option than the other, and is deficient only in the case where it assigns the same utility to two different options that you value differently. But it is very unlikely for two arbitrary reals to be exactly the same, and even when this does happen, the difference is infinitesimally unimportant compared to other preferences, so this isn’t a big loss.