I think the most “native” representation of utility functions is actually as a function from ordered triples of outcomes to real numbers. Rather than having an arbitrary (affine symmetry breaking) scale for strength of preference, set the scale of a preference by comparing to a third possible outcome.
The function is the “how much better?” function. Given possible outcomes A, B, and X, how many times better is A (relative to X) than B (relative to X).
If A is chocolate cake, and B is ice cream, and X is going hungry, maybe the chocolate cake preference is 1.25 times stronger, so the function Betterness(chocolate cake, ice cream, going hungry) = 1.25.
This is the sort of preference that you would elicit from a gamble (at least from a rational agent, not necessarily from a human). If I am indifferent to a gamble with a probability 1 of ice cream, and a probability 0.8 of chocolate cake and 0.2 of going hungry, this tells you that betterness-value above.
Anyhow, interesting post, I’m just idly commenting.
If I am indifferent to a gamble with a probability 1 of ice cream, and a probability 0.8 of chocolate cake and 0.2 of going hungry
To check I understand correctly, you mean the agent is indifferent between the gambles
(probability 1 of ice cream) and (probability 0.8 of chocolate cake, probability 0.2 of going hungry)?
If I understand correctly, you’re describing a variant of Von Neumann–Morgenstern where instead of giving preferences among all lotteries, you’re specifying a certain collection of special type of pairs of lotteries between which the agent is indifferent1, together with a sign to say in which `direction’ things become preferred? It seems then likely to me that the data you give can be used to reconstruct preferences between all lotteries...
If one is given information in the form you propose but only for an incomplete' set of special triples (c.f.weak preferences’ above), then one can again ask whether and in how many ways it can be extended to a complete set of preferences. It feels to me as if there is an extra ambiguity coming in with your description, for example if the set of possible outcomes has 6 elements and I am given the value of the Betterness function on two disjoint triples, then to generate a utility function I have to not only choose a `translation’ between the two triples, but also a scaling. But maybe this is better/more realistic!
1. By `special types’, I mean indifference between pairs of gambles of the form
(probability 1 of A) vs (probability p of B and probability (1−p) of C)
for some 0≤p≤1, and possible outcomes A, B, C. Then the sign says that I prefer higher probability of B (say).
I think the most “native” representation of utility functions is actually as a function from ordered triples of outcomes to real numbers. Rather than having an arbitrary (affine symmetry breaking) scale for strength of preference, set the scale of a preference by comparing to a third possible outcome.
The function is the “how much better?” function. Given possible outcomes A, B, and X, how many times better is A (relative to X) than B (relative to X).
If A is chocolate cake, and B is ice cream, and X is going hungry, maybe the chocolate cake preference is 1.25 times stronger, so the function Betterness(chocolate cake, ice cream, going hungry) = 1.25.
This is the sort of preference that you would elicit from a gamble (at least from a rational agent, not necessarily from a human). If I am indifferent to a gamble with a probability 1 of ice cream, and a probability 0.8 of chocolate cake and 0.2 of going hungry, this tells you that betterness-value above.
Anyhow, interesting post, I’m just idly commenting.
Thanks for the comment Charlie.
To check I understand correctly, you mean the agent is indifferent between the gambles (probability 1 of ice cream) and (probability 0.8 of chocolate cake, probability 0.2 of going hungry)?
If I understand correctly, you’re describing a variant of Von Neumann–Morgenstern where instead of giving preferences among all lotteries, you’re specifying a certain collection of special type of pairs of lotteries between which the agent is indifferent1, together with a sign to say in which `direction’ things become preferred? It seems then likely to me that the data you give can be used to reconstruct preferences between all lotteries...
If one is given information in the form you propose but only for an
incomplete' set of special triples (c.f.weak preferences’ above), then one can again ask whether and in how many ways it can be extended to a complete set of preferences. It feels to me as if there is an extra ambiguity coming in with your description, for example if the set of possible outcomes has 6 elements and I am given the value of theBetternessfunction on two disjoint triples, then to generate a utility function I have to not only choose a `translation’ between the two triples, but also a scaling. But maybe this is better/more realistic!1. By `special types’, I mean indifference between pairs of gambles of the form
(probability 1 of A) vs (probability p of B and probability (1−p) of C)
for some 0≤p≤1, and possible outcomes A, B, C. Then the sign says that I prefer higher probability of B (say).