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).
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).