I am confused. My current understanding is that we’re starting with only a preference relation, and no assumptions on probability (so no lotteries, as in the VNM theorem). In that case, there are tons of utility functions that can model any given arbitrary preference relation. It seems like I could get a result like this by saying “take the preference relation, write down a utility function that encodes it, decompose it into the ratio of two parts, call one of them ‘probability’ and the other ‘probability*utility’, and now note that there are transformations to other utility functions that encode the same preference relation and unsurprisingly they change the relative amounts of each of the parts—therefore probability and utility are inextricably linked”. (This is almost certainly either wrong or a strawman, but I don’t know how.) But in all of this there’s no reason to think of the denominator of the ratio as “probability”, we just called it that suggestively. Perhaps my critique is that if we start with _just_ a preference relation and only need to keep the preference relation intact, we shouldn’t expect to recover anything like normal expected utility theory, because there’s no formal reason to have anything like probabilities. Even if you want to interpret probability as a “caring measure” instead of “magical reality fluid” it should still show up before you work through the math and interpret one of the quantities as “caring measure”. But mostly I’m confused so who knows, this may all be incoherent.
I am confused. My current understanding is that we’re starting with only a preference relation, and no assumptions on probability (so no lotteries, as in the VNM theorem). In that case, there are tons of utility functions that can model any given arbitrary preference relation. It seems like I could get a result like this by saying “take the preference relation, write down a utility function that encodes it, decompose it into the ratio of two parts, call one of them ‘probability’ and the other ‘probability*utility’, and now note that there are transformations to other utility functions that encode the same preference relation and unsurprisingly they change the relative amounts of each of the parts—therefore probability and utility are inextricably linked”. (This is almost certainly either wrong or a strawman, but I don’t know how.) But in all of this there’s no reason to think of the denominator of the ratio as “probability”, we just called it that suggestively. Perhaps my critique is that if we start with _just_ a preference relation and only need to keep the preference relation intact, we shouldn’t expect to recover anything like normal expected utility theory, because there’s no formal reason to have anything like probabilities. Even if you want to interpret probability as a “caring measure” instead of “magical reality fluid” it should still show up before you work through the math and interpret one of the quantities as “caring measure”. But mostly I’m confused so who knows, this may all be incoherent.