I think a reasonable question to ask is what value having relative strength of the preference ordering matters to questions we want to answer. That is, I suspect the reason we normally only consider a preference ordering and not a preference measure is that it’s not relevant to observed behavior, since in that case we presume the most preferred possible action will always be taken, thus more information than the order is not relevant to the decision.
I can imagine we might care about measure in cases like modeling how human preferences are actually manifested where they might have relative weights and can be updated, although personally I prefer the idea of avoiding this by making updating appear immutable by conditioning each preference on the entire causal history that came prior to its realization, although this has its own problems.
Sure, in the end we only really care about what comes top, as that’s the thing we choose. My feeling is that information on (relative) strengths of preferences is often available, and when it is available it seems to make sense to use it (e.g. allowing circumvention of Arrow’s theorem).
In particular, I worry that, when we only have ordinal preferences, the outcome of attempts to combine various preferences will depend heavily on how finely we divide up the world; by using information on strengths of preferences we can mitigate this.
I think a reasonable question to ask is what value having relative strength of the preference ordering matters to questions we want to answer. That is, I suspect the reason we normally only consider a preference ordering and not a preference measure is that it’s not relevant to observed behavior, since in that case we presume the most preferred possible action will always be taken, thus more information than the order is not relevant to the decision.
I can imagine we might care about measure in cases like modeling how human preferences are actually manifested where they might have relative weights and can be updated, although personally I prefer the idea of avoiding this by making updating appear immutable by conditioning each preference on the entire causal history that came prior to its realization, although this has its own problems.
Sure, in the end we only really care about what comes top, as that’s the thing we choose. My feeling is that information on (relative) strengths of preferences is often available, and when it is available it seems to make sense to use it (e.g. allowing circumvention of Arrow’s theorem).
In particular, I worry that, when we only have ordinal preferences, the outcome of attempts to combine various preferences will depend heavily on how finely we divide up the world; by using information on strengths of preferences we can mitigate this.