1. Suppose your set of values are lattice-ordered, and 2. Suppose they admit some sort of group structure that preserves this ordering: if you prefer apples to oranges, then you prefer two apples to two oranges, and so forth.
Then:
1. As long as you don’t have “infinitely” good outcomes, your preferences can be represented by a utility function. 2. If you have “infinitely” good outcomes, your preferences can be represented by a set of agents, each of which has a utility function, and your overall preference is equivalent to these subagents “unanimously agreeing”.
Wouldn’t the Hahn embedding theorem result in a ranking of the subagents themselves, rather than requiring unanimous agreement? Whichever subagent corresponds to the “largest infinities” (in the sense of ordinals) makes its choice, the choice of the next agent only matters if that first subagent is indifferent, and so on down the line.
Anyway, I find the general idea here interesting. Assuming a group structure seems unrealistic as a starting point, but there’s a bunch of theorems of the form “any abelian operation with properties X, Y, Z is equivalent to real/vector addition”, so it might not be an issue.
This can be formalized in the following sense:
1. Suppose your set of values are lattice-ordered, and
2. Suppose they admit some sort of group structure that preserves this ordering: if you prefer apples to oranges, then you prefer two apples to two oranges, and so forth.
Then:
1. As long as you don’t have “infinitely” good outcomes, your preferences can be represented by a utility function.
2. If you have “infinitely” good outcomes, your preferences can be represented by a set of agents, each of which has a utility function, and your overall preference is equivalent to these subagents “unanimously agreeing”.
The former claim is due to Holders theorem, and the latter is a result of the Hahn embedding theorem. I wrote a little bit more about this here.
Wouldn’t the Hahn embedding theorem result in a ranking of the subagents themselves, rather than requiring unanimous agreement? Whichever subagent corresponds to the “largest infinities” (in the sense of ordinals) makes its choice, the choice of the next agent only matters if that first subagent is indifferent, and so on down the line.
Anyway, I find the general idea here interesting. Assuming a group structure seems unrealistic as a starting point, but there’s a bunch of theorems of the form “any abelian operation with properties X, Y, Z is equivalent to real/vector addition”, so it might not be an issue.
Good point, yeah – it’s a lexical ordering, not a unanimous agreement.