Each agent has a computable partial preference ordering x≤y that decides if it prefers x to y.
We’d like this partial relation to be complete (i.e., defined for all x,y) and transitive (i.e., x≤y and y≤z implies x≤z).
Now, if the relation is sufficiently non-trivial, it will be expensive to compute for some x,y. So it’s better left undefined...?
If so, I can surely relate to that, as I often struggle computing my preferences. Even if they are theoretically complete. But it seems to me the relationship is still defined, but might not be practical to compute.
It’s also possible to think of it in this way: You start out with partial preference ordering, and need to calculate one of its transitive closures. But that is computationally difficult, and not unique either.
I’m unsure what these observations add to the discussion, though.
Do I understand you correctly here?
Each agent has a computable partial preference ordering x≤y that decides if it prefers x to y.
We’d like this partial relation to be complete (i.e., defined for all x,y) and transitive (i.e., x≤y and y≤z implies x≤z).
Now, if the relation is sufficiently non-trivial, it will be expensive to compute for some x,y. So it’s better left undefined...?
If so, I can surely relate to that, as I often struggle computing my preferences. Even if they are theoretically complete. But it seems to me the relationship is still defined, but might not be practical to compute.
It’s also possible to think of it in this way: You start out with partial preference ordering, and need to calculate one of its transitive closures. But that is computationally difficult, and not unique either.
I’m unsure what these observations add to the discussion, though.