Though I should mention that my current version of partial preferences does not assume all cycles are closed—the constrained optimisation can be seen as trying to get “as close as possible” to that, given non-closed cycles.
Thanks! I like the way your optimisation problem handles non-closed cycles.
I think I’m less comfortable with how it treats disconnected components—as I understand it you just translate each separately to have `centre of mass’ at 0. If one wants to get a utility function out at the end one has to make some kind of choice in this situation, and the choice you make is probably the best one, so in that sense it seems very good.
But for example it seems vulnerable to creating ‘virtual copies’ of worlds in order to shift the centre of mass and push connected components one way or the other. That was what started me thinking about including strength of preference—if one adds to your setup a bunch of virtual copies of a world between which one is `almost indifferent’ then it seems it will shift the centre of mass, and thus the utility relative to come other chain. Of course, if one is actually indifferent then the ‘virtual copies’ will be collapsed to a single point in your ¯¯¯¯¯¯W, but if they are just extremely close then it seems it will affect the utility relative to some other chain. I’ll try to explain this more clearly in a comment to your post.
Neat!
Though I should mention that my current version of partial preferences does not assume all cycles are closed—the constrained optimisation can be seen as trying to get “as close as possible” to that, given non-closed cycles.
Thanks! I like the way your optimisation problem handles non-closed cycles.
I think I’m less comfortable with how it treats disconnected components—as I understand it you just translate each separately to have `centre of mass’ at 0. If one wants to get a utility function out at the end one has to make some kind of choice in this situation, and the choice you make is probably the best one, so in that sense it seems very good.
But for example it seems vulnerable to creating ‘virtual copies’ of worlds in order to shift the centre of mass and push connected components one way or the other. That was what started me thinking about including strength of preference—if one adds to your setup a bunch of virtual copies of a world between which one is `almost indifferent’ then it seems it will shift the centre of mass, and thus the utility relative to come other chain. Of course, if one is actually indifferent then the ‘virtual copies’ will be collapsed to a single point in your ¯¯¯¯¯¯W, but if they are just extremely close then it seems it will affect the utility relative to some other chain. I’ll try to explain this more clearly in a comment to your post.