Averaging by taking the mean sucks. Averaging by taking the median sucks less. It is a procedure relatively immune to gaming by would-be utility monsters.
The median is usually the ‘right’ utilitarian algorithm in any case. It minimizes total collective distance from the ‘average’. The mean minimizes total collective distance^2 from the ‘average’. There is no justification for squaring.
What’s the appropriate metric on the space of preferences? This seems like something people would have different opinions about; i.e. “People who are smart should have more say!” “People who have spent more time self-reflecting should have more say!” “People who make lifestyle choices like this should be weighted more heavily!” “People who agree with me should have more say!”
Depending on the distribution, squaring could be better, because more (might be) lost as you get further away. And of course you can only take the median if your preferences are one dimensional.
Personally, I am unconvinced that there is any fundamental justification for considering anyone’s utility but one’s own. But, if you have reason to respect the principles of democracy, the median stands out as the unique point acceptable to a majority. That is, if you specify any other point, a majority would vote to replace that point by the median.
What’s the appropriate metric on the space of preferences?
That depends on what kinds of preferences you are comparing. If you are looking at the preferences of a single person, the standard construction of that person’s utility function sets the “metric”. But if you attempt to combine the preferences of two people, you either need to use the Nash Bargaining solution or Harsanyi’s procedure for interpersonal comparison. The first gives a result that is vaguely median-like. The second gives an answer that is suitable for use with the mean.
Averaging by taking the mean sucks. Averaging by taking the median sucks less. It is a procedure relatively immune to gaming by would-be utility monsters.
The median is usually the ‘right’ utilitarian algorithm in any case. It minimizes total collective distance from the ‘average’. The mean minimizes total collective distance^2 from the ‘average’. There is no justification for squaring.
Is there a justification for not-squaring?
What’s the appropriate metric on the space of preferences? This seems like something people would have different opinions about; i.e. “People who are smart should have more say!” “People who have spent more time self-reflecting should have more say!” “People who make lifestyle choices like this should be weighted more heavily!” “People who agree with me should have more say!”
Depending on the distribution, squaring could be better, because more (might be) lost as you get further away. And of course you can only take the median if your preferences are one dimensional.
Personally, I am unconvinced that there is any fundamental justification for considering anyone’s utility but one’s own. But, if you have reason to respect the principles of democracy, the median stands out as the unique point acceptable to a majority. That is, if you specify any other point, a majority would vote to replace that point by the median.
That depends on what kinds of preferences you are comparing. If you are looking at the preferences of a single person, the standard construction of that person’s utility function sets the “metric”. But if you attempt to combine the preferences of two people, you either need to use the Nash Bargaining solution or Harsanyi’s procedure for interpersonal comparison. The first gives a result that is vaguely median-like. The second gives an answer that is suitable for use with the mean.