What do you think Pareto efficiency in a multi-system means/should mean? And is it desirable?
According to the voting wiki:
If every voter prefers alternative X over alternative Y, then the system prefers X over Y
With a multi-winner election, I would take that to mean that if Y is elected, then X must be as well. However, I don’t think that definition is strong enough. I would say that if every voter rates X at least equal to Y and at least one voter rates X ahead of Y, then the system prefers X over Y. And I think it’s desirable for a voting system to meet this criterion.
But arguably you can go further with cardinal multi-winner systems. You can define how much someone likes a set of candidates by totalling the scores (or approvals) they gave to the candidates in the set. So for a system to obey this criterion, if every voter rates set X at least equal to set Y and at least one voter prefers X to Y, then Y should not be elected. And in a an optimal system capable of giving scores to every set of candidates, X should have a higher score than Y. That last might not be so important as long as it never affects the winning set.
Is this desirable? Let’s look at an example:
2 to elect (approval voting)
5 voters: AC
4 voters: BC
1 voter: BD
I think that most people would agree that BC is the best set to elect. But compare AB with CD. In both cases every voter has one elected candidate each, so from a “Pareto” point of view they are equal. But under AB, 5 have approved A and 5 have approved B, whereas under CD, it’s 9 for C and 1 for D, which doesn’t seem very proportional. So then if you have this example:
500 voters: AC
400 voters: BC
100 voters: BD
1 voter: CD
CD now Pareto dominates AB. But is it actually the better set? You might argue that it doesn’t matter because BC is better than either, but it does at least raise the possibility that something similar could arise in the competition for best set. In such a case, which set would be better? Let’s say for some reason, it has to be AB or CD. Which set wins in your opinion?
In any case if you think this is too contrived, I will see if I can come up with an example for the winning set.