Wednesday, April 19, 2006

Social Choice: Ties and Indifference

My friend John has this interesting post on Arrow's R-notation (cross-posted here). I don't have a problem with the R-notation myself, which effectively represents preferences using a 'greater than or equal to' relation, rather than merely 'greater than' (Preference) and 'equal to' (Indifference).

John also, however, repeats one of his regular criticisms, about Arrow's equivalence of ties and indifference. One he's made before here (see points 2, 11, and 17) and here, and something I had a go at responding to in this blog previously here (responses I now think could be much improved).

Anyway, this stimulated me to re-arrange my present thoughts on social choice and utilitarianism - not so much new thoughts, but a new way of fitting them together and understanding the whole. I re-produce the relevant section of my comment:
Then there's the issue of social indifference. You complain Arrow interprets half aPib and half bPia as aIb.
I can see what you're complaining about. It matters to each individual whether a or b, so I think there's a need for some just resolution mechanism, e.g. flipping a coin.
Arrow's position is (I think) a quasi-utilitarian one. It may matter to each individual whether a or b, but it doesn't matter to 'society', since aggregate net satisfaction is the same. He's adopting some impartial point of view, which is indifferent between satisfying aPib and bPja.
If you reject this, which I think one committed to a non-utilitarian individualist viewpoint might, then I think your problem isn't just with indifference but the whole idea of social preference. Suppose two-thirds of people vote aPib and one third bPia. On what basis is that turned from a two-third/one-third split (which might demand some sort of proportionality of result) into a seemingly monolithic aPb?
Put that way (which I wish I had earlier), you could say it's a major motivation of my own research.


At 6:54 pm, Anonymous John Lawrence said...

I agree with you, Ben, that aggregate net social satisfaction is the same if half the individuals have aPib and half have bPia irregardless of whether society chooses a or b. In that case society could flip a coin with heads being a and tails being b and societal satisfaction would be the same regardless of the outcome.

I think Arrow should have called this case a tie not an indifference. Indeed why even intoduce the concept of indifference if it's not needed. It gives a misleading idea that the individuals are indifferent as to the outcome when in fact they're deeply divided in this case at least.

Wouldn't individual indifferences be thrown out anyway and the election decided by the number of preferences involved? I don't have a problem with introducing some mechanism to resolve a tie. However, I do think that, if ties are allowed, Arrow's analysis goes down the drain and social choice is not impossible especially if a tie solution is acceptable followed by a resolution-of-tie mechanism if desired such as flipping a coin.

You are right that if the solution can allow 2 outcomes, then a two-thirds, one-third split could be resolved by giving one party two thirds of the representatives and the other party one third. But it all depends on the nature of the election and whom you're voting for. If you're voting for a position that can only be occupied by one guy, then you can't have a proportional outcome. It must be one or the other. On the other hand, if you're voting to elect a parliament, you could take the proportionality approach. It's all in how you define the problem at the outset.

I'm preparing a more detailed analysis to try and resolve (in my mind at least)your point about R being merely a representation mechanism. Look for it soon at a blog near you!


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