One of the highlights of my graduate school years was the lecture where I learned about Arrow's Impossibility Theorem. It's hard to imagine an question more important to the social sciences that the social choice problem: how can we aggregate individual preferences into a coherent social order? Arrow's answer is that we can't, but that doesn't make the issue less important. Of course, from the perspective of a graduate student looking for a thesis topic, Arrow did his job rather too well: you had to be a serious math jock to identify and solve an issue he hadn't already dealt with. And I'm by no means a math jock.
But it turns out that my modest skills as an econometrician are useful in attacking the social choice problem using another approach, one that is as old as the social choice problem itself, but of which I was unaware until it was explained to me by my colleague Michel Truchon.
The 18th-century French scholar Nicolas de Condorcet set up the problem in the following way: suppose that a true social ordering exists, but that individuals observe it with random error. If there are two alternatives, and if the probability that an individual voter will make the correct ranking is greater than 0.5, then a majority vote is a maximum likelihood estimator for the true social order. In addition to the development of the social choice problem itself, this result is one of the first applications of probability and of the maximum likelihood principle.
For reasons I don't quite understand – although there is a lot of work in the statistics literature on ranking methods, especially in sports – this approach hasn't received much in the way of attention by econometricians. Michel and I started working in this project a couple of years ago, and we've finally finished two papers (here and here). [Update: published here and here.] Some things we discuss include:
- More flexible models for vote probabilities. For example, two alternatives that are 'close' make be more difficult to rank correctly than the first- and last-placed alternatives. And in an empirical application to figure skating, we include a national bias parameter.
- Loss functions for the choice problem. This is an issue that is – to the best of our knowledge – not been addressed in this literature.
- The ex ante vs the ex post problems. If there's no reason to commit yourself to a fixed aggregation rule before the votes are cast, then the problem is one of ex post inference. If you are, then you have to perform an ex ante evaluation of available rules, and choose the one that does best in repeated samples.
[I'm taking a break from blogging over the holidays, and recycling
some earlier posts. This one was first published on July 5, 2006.]
Worthwhile Canadian Initiative 
One underlying assumption is that human preferences are ordered. However, the empirical literature suggests that they are only partially ordered. For instance, transitivity may fail. This is usually interpreted as irrationality, but may simply indicate that context matters for preferences that are confused with each other.
Now, preferences may still have mean values that are ordered, and which will enable us to predict choices in most contexts. Preferences with close mean values are more likely (in an informal sense) to be confused than preferences with large differences between their mean values.
I think that confusion makes more sense than error. This can be tested empirically by seeing if non-transitive choices are consistent when the contexts are repeated. If it is a matter of error, they would not be consistent.
BTW, did Keynes say anything about this? I know that he considered probabilities to be only partially ordered.
Stephen: This is fascinating!. I was totally unaware that Arrow’s problem could be looked at in this way. And that Condorcet had looked at it this way so long ago!
If I’ve got my head around this right, what you are then looking for is a Bayesian estimator that can use “data” on individual’s rankings to create an estimate of the “true” best ranking. And your estimator must then violate one (or more) of Arrow’s assumptions. Independence of irrelevant alternatives?
It never occurred to me to ask Michel that, but I think IIA is one of the things that is violated.