...And A Little Bit More

It's clear from the conversations I've had about what I wrote last that the picture of choice that I presented is too stark to be realistic. I agree. It was an illustration of how social choice theory goes about the subject. The plan is to try and broaden this view in one particular direction (my own), but alternative scenarios abound.

See, in truth, choice is a capricious thing. Without getting too far into the philosophy of it all, it is safe to say that we don't always understand the choices we make, nor do we always make them sensibly. Outside events too play a large but under-appreciated role in determining what we end up doing. Our basis for judgement and decision making is constantly evolving, mostly always improving, and is highly variable from context to context. The inside of the mind, in short, is a wonderland. Especially other people's minds.

But when we cannot say very much about what goes on inside the heads of various decision makers, we can nevertheless observe their actions. Look around you. You do by and large have firm opinions about your family, friends, and colleagues, based on how sensible you think they are, in turn based on their behaviour that you observe. If you were to see me consistently doing the foolish thing, how far the conclusion that I am indeed a fool? That relationship, intuitively reached for the most part, but possibly backed by with observation, is the object under consideration.

Revealed preference, they call it. Revealed by observation. When I am seen choosing A over B when being able to get either, and I make this choice once, or even many times over, the conclusion drawn (alternatively, the assumption made) is that I prefer A to B. There are various ways to complicate this argument, talking about things like context or time or mood, but the basic point is simple. I have a preference. So far this is not very controversial, but things get tricky when trying to pin down the nature of this preference, because now we are again talking about wonderland.

In trying to systematise, a certain amount of generality is lost, and theoretical preferences adopt a somewhat more logical structure than how people, in reality, manifest them. It is that logical view that I am trying to broaden, with the clear knowledge of the limits of the argument. Naturally, the best picture will still be too simplistic. But swallow your disbelief, take a deep breath, and come along.

On The Theory Of Social Choice

In the previous post, I mentioned how sections of economic theory make behavioural assumptions and then build on that edifice. This time I'm going to look at one of those fields of economics, called the theory of social choice.

Social choice theory concerns itself with the question of how a society should make decisions based on the view of the individuals that comprise it. In particular, when faced with a set of options, when individuals have differing opinions on them (ie, different preferences over the set of options), the questions boils down to how best to aggregate those individual opinions to form a composite social opinion, or preference ranking, of those options.

Why is this even relevant? Well, it is, whether you consider the case of committees discussing an agenda, or governments seeking to outline policy, or even a few friends getting together and deciding what to order for dinner. In the committee case, say there are three issues on the table (A, B and C) and each of the committee members has a view on which is the most important and should be discussed first. A final agenda must be drawn up, which consists of a sequence of issues (ABC, BCA, CAB, or something like that), and which ideally should be constructed based on the opinions of the committee members.

In the government example, say that policy needs to determine the order in which the government spends money on three competing projects (say building a road, a dam, and improving primary education). Voters (or government members) have differing views on these options, but still a final policy order needs to be specified.

The friends example is a straightforward one; we have all at some point or the other had to decide with friends which restaurant, say, to go for dinner, and had to resolve and reconcile the differing views. So the question isn't a trivial one, although there might be better ways of looking at it.

Social choice theory proceeds, by and large, in the following way. It assumes that individual preferences satisfy certain properties (e.g rationality, transitivity etc. most of which are intuitive and some of which are discussed below), and then tries to outline mechanisms by which these preferences are to be aggregated into a composite preference ordering (the agenda, policy or restaurant ranking), which too is expected to satisfy certain properties (e.g fairness, collective rationality and suchlike). Naturally, different assumptions on individual behaviour lead to different conclusions, as do different requirements of the aggregation mechanism.

A seminal result in this field is what is called Arrow's Impossibility Theorem, proposed by Kenneth Arrow sometime in the 1950s and causing great grief (and sustenance) to economists over the last sixty years. He showed that, under certain plausible conditions, the only social mechanism that 'works' is a dictatorial one, ie, one in which the final social ranking is determined by the views of only one, fixed, individual. This means that regardless of the views of everyone else, even if they are polar opposite in nature, only this guy determines how society ranks things. Now while this might come as no surprise to dictators around the world, or even to the sort of dominating friends that we all have, this is anathema to any proponent of democracy and liberalism, and so this theorem has come under great scrutiny.

And now it is my turn.

The great allure of this theorem is in the frugality of its assumptions. About individual behaviour, the only requirement is that each individual has a preference ordering, ie, a ladder-like ranking of options. This hinges on the assumption that if an individual prefers option A to B, and also prefers option B to C, then it must follow that he or she prefers A to C. This is called transitivity, and it doesn't always work. But it is in general an innocuous condition.

The social aggregation mechanism (let's called it a social welfare function, or SWF, as Arrow and Amartya Sen did) also has certain requirements placed on it. One, that it must admit any sort of preference ranking that the individuals might have. In other words, people can have any kind of ranking that they like over those options, and the SWF must be able to process it. There's not much to argue with this, unless one wants to keep the crazies out of the decision process, and thus limit the preference rankings that the SWF needs to process. But we can argue forever about who qualifies as crazy, so for now let's leave this one alone. Everyone should be considered, fair enough.

The second requirement is that the SWF must respect unanimity. That is to say, if everybody agrees that option A is better than option B, then so must the SWF. There's little to find fault with this one, because the alternative is that the SWF picks B over A, even though everyone thinks A is better, which makes it a pretty silly society to live in.

The third requirement is a kind of independence condition. Since preferences are constructed in a binary manner (through pairwise ranking; and this is an implicit assumption that I will dicuss a little bit later), this merely says that when the SWF is deciding how to rank A and B relative to each other, all the information that is necessary to make the decision is how the individuals rank those two options vis-a-vis each other. For example, when the government needs to decide how to rank the road and the dam, all it needs to consider is how the people rank the road and the dam in relation to each other, and not consider, say, their views on education. Those views will need to be considered when education is being ranked, of course, but for the purpose of the present choice, it is irrelevant. Pairwise should remain pairwise.

That's it.

Put all these requirements together, and the theorem says that the only SWF that satisfies these three properties (assuming individuals behave as specified earier) is a dictatorial one. There is some guy, whose rankings always determine the social ranking, irrespective of what anyone else thinks. To put it differently, if we all agree that any social mechanism should satisfy these three properties, then we must live with the fact that there will be a dictator. The only way to avoid dictatorship is to relax one of these conditions.

Let me try and respecify what the theorem is saying. If we want to avoid our social decision mechanism being a dictatorial one, then we must either (a) keep the crazies out of the decision process, (b) live in a silly society that functions oddly, or (c) allow the ranking of alternatives to depend on other alternatives too, and not remain solely binary in nature. There is no other way. Make up your own minds about which of the three is relaxed in various societies (since we don't live in dictatorial ones) and, indeed, which of the three should be relaxed in an ideal one. Or be prepared to live with dictatorship.

Economists have been banging away at this theorem and related stuff for a very long time now, so in all probability most modifications and reinterpretations of this theorem have long since been exhausted. Social choice theory itself has moved on other questions, but no matter how far you look, the link back to this theorem, this result, and a related way of looking at the question remains. As I said earlier, there might be other useful ways of thinking about this aggregation problem, and I have a couple in mind, which I shall get around to discussing in the next post.