Kenneth Arrow and Social Choice Theory - An Introduction

What is the best way of ranking a group of alternatives with regard to any social sector issue? Kenneth Arrow’s pioneering work could offer a solution.

If there is one economist who has single-handedly opened more than one discipline in economics and set milestones that will probably not be surpassed by any other economist until most of us are dead, it would be Kenneth Arrow. Nearly every discipline in economics that is an area of research today has had some major contribution from Arrow at some point of time. And like true geniuses, some of whom we will see later in this series, he essentially opened the field for others to explore and moved on to something more exciting.

But of all that Arrow is famous for, he is probably the most famous for his Fundamental Theorems of Welfare Economics and The Impossibility Theorem. We will talk about the latter today. To get a sense of how big a genius Arrow is, bear in mind that he wrote this as a PhD student in Columbia.

It is the 1950s. People are grappling with the ideas of voting systems, if a group of people want to rank the alternatives, what should they do? How should they design a scheme to do it best? And what do we mean by the best?  The problem — and its beautiful answer — can be best exposited using an example.

Imagine a researcher ranking states from best to worst. There exist various parameters according to which the states can be ranked. Let’s say GDP per capita gives one ranking of states, literacy rate gives another ranking, nutrition per capita gives yet another ranking. It is quite conceivable that no two parameters produce the same rankings. That is, maybe according to GDP we have Maharashtra, Gujarat, Karnataka, Delhi and so on. But say, on literacy we have Karnataka, Tamil Nadu, Maharashtra etc.

This researcher is now in a dilemma. How should he aggregate all these individual rankings into one unified ranking? As is often done in practice, every position in an individual ranking is given a score and then all the scores are aggregated. While I will say precisely what goes “wrong” with this approach. Suppose we have 20 states and we give points from 20 to 1 from best to worst in individual rankings. So Maharashtra receives 20 for GDP, 16 for literacy etc. Now, we sum all these up. But what is so holy about 20 to 1? Why not choose the square of each of those numbers? Ordinarily, each ranking remains the same. But in many scenarios adding all those squared numbers is going to change the ranking. Therefore, we cannot possibly rely on such methods. But then what do we do?

What makes this paper by Arrow foundational, remarkable and insightful is not just the answer that he gives to this problem (which we will see in a while) but his approach too. This is one of the first papers following what is known as the axiomatic approach. In this approach what we do is since we don’t have a ready answer at hand, we list down the desirable properties that we want and then try to show that a certain class is the only class of rules that satisfies those properties. So here’s what Arrow lists down as what’s desirable.

1. Well-defined. The method should give a sensible ranking for any possible individual rankings we may have. For example, it shouldn’t say Karnataka is better than Tamil Nadu is better than Maharashtra is better than Karnataka. Such cycles aren’t allowed. No matter what the “input” rankings are the output should be one unified sensible ranking.
2. Pareto Principle. This says, if according to every ranking if x is better than y, then the aggregate ranking should also say x is better than y. In our example, if according to every ranking Tamil Nadu is better than Haryana, then aggregate ranking should also say the same.
3. Non-Dictatorship. The name is self-explanatory. There shouldn’t be one ranking or dimension that dictates the social ranking. That is, in the context of our example, we don’t want a rule that says, no matter what other rankings say we will only rank according to GDP per capita.
4. Independence of Irrelevant Alternatives. This is the most discussed axiom in Arrow’s theorem. It is best to explain this through an example of our states. Suppose we do this ranking business every year. So we receive some rankings on various dimensions in 2012 and 2013, say. Suppose it so happens that on all the dimensions  Madhya Pradesh outperformed Rajasthan in 2012, it did so in 2013, too. Moreover, on all the dimensions on which Rajasthan outperformed Madhya Pradesh, it continues to do so. This is not to say that Madhya Pradesh is better than Rajasthan on every ground. On some it did better than Rajasthan in 2012, let’s say on GDP, literacy and sanitation it did better than Rajasthan. On nutrition and agriculture Rajasthan did better than Madhya Pradesh, in 2012. In 2013 they continue to do the same. That is, in 2013 Madhya Pradesh does better than Rajasthan on GDP, literacy and sanitation and does worse on nutrition and agriculture.

Then, the axiom says, if we said Madhya Pradesh is better than Rajasthan in 2012, we continue to say the same in 2013 or vice-versa. Again, this is not saying that we must choose Madhya Pradesh better than Rajasthan. Whatever we said between those two states in 2012, we don’t reverse that answer in 2013. This is very natural. All this demands is we should have some consistency in our method if the relative rankings remain the same for two alternatives.

Now, here comes Arrow’s masterpiece. Funnily enough, Arrow calls it the General Possibility Theorem. The theorem says, if the number of alternatives that we are ranking is three or more then there cannot exist any method that satisfies the four properties listed above.

A negative result of this sort is not the most discouraging. This then prompted researchers to look at each condition of the theorem to see what was driving the result and in practical situations can we say something hopeful. The field of social choice theory was born. And as it happens to the greatest of works in their respective fields, the work is chewed, digested and assimilated. Dozens of proofs of this result have been given. One key ingredient in the proof of Arrow’s theorem is it requires all possible rankings to exist. That is, on certain dimensions we can have Meghalaya better than Tripura better than Chattisgarh better than Maharashtra. In a practical problem when we are ranking states on economic conditions this sort of a ranking seems unlikely. People have therefore examined the restrictions one can put on what kind of rankings we can have so that Arrow’s negative result doesn’t hold.

This is what great work is. It prompts hundreds of other smart people to push the boundaries of the field that a genius opens in one stroke. Today, this combined effort has made it possible for people to teach Arrow in an intermediate microeconomics course. When seemingly uninterested students too express shock and disbelief in class upon hearing this result and start thinking about what just hit them that you realize what phenomenal stuff Arrow has achieved. And this is just one of this great works. A post on his other extraordinary work will have to wait