Arrow's Impossibility Theorem
Arrow's Impossibility Theorem
The most stunning result in economics?
Einstein’s theory of relativity is quite well-known, even in non-physicist circles. Unfortunately, various equally fascinating theorems in economic theory have not caught on in popular culture. Even many undergraduate students of the subject are not aware of the First Fundamental Theorem of Welfare Economics (though a reduced form is included in most standard microeconomics texts) or Arrow’s Impossibility Theorem. Both may be mathematically dense, but their results are easily discernible even to the non-expert. There are other beautiful concepts whose proofs are less difficult to grasp, like the Vickrey-Clarke-Groves auction system. Yet it is not well-known. This post, and maybe some more in the future, is my little contribution towards correcting this wrong.
This particular post is a simple explanation of the Arrow Impossibility Theorem, a truly stunning result. The original proof can be found in Arrow’s Social Choice and Individual Values (1951), and a slightly amended one in Amartya Sen’s Collective Choice and Social Welfare (1970). Both are fairly mathematically dense (though accessible to an interested reader) but enjoyable reads. I have made a video series on the theorem which follows Sen’s book. It can be found here. It can be followed by anybody with Indian 12th standard mathematics.
To understand Arrow’s theorem, one needs to understand social choice theory.
A social state is any description of the state of society. The particular details are not important, but every individual in society must have some preferences over these social states, and order these preferences.
Example 1: There may be 3 social states x, y and z and 3 individuals in society.
x: high income inequality, high growth, communalised politics.
y: high income equality, low growth, secular politics.
z: high income inequality, high growth, secular politics.
x, y and z may fit the social states associated with the right, left and centrist parties in India (resp. BJP, Left, Congress).
Then, the first individual may prefer x to y, y to z and z to x. The second and third individuals may have some other ranking of the three states.
The field of social choice theory (henceforth SCT) studies how we can “input” the rankings of individuals, and decide what “Society” ranks different social states as. In a sense, it is like the study of voting systems (but not exactly).
In SCT, we define something called a social welfare function. A function is simply something that takes in input(s) and returns an output. Here, individual rankings are inputted and a social ranking is outputted. Each function corresponds to a particular method of creating social rankings.
Example 2: A social welfare function could be the method of majority ranking. It would be defined as -
Ask every individual their rankings
Check which social state is ranked 1st by the largest number of people
Declare this social state as rank 1 for Society
Similarly, check which social state is ranked highest most after the above social state
Declare this state as rank 2 for Society
Continue until you rank all social states
Example 3: Another possible social welfare function could be a unanimous function. The method is -
For any two social states, check if everybody prefers one to the other
If everybody prefers one to the other, then Society does as well
If even one person does not prefer one of them to the other, then Society is indifferent between the two
In both these examples, we were given individual rankings and we returned a social ranking.
We would like our social welfare functions to have some desirable characteristics, and then see if we can find a function that satisfies these characteristics. This is where Arrow’s Impossibility Theorem comes into play.
We define 5 “desirable” characteristics of any such function.
Ordering: The method of social ranking must be complete, reflexive and transitive. These are fairly simple conditions to understand.
a. Completeness: for any two social states, we should be able make some comparison. Either one is preferred to another, or they are same.
b. Reflexive: every social state should be indifferent to itself
c. Transitivity: suppose social state x is preferred by Society to y, and y to z. Then Society should prefer x to z as well.
Unrestricted Domain: Very simply put, we should be able to output a social ranking for any possible collection and combination of individual rankings. If this were not true, for instance, it could be that an election will not return any result if people vote in a certain way.
Pareto Principle: If everybody in society prefers one social state to the other, so should Society.
Independence of Irrelevant Alternatives: When we check if Society prefers social state x to y, we should only be considering each individual’s rankings between x and y. For example if we wish to know if the American Society prefers Trump to Biden, we should not care how individuals compare Trump to Obama.
Non-Dictatorship: There should be no person who can unilaterally decide what society prefers. More precisely, there should be individual such that if he prefers social state x to y for any two x and y, then society prefers x to y as well (irrespective of others’ preferences).
These characteristics appear fairly light and reasonable. One would expect any good social ranking system to satisfy them. If they did not, we would seriously doubt the sanity of the system. Well, so we must:
(Arrow’s Impossibility Theorem) No social welfare function satisfies the above 5 conditions.
This is an incredibly stunning result indeed. Here, we have 5 very basic conditions and yet no social welfare function can satisfy it.
We can see how these conditions fail for the majority voting method.
Example 4: Suppose there are 3 social states, x, y and z and 3 individuals, 1, 2 and 3.
1’s rankings: x>y>z
2’s rankings: y>x>z
3’s rankings: z>y>x
Then, 2 out of 3 people prefer x to y. So Society should prefer x to y.
Similarly, 2 people prefer y to z. So Society should prefer y to z.
But then by transitivity (Condition 1c), Society should prefer x to z.
Yet, 2 people also prefer z to x. So Society must prefer z to x according to our method.
This is naturally a violation of condition 1c.
A visual explanation of this can be found here.
The last example was discovered in the 18th century by a French mathematician Condorcet. It is thus called Condorcet’s paradox.
Condorcet found that transitivity was violated for one particular method of social choice. Arrow’s intellectual leap was to first define 5 simple conditions, among them transitivity, and then show that no method satisfies these conditions. In fact, the conditions are so well-defined that if you drop even one of them, the theorem falls apart. This makes the theorem extremely mathematically elegant.
Unfortunately, the proof of the theorem cannot be done here, since it requires quite a bit of math. For that one may read the original books, or check out my video series. Nevertheless, the beauty and elegance of the theorem can be understood without this, too.
In a sense, Arrow dismantled the idea of “rational democracy” with this method, since one can barely call any social ranking system which does not satisfy these conditions rational (or sane). After the publication of this theorem, Sen also proved the Paretian liberal paradox, another depressing result.
This sense of doom partially lifts when we ease some of the conditions, especially 1 and 2, and see that we can find functions which satisfy them. A lot of it is in Sen’s book, and some of it in the video series as well.
Notes
VIDEO SERIES LINK: https://www.youtube.com/playlist?list=PLQ7w-b6FKrM4azoGOV12XIW7tyI7f4usm
While I don’t think anybody who has read the original theorems will be reading this post in the near future, they would be right to point out that I have been imprecise with my terminology (esp. that of social welfare functions and “rankings”). It was done to ease the explanation.