This article is the first in a series of many (I hope), that explores the theories and ideas around social choice and welfare and the role of ethics in economics. This short article discusses the Arrow Impossibility Theorem, first proved by the economist Kenneth Arrow, which lies at heart of the field of collective choice and social welfare.

First let’s laydown the setup and four axioms that form the basis for the theorem:

Given a set n {1,….., n} individuals who have individual preferences {R1……..Rn}, a welfare function w() which maps the set of individual  preferences to a social preference R.

  • Unrestricted Domain (U): For any logically possible set of individual preferences, there is a social ordering R.
  • Independence of irrelevant alternatives (I): The social ranking of any pair {x, y} will depend only on individual rankings of x and y.
  • Pareto Principle (P): If everyone prefers x to y, then x is socially preferred to y.
  • Non-dictatorship (D): There is no such person i such that, whenever this person prefers x to any y, the x is socially preferred to y, no matter what other people prefer.

Now let’s state the theorem,

The General Possibility Theorem (as Arrow called it): If there are at least three distinct social states, and finite number of individuals, then no social welfare function can satisfy  all U, I, D, and P.

It is often stated as that, a social welfare function that satisfies unrestricted domain, independence, and Pareto principle has to be dictatorial. The version of the theorem discussed is a watered down one and the proof is more logical than mathematical.

We begin by proving two lemmas:

Lemma 1: Spread of Decisiveness

If G (a set of individuals) is decisive over a pair {x, y}, then G is globally decisive.

Proof:  Take any pair {a, b} different form {x, y}, and assume that everyone prefers a to x and y to b. For those individuals not in G there are no other restrictions, but for those who belong to G we further impose that all people prefer x to y (remember G is decisive over {x, y}), hence the preference of people in G is a,x,y,b.

Now using the Pareto Principle a is socially preferred to x and y is socially preferred to b. Logically we can conclude a is socially preferred to b. Hence we have shown if G is decisive on {x, y} it is also decisive on {a, b}. This applies to all pairs {a, b} distinct from {x, y}. So G is globally decisive.

 Lemma 2: Contraction of Decisive Sets

If a set G of individuals is decisive (and G has more than one individual), then some reduced part of G (or a proper subset) of G is decisive as well.

Proof: Assume preferences over {x, y, z}.  Again we have x is social preferable to y as G is decisive. Now partition G in to two subsets G1 and G2. In G1 individuals prefer x to y and x to z with no specification on ranking between y and z, further; in G2 individuals prefer x to y and y to z. Putting above statements together we can conclude that z must be socially preferable to y. Since no one’s preference over {z, y} is specified except for the individuals in G2, who prefer z to y, hence G2 is decisive on {z, y} and by Spread of Decisiveness G2 is globally decisive. Similar argument goes for the set {x, z} and the set G1.

Hence we get either G1 or G2 is globally decisive. This concludes the proof for Contraction of Decisive Sets.

Now putting it all together we can arrive at Arrow’s Impossibility Theorem.

Proof of theorem:

By the Pareto Principle the set of all individuals is decisive. By Contraction of Decisive Sets we can arrive at a smaller subset that is also decisive. Take that set and arrive at yet another smaller subset of decisive individuals and continue till there is a set of one individual only this violates Non-dictatorship. Hence the impossibility of any welfare function satisfying all the above axioms (U, I, D, P).

The above proof was presented by Amartya Sen in at the Arrow lectures at Columbia University in 2012. It is somewhat non-rigorous, but proves the point that it is impossible to have a welfare function that is completely ‘fair’ i.e. satisfies all the above axioms at once. There is a longer and more rigorous proof available here. It requires basic set theory, logical deduction, and some patience on the part of the reader to understand.

Arrow’s Impossibility Theorem has major consequences for how a society collectively makes decisions. In the real world, a welfare function may refer to a voting rule that a democracy may use to choose between candidates in an election. We now know that there is no one perfect voting rule that will fairly reflects a society’s choice, but this does not mean we cannot improve on existing rules. In fact, it is an area of major theoretical research to find better welfare functions, by relaxing one or more of the above axioms.

Now that we know the inherent contradiction at the heart of the problem of social choice we can begin to better understand and analyze the outcomes in our democratic societies.

References & Further Exploration:

Discussion and Proof of the theorem Eric Pacuit on Youtube.

Arrow’s Impossibility Theorem by Amartya Sen and Eric Maskin

Amartya Sen’s Lecture at Columbia University

Note: The great economist Kenneth J. Arrow who first proved the theorem passed away on February 21, 2017. His work lies at the very foundation of modern economics and has influenced many a intellectual giants of our times. His timeless legacy will continue to live on not just in university departments, but in every democratic society across the world. He will be remembered.