The non-dictatorship axiom

Kenneth J Arrow remains the youngest economist to win the Nobel Prize, which he did along with John Hicks in 1972. But the sterling quality that sets him apart was his profound sense of humanity, his appeal really, to our collective responsibility toward one another, to the imperative of equality, and his distaste for discrimination.

Kenneth J Arrow ranks as one of the most influential economists of the 20th century. He made original contributions to social choice theory, general equilibrium theory, growth theory, and the analysis of discrimination against groups. He remains the youngest economist to win the Nobel Prize, which he did along with John Hicks in 1972. But the sterling quality that sets him apart was his profound sense of humanity, his appeal really, to our collective responsibility toward one another, to the imperative of equality, and his distaste for discrimination.

In his classic 1951 monograph ‘Social Choice and Individual Values’ (SCIV), Arrow explains, with humour, the ethical principle of treating others the way you would want them to treat you, quoting from an epitaph on an English gravestone: ‘Here lies Martin Engelbrodde/Have mercy on my soul, Lord God/As I would do were I Lord God/And thou were Martin Engelbrodde.’ Simple and profound.

Born in 1921, in New York City and growing up during the Great Depression, the young Arrow embraced socialism in his youth. He would later move away from socialism, but his views retained a Left-leaning philosophy. He earned a Bachelor’s degree in 1940 in mathematics, and then a Master’s degree at Columbia University. While at Columbia, he decided to change his field to economics.

Arrow’s great contribution is the famous Impossibility Theorem, in which he asks, “if it is formally possible to construct a procedure for passing from a set of known individual tastes to a pattern of social decision-making” . He goes on to illustrate this problem with the example of the paradox of voting in democracies: It is impossible to have a fair rule for aggregating the preferences of a group of individuals. The only rule that satisfies some combination of reasonable restrictions is to take one of the individuals to be a ‘dictator’, and to just always follow that individual’s preferences. That clearly is not a fair rule.

Here is a basic version of Arrow’s Impossibility Theorem: Let there be a finite number N of individuals that have to rank a finite set of alternatives A. There is need for a rule to combine the preferences of each individual into a preference for the whole group. Preferences constitute a linear order of the set of alternatives, from least to most acceptable. The rule for combining these preferences must be seen as the constitution of a democratic process of combining the individual preferences.

The rule must satisfy two requirements: (i) If for two alternatives a, b ∈ A, all the individuals prefer b to a, then the outcome for the rule should be that the group as a whole also prefers b to a. This is the requirement of respecting unanimity; (ii) It should not be possible for an individual to change the group decision on the order of a and b by manipulating his or her preference for a third alternative, c. This is the requirement of independence of irrelevant alternatives.

Arrow establishes that the only possibility for a rule satisfying these two requirements is if there is an individual n, called the ‘dictator’, such that the rule is to take the group preference always to be identical to the preference of this individual. Arrow’s theorem points to the fact that in collective decision-making, individual rationality can often result in societal irrationality.

Arrow’s theorem is brilliant and, considering it was formulated 73 years ago, prescient. It is brilliant because to prove it, you do not need any complicated mathematics, just an ability for sustained reasoning; prescient because it is relevant today as never before in democracies around the world, positing that though a democracy’s fundamental task is to aggregate the diverse preferences of the population into some coherent social preference, which could be used to make rational policy choices, this is not always the case.

Read SCIV, a fascinating, relevant, and easy book to read. The impossibility paradox led to the birth of an entirely new field of research for economists and political scientists, called social choice theory, and formed the basis of Amartya Sen’s work. Take any model to analyse political competition or to explore voting patterns from any perspective, in one way or the other, it will bring you back to Arrow’s impossibility result.

Arrow’s theorem is not far removed from the real world of politics in our own times. Look closely at the political discourse in the democracies of the world, and the promises being made by politicians, and the irrationality that prevails in our collective life manifests: Deception, self-deception, and post-truth. Kenneth Arrow’s presence looms large even today.