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There was a lot of fuss when Narendra Modi became the prime minister even though less than a third of those who trudged to the polling booths during the 2014 general election voted for the Bharatiya Janata Party (BJP). The truth is that no Indian prime minister has ever got the top job with the approval of more than half the electorate. Parliamentary majorities are rarely perfect representations of vote share in the voting system that India has.

A similar issue cropped up after Donald Trump began to gain popular support in the months leading to the US presidential election due later this year. There were fears that the voting system in the primaries could help Trump eventually move into the White House. Questions have thus been raised about the way US presidential candidates are chosen.

A few economists have jumped into the debate. They have good reasons to. One fascinating part of economics—social choice theory—shows it is never easy to aggregate the different preferences of individual voters into a coherent collective result that leaves citizens satisfied. In short, there is no perfect voting system in a democracy. The individual choices of millions of voters often get aggregated in puzzling ways.

Two Nobel laureates from Harvard University—Amartya Sen and Eric Maskin—used social choice theory to argue in The New York Times that the way voting in the Republican Party primaries is structured allowed Trump to gain, even though several candidates who stood against him in the primaries might well have beaten him in one-on-one contests. The vote against Trump was split, so he could win the primaries despite getting less than half the votes cast.

India chose the British system of giving a parliamentary seat to that candidate who wins the maximum number of votes, even though more people have collectively voted for other candidates—a system of plurality voting instead of majority voting, in which the winning candidate has to get more votes than all his opponents combined.

There is some history to this decision by the writers of the Indian Constitution to choose the first-past-the-post system over alternatives such as proportional voting. One person whose advice they sought out on which voting system to choose was Eamon de Valera, the Irish patriot and passionate supporter of Indian independence. De Valera told his Indian friends, including Rajendra Prasad, that an alternative system such as proportional representation would not work in India because of the sheer political complexity of the country. No single party would get a parliamentary majority given our social divisions. India would be cursed with unstable governments.

Then in 1951, a few years after de Valera advised the Indian political leadership about the ideal voting system, a young economist named Kenneth Arrow wrote a stunning paper that has had profound implications on how we think about society.

Arrow, who won the Nobel Prize for economics in 1972, placed a few sticks of dynamite under the popular view that democracies do a good job in terms of aggregating individual preferences. A very good explanation is available in the Stanford Encyclopaedia of Philosophy.

Arrow showed that there is no voting rule that satisfies the four rigorous axioms of decisiveness, consensus, non-dictatorship and independence. The Arrow Impossibility Theorem—the man himself more cheerfully named it the General Possibility Theorem—not only inaugurated the modern era of social choice theory, but also led to a rich vein of academic research as other economists tried to find ways around essentially pessimistic conclusions of the impossibility theorem.

This Nobel speech on the possibility of social choice by Sen, one of the giants of modern social choice theory, is one of the clearest introductions to its paradoxical insights.

The Arrow theorem was preceded several centuries ago by the work of Marquis de Condorcet, an 18th century French polymath. The Condorcet Paradox provides an easy way to illustrate the problem of social choice. Assume there are three voters who have to choose between three political candidates: Modi, Rahul Gandhi and Arvind Kejriwal. Now, here is how their political preferences look.

Two voters prefer Modi to Gandhi. Two voters prefer Gandhi to Kejriwal. Two voters prefer Kejriwal to Modi. This ordering does not satisfy the key mathematical requirement of transitivity, where if x is greater than y and y is greater than z, then x has to be greater than z. The result in social choice is cyclical rather than transitive—and hence confusing. There is no clear winner.

Now, look at the Condorcet Paradox in another way. Let us assume that the first preference gets three points, the second preference gets two points and the third preference gets one point. All three candidates end up with six points each.

In other words, the result is not coherent. Nobody is happy. For example, a Modi win leaves the two voters who prefer Kejriwal over Modi unhappy. It is the same with a Kejriwal win or a Gandhi win.

In real life, much depends on the system of voting that a group—be it a country, a committee, a group of friends or a family—depend on to come to a collective choice. Is it by consensus or plurality voting or proportional voting or some other rule? One interesting idea that has come up in recent years is quadratic voting.

Legal scholar Eric Posner describes the voting system with great clarity: “Quadratic voting is a procedure that a group of people can use to jointly choose a collective good for themselves. Each person can buy votes for or against a proposal by paying into a fund the square of the number of votes that he or she buys. The money is then returned to voters on a per capita basis."

The efficiency of quadratic voting increases as the number of voters increases. The reason that they propose quadratic payments (rather than, say, cubic) is that under the quadratic rule, the marginal cost of each vote increases proportionately with the number of votes (or marginal benefits).

In a recent paper, Posner and his Chicago University colleague E. Glen Weyl use the example of gay rights to make the case for quadratic voting, where those who passionately believe in equal rights for gays could use quadratic voting to buy out the opposition. Closer to home, such a system could be used to deal with the problems of land acquisition in particular or eminent domain in general.

Not everyone, however, believes quadratic voting may improve the democratic process. American economist and polymath Tyler Cowen argued that one of the important aspects of democratic choice is the process of deliberation through which some groups (even if they are in the minority) may be able to convince others of their position.

Tyler fears that quadratic voting may help cement extremist positions and urges us to consider the anti-abortion movement in lieu of the gay rights movement as the relevant minority group: what happens if a group such as the anti-abortion camp buys out the opposition and succeeds in getting its way?

“By elevating persuasion over trading in politics (at some margins, at least), we encourage centrist and majoritarian groups," writes Cowen. “We encourage groups which think they can persuade others to accept their points of view. This may not work well in every society but it does seem to work well in many."

How do groups generally deal with the paradox of social choice? There are three standard responses that economists tell us about.

First, someone may set an agenda that includes voting in steps. Readers can spend a few minutes to figure out what happens in the Condorcet example given above: Say, first a choice between Gandhi and Kejriwal, followed by another between the winner of that vote and Modi. The results can be interesting as the agenda is changed.

Or suppose there is a corporate board meeting where board members have to choose between three alternative strategies. The chairman of the board can set an agenda. He first asks board members to choose between two competing strategies—and then again choose between the winning strategy and the third option. Such strategic agenda setting allows social groups to bypass the Condorcet Paradox, but also gives the agenda setter immense discretionary power. In other words, the person who sits at the head of the table can get his way.

Second, voters themselves decide to think strategically. So, all those who do not want Kejriwal but feel that Modi has a better chance at winning than Gandhi will transfer their vote to the candidate most likely to win against Kejriwal.

One often sees this in Indian elections, where third candidates get hammered at the polling booth when there is a very polarized election. Take the recent Assam assembly election. Is it possible that the people who voted for the All India United Democratic Front decided to strategically back the Congress because they thought it had the most realistic chance of stopping the BJP? Think about it.

Third, the voting paradox can be solved if one person has a dictatorial veto. That seems politically incorrect, so it is best to give a less contentious example. Suppose four members of a family want to go out for an evening meal. They cannot decide on the four options: Indian, Italian, Thai and Chinese. They face a culinary version of the Condorcet Paradox. One person then has to take the decision—and everybody agrees or stays at home.

And that is how Saturday night family squabbles are dealt with by domestic dictators, unless the other members of the family learn to vote strategically.

Economics Express runs weekly, and features interesting reads from the world of economics and finance.

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