Kenneth Arrow and the age of empires
Applying Nobel-Prize winning economist Kenneth Arrow’s Impossibility Theorem to Uttar Pradesh polls. Does it throw up an election result?
The most powerful nations of the world, including those that are democracies, are increasingly being controlled by leaders with overweening power. The work of the Nobel Prize winner Kenneth Arrow offers some fascinating insights into this phenomenon. But, taking the help of some hypothetical figures, let’s start at the beginning, with his remarkable Impossibility Theorem, applied to the mother of all elections—Uttar Pradesh.
Let us suppose all voters are asked to rank the three main leaders in the fray—Narendra Modi (or his surrogate), Akhilesh Yadav and Mayawati, from one to three. Then a social planner is given 100 million ballot papers and asked to choose the next chief minister of Uttar Pradesh. What would be the right way of doing this?
The social planner could use a simple majority rule in which the candidate that wins in one-on-one face-offs against every other candidate wins the election.
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Let us assume that 40 of a total electorate of 100 voters, comprising the Dalit-Muslim coalition, prefer Mayawati to Yadav and Yadav to Modi.
Another 30 comprising upper-caste Hindus and non-Yadav Other Backward Classes (OBC) prefer Modi to Mayawati and Mayawati to Yadav.
The last 30, comprising Yadavs, Muslims and the youth, prefer Yadav to Modi and Modi to Mayawati.
Then, in a straight fight, Mayawati defeats Yadav, who in turn defeats Modi. But in a contest between Modi and Mayawati, Modi wins. So the majority rule fails to yield a decisive victor.
If we revert to the first-past-the-post system in which the candidate with the maximum number of first-place ranks wins, then Mayawati wins decisively. But that is only because Yadav is acting as a spoiler by pipping Modi in the Yadav-Muslim-youth segment. If Yadav withdraws from the contest, then in a straight fight Modi defeats Mayawati.
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This prospect may seem irrelevant, but recall the 2000 US presidential election in which Al Gore lost the presidency to George Bush because of his 600-vote defeat in Florida, while Ralph Nader played spoiler, taking 100,000 votes that would have gone mostly to Gore. Now we begin to see the need for a voting rule that is not susceptible to such a reversal of outcomes, or as Arrow put it, a rule that satisfies the property of the “independence of irrelevant alternatives”.
Finally, a voting rule should reflect the preferences of voters, at least in the weak sense that if every voter prefers one candidate to another, then that candidate should be ranked higher by the voting rule.
With several voting rules on offer, surely there should be some that satisfy the reasonable criteria of being decisive, independent of irrelevant alternatives and respectful of the consensus view? And yet, Arrow proved, stunningly, that any social choice rule that satisfied all three conditions could only do so because of the presence of a dictator whose individual preference would be the social preference irrespective of the rankings of others. This is the celebrated Impossibility Theorem that has spawned a whole branch of research called social choice theory.
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While Arrow’s mathematical proof was rather long-winded, his student John Geanakoplos, now a professor at Yale University, has come up with three extremely elegant and short proofs. One of these provides a clue to understand the rise of supremely powerful leaders around the world. To understand this, we move from a model where voters choose between political candidates to a model where they choose between different social alternatives.
In the first step of the proof, Prof. Geanakoplos shows that any social agenda that attracts strong emotions, for example “minority appeasement”, or the immigration ban, and that, therefore, is placed either at the top or bottom of a voter’s ranking, must rank either at the top or bottom of the social preference. We will not go into the details of the proof but merely highlight the surprising result that even if Trump’s travel ban is at the top of half the voters’ preferences and at the bottom of the other half, the social ranking must put it either at the top or the bottom, never in the middle (details of proof at here
The next step of the proof shows that for every such extreme alternative, there must be a pivotal voter who, by changing his individual preference for this alternative from the bottom of his ranking to the top, would cause the social ranking of this alternative to jump from the bottom to the top. In the final step, it is shown that this pivotal voter must be a dictator.
Drawing a loose parallel with the real world, one might say that a citizen who can become pivotal in changing the social ranking of an emotive subject from the bottom to the top, must come to possess dictatorial powers. The mathematical proof merely requires him to flip the alternative from the bottom to the top in his personal preference.
However, in the real world, such pivotal players usually have to successfully articulate the dearly held but unexpressed beliefs of a large number of people about these emotive issues, creating sharp schisms in society. These latent beliefs reflect a mixture of suppressed voices, regressive ideologies and untamed instincts, and cannot be assumed to conform to any social norm other than the democratic norm that every voice matters.
However, such a dictator must face the problems that come with opening the Pandora’s box of social tension. On his ability to control the forces he has let loose rests the long-term success of his meteoric rise.
Rohit Prasad is a professor at MDI, Gurgaon and author of Blood Red River. Game Sutra is a fortnightly column based on game theory.
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