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There is no perfect voting rule for three or more options. Shutterstock/Constantin Stanciu shutterstock

Kenneth Arrow’s legacy and why elections can be flawed

Stanford professor Kenneth Arrow was considered one of the most influential economists in history with monumental and lasting contributions to the field. He died in February this year, aged 95.

His work included some explanation for why election results can turn out as they do, not always the way most voters would prefer.

In 1972, Arrow was awarded the Nobel Memorial Prize in Economic Sciences for his fundamental work on demand and supply in competitive markets.

But it has been noted that Arrow could well have won multiple Nobel prizes for his contributions to social choice, financial securities and health economics.

Although his approach was highly abstract, the insights derived from his mathematical results have had a broad and deep impact. For example, his work on capturing risk and uncertainty has wide application in the finance sector.

Social choice

My own interest in Arrow’s work stems from his contribution to social choice theory – a field of economics that he helped to establish.

Social choice concerns making principled collective decisions by taking people’s preferences into account. These types of collective decisions are not just made in political elections and the Oscars but come up in our everyday lives, such as any panel decision when several candidates are interviewed for a job.

Insights from the field are also making their way into computer science as we consider building systems in which multiple agents – be they human, virtual or robots – need to coordinate and make joint decisions.

Arrow’s most profound result in social choice was achieved in the late 1940s when he was working towards a PhD from Columbia University.

Suppose all the available voters provide rankings of all the possible candidates and we want to use those reported rankings to come up with an aggregate ranking.

If we are to use some voting rule to find the aggregate ranking, the following seem to be pretty reasonable properties that one would like the rule to satisfy:

  1. If every voter prefers candidate A over candidate B, then the aggregate ranking puts A before B.

  2. If every voter’s preference between A and B remains unchanged, then the relative order of A and B in the aggregate ranking should not change either.

  3. There should not be a voter such that the aggregate ranking is always the same as that voter’s ranking. In other words, no single voter can act as a dictator and override the preferences of the other voters.

Arrow made the startling discovery that it is mathematically impossible for a voting rule to simultaneously satisfy these three properties.

One possible way to interpret Arrow’s impossibility theorem is that no voting rule is perfect and that even if each individual is rational, one cannot rationalise group choices.

Interestingly, the paradox identified by Arrow disappears when there are fewer than three candidates. For two candidates, selecting the candidate preferred by a majority satisfies all the three conditions.

Not as simple as A, B, C

Consider the case in which there are seven voters who have the following rankings for two candidates, Alice and Bob:

4 voters rank: Alice > Bob

3 voters rank: Bob > Alice

For any reasonable voting rule, Alice is higher than Bob in the aggregate ranking.

But if we add a third candidate, Charles, then mathematical havoc can occur. Say Charles also runs for election so that the seven voters now rank the candidates as follows.

2 voters rank : Alice > Bob > Charles

2 voters rank : Charles > Alice > Bob

3 voters rank : Bob > Alice > Charles

Consider plurality, the widely used rule that ranks candidates according to the number of times they are ranked first (in a simple election, for example, whoever gets the most votes wins). The rule satisfies properties 1 and 3, but not 2.

Why? That the rule does not satisfy 2 is evident from observing that when Charles participates in the election, Bob has the highest number of first positions – three times, where as both Alice and Charles are only in first position on two occasions each.

In this way, Charles acts as a spoiler for Alice, despite the fact that a majority of voters prefer her over Bob and Charles.

Election troubles

The fact that no rule satisfies all the three properties can wreak political havoc. In particular, the plurality rule is highly vulnerable to the spoiler effect.

Consider the 2000 US presidential elections, in which a majority of people from the crucial state of Florida preferred Democrat Al Gore over Republican George W. Bush.

But Ralph Nader’s presence as a presidential candidate resulted in Bush winning Florida. The rest, as they say, is history.

Although Arrow’s theorem can be viewed as rather dismal news, it popularised an axiomatic approach where procedures are judged on the basis of what logical properties they satisfy.

Although philosophers have been thinking about collective decision making for ages, Arrow helped approach these issues in a mathematical manner. More generally, Arrow’s insights have kick started new fields that have impacted the theory and practice of economics and political science.

It is said that great academics are not just original thinkers but inspire future generations as well. Arrow did not only come up with deeply profound ideas, five of his own students went on to win Nobel Memorial Prizes. This is no small feat and is just one measure of this great man.

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