This "cycling" seems very undesirable. Worse still, if we have a runoff (say A versus B and then the winner versus C), the order of the voting matters. With this paradox, Condorcet showed that some voting rules seem to be problematic.

Arrow, however, went much further. He stipulated that any rule that aggregated individual choices into a collective (or "social") choice should satisfy four principles or axioms. Roughly put: (i) no person's choice should exclusively determine the social choice; (ii) the social choice should account for all possible individual preferences; (iii) in comparing any two alternatives another option should not matter; and (iv) if all individuals rank one option above another the social ordering agrees.

Hard to disagree with any of those. Yet, in an epoch-making monograph published in 1951, Arrow proved that it is impossible to aggregate individual preferences into a social choice without violating at least one of those four axioms. No rule – not assigning points to choices, not Australia's "Hare-Clarke" system, nothing you can think of – will do the trick.

Massive contribution

The result, known as 'Arrow's Impossibility Theorem' has had breathtakingly large implications for democratic systems. Any system has at least one major problem. We have to give up on something important to make collective decisions. Political scientists and economists have been in a tailspin ever since – trying to figure out which is the least bad thing to give up on, why, and when.

It is no overstatement to say this is probably the second-greatest intellectual contribution of the 20th century – behind Einstein's theory of General Relativity. But Arrow didn't stop there.

The work for which he received the Nobel Prize in Economic Sciences in 1972 (the youngest ever recipient at age 51) concerned another age-old problem. Arrow, along with French economist Gerard Debreu, provided an elegant and comprehensive mathematical proof of Adam Smith's notion that competitive markets lead to an efficient allocation of resources.

As with all economic theories, the conclusion is true under a set of assumptions. The importance of the mathematical rigour, though, is that one can't question the logic – it is inescapable – only the assumptions.


Two of the most important assumptions behind Arrow's proof of what became known as the "First Fundamental Theorem of Welfare Economics" are that markets are complete and that all parties have symmetric information. In practice, of course, neither of these two things hold.

But by highlighting the way in which these assumptions are crucial, it focused the attention of economists on settings in which they are violated. Pollution is a notable example. Without a market for, say, carbon-dioxide emissions, the overall economy will not deliver an efficient allocation of resources. In particular, there will be too much carbon emitted. A carbon tax (or emissions trading scheme) creates this missing market, and rescues Adam Smith's invisible hand.

Markets need help, too, when it comes to information asymmetries. As fellow Nobel Laureate George Akerlof subsequently showed, if one party knows more than another (perhaps about their health, or the quality of the car they wish to sell) then markets can break down. Modern health systems, like "lifetime community rating" in Australia or "Obama-Romney care" in the US are designed to address this informational problem.

Far from being an apologist for unfettered markets, Arrow's work showed when government interventions can help, and when they hurt. No ideology – just the immutable logic of mathematics. Our current crop of politicians would do well to read his work.

Kenneth Arrow was a social scientist the likes of which we will not see again any time soon. He illuminated the biggest issues of the age: voting and markets, and counted four fellow Nobel Laureates among his PhD students. He was the very model of modern social scientist.

If Ken Arrow had been a religion, I would have converted.

Richard Holden is professor of economics at UNSW Business School.