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The Quest for the Holy Will

November 16, 2007

Democratic fundamentalism ranges from trying to export democracy at gun-point to voting on whether Californians should be allowed to buy horse meat for consumption. What’s worse is that more than 50 years ago an economist named Kenneth Arrow showed that most voting schemes necessarily yield very unpleasant outcomes. Since then the problem of aggregating knowledge that is dispersed among all individuals into a societal preference that could be called “the people’s will” has proven more difficult than expected.

In a very readable book: Decisions and Elections, mathematician Donald Saari takes the reader on a panoramic view of the many paradoxes and inherent injustices that arise when a group of people tries to make a common choice, what is sometime called “social choice”. Aside from Arrow’s Theorem, Saari also discusses Sen’s Theorem about the fact that the hypothesis of Minimal Liberalism where each individual is allowed to have priority of judgment over certain pairwise decisions necessarily implies that the resulting will of the people develops cycles, e.g. prefers A to B, B to C, and C to A. Saari shows how common such problems are in our daily lives and how unsuspecting people are towards these inherent flaws of democracy.

He however intends to try and resolve these riddles in a positive way. First of all, Arrow’s and Sen’s theorems are dissected to see what causes the cyclic outcomes. For instance, Arrow’s theorem only applies to voting schemes that satisfy among other things two conditions: that individuals be themselves rational, i.e. don’t have cycles in their preferences, and that the voting scheme satisfy “binary independence”, i.e. that when people choose between two alternatives, their preferences over other alternatives should be irrelevant. The problem is that this second hypothesis does not differentiate between rational voters and irrational voters. So it in effect nullifies the requirement that the individuals be rational. The heuristic is that “binary independence” is a condition about the “parts” of the society and ends up missing the peculiar characteristics of rationality present in the “whole”. In other words, the “whole is more than the sum of the parts”. This is a very appealing notion. For instance in economics it’s understood that prices (which are a way to aggregate knowledge) depend on a very complex web of interactions and interrelations.

The book ends with Saari’s own theorem which advocates the use of the Borda Count (after the French mathematician Jean Charles de Borda, a precursor of Condorcet). This voting schemes requires that when deciding between, say, 5 alternatives, each person give 4 points to their top choice, 3 to the second-best, etc…and 0 to the last one. This system does not satisfy binary independence but it does satisfy a modified “Intensity binary indipendence” which allows to distinguish between rational and irrational preferences. In this case the paradoxes vanish and a popular consensus that is rational arises. Likewise, Sen’s dilemma is resolved by introducing negative externalities.

What these theorems demonstrate, in my view, is not so much that democracy is ok after all, but rather that the plurality voting now in vogue is inherently flawed, and to really capture the complexity of society’s knowledge one cannot really isolate and analyze smaller parts (say one side of the coin, like trade deficits). Thus, for instance, prices which take into account myriads of interrelations have a better claim at being an approximation to the will of the people, than straightforward elections.

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