Cognitive Democracy: Condorcet with Competence

We usually think of democracy as a way of aggregating diverse preferences but we can also imagine that we share similar preferences and that what we disagree about is the best way to achieve those preferences. From this perspective, democracy can be thought of as a tool for information aggregation. Using simple probability theory, Condorcet showed in 1785 that even when each individual voter has only a slightly better than chance probability of choosing the bettier of two options the probability that majority rule chooses the better outcome quickly goes to 1 as the number of voters increases (the wisdom of the crowds).

A number of writers at Crooked Timber have been discussing Knight and Johnson’s The Priority of Democracy, one strand of which involves such an cognitive defense of democracy. Cosma Shalizi, for example, writes:

Democratic debate is a tool for cognition, for harnessing the dispersed knowledge of the citizens and their diversity of perspectives and insights.

But does an cognitive defense of democracy lead to universal suffrage? Or does it suggest what Melissa Schwartzberg calls “epistocracy”, rule by the educated? (See also Henry Farrell’s comments). The wisdom of the crowds breaks down when the crowd’s errors are systematically biased rather than random. As Peter Boettke notes, Bryan Caplan makes a strong case in The Myth of the Rational Voter that better educated voters are less systematically biased than the average voter and more likely to agree with experts on questions of fact.

When voters are not equally competent some remarkable mathematical results show that the best cognitive democracy is not universal suffrage and one-person, one-vote but a specific form of weighted voting.

Begin with a simple example. Suppose there is one correct decision and there are three voters each trying to reach the correct decision with competence levels of {.55, .55, .55}, where the competence levels are just the probabilities that each voter chooses the correct decision. The best a dictator could do in choosing the correct decision is .55 but if use majority rule the probability of reaching the correct decision is 0.57475, higher than that of any individual voter. (We reach the correct decision if all three voters reach the correct decision which has prob .55^3 or if two voters reach the correct decision and one does not, as this can happen in three ways the probability of the latter is 3*.55*.55*(1-.55) for a grand total of .57475.) Moreover, if we were to increase the number of voters to 100, the probability of majority rule reaching the correct decision goes to 84%–far above that of any dictator, this is the essence of Condorcet’s theorem.

Now let’s assume that the voters have competences of {.55,.60,.70}. Majority rule, using the same reasoning as before, gets us a democratic competence level of .673, not bad but notice that this is less than the competence level of the highest competence individual. The ideal voting system in this case would weight voter three enough so that she determines the outcome, thus giving democracy a competence level of .7.

More generally, if the voter competences levels are {p1,p2,p3} then the cognitively most efficient voting scheme gives each voter a weight of Log[pi/(1-pi)]–the result is remarkable for a being such a simple formula of the voter’s own competence level (note that the individual’s weighting is not a function of the competency levels of the other voters.) The result was shown first in this context by Nitzan and Paroush, Nobel-prize winner Lloyd Shapely and Bernard Grofman also made important contributions and see Grofman, Owen, Feld for some related results.)

Democracies make many decisions which are information based (Does Iraq have weapons of mass destruction? Will an invasion make the US safer? Do phthalates cause significant health risks?). Note also that we might also use this method for many committee decisions. Which scientific approach is deserving of greater funding? Which marketing plan should we adopt? Is surgery the best option? and in these decisions weighting votes by a measure of competence, which can be estimated from past decisions, may lead to significant improvements in outcomes.

Voters have diverse preferences not just competences but we could combine cognitive and preference aggregation theories of democracy by using high competence voters from different demographics categories to estimate what people would think about issues if only they were better informed. In this way we can distinguish differences due to knowledge from those due to preferences and we could upweight the competent while maintaining demographic balance thus creating a cognitive democracy based on enlightened preferences.