Condorcet jury theorem

M.J.A.N. de Caritat, Marquis de Condorcet, studied the mathematical problem of how best to combine the opinions of several individuals so as to form a group decision.

Under Rousseau's theory of the general will, it is assumed that all group members wish to obtain what is best for the group; the problem is that they differ as to their opinion of what the best decision should be. The situation is best exemplified nowadays by a trial jury, in which all members have the same desire — to convict a guilty party, and acquit an innocent one — but have different opinions as to the accused party's innocence or guilt (cf. also Social choice).

Condorcet makes the simplifying assumption that all the individuals have equal competence (probability of making the correct choice), that this competence is greater than $0.5$, and that these probabilities are independent (cf. also Independence). He also assumes that there are only two alternatives available. Moreover, he implicitly assumes that, for each individual, the probability of a type-I error (convicting an innocent man) is the same as that of a type-II error (freeing a guilty man); see also Statistical test. Under these circumstances, it is not difficult to prove that the decision of a majority of the voters is more likely to be correct than that of the minority. Moreover, as the number of jury members increases, the probability that the group majority will make the correct decision approaches $1$.

It is reasonable to look for modifications of the assumptions. The easiest modification assumes that different individuals have different levels of competence: each individual, $i$, has probability $p_i$ of making the correct choice. In this case, the probability of a correct group decision is maximized by weighted voting, in which individual $i$ is give a weight $w_i$ proportional to the logarithm of $p_i/(1-p_i)$.

Another modification assumes different probabilities for type-I and type-II errors. This can be handled by, essentially, giving an artificial advantage to one of the two sides: alternative $A$ will be chosen if its vote tally surpasses that of $B$ by a sufficiently large margin.

A more complicated modification assumes that the different individuals' competences are not independent. In this case it is still possible, on the basis of voting, to decide which alternative is more likely to be correct, but the formulas for this can be quite complex.

Finally, Condorcet tried to generalize the method to the case of three or more alternatives. In this case, he found that his method can easily lead to contradictions: this is known as the Condorcet paradox.

References

[a1]	N.C. de Condorcet, "Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix" , Paris (1785)
[a2]	B. Grofman, "Judgmental competence of individuals and groups in a dichotomous choice situation" J. Math. Sociology , 6 (1978) pp. 47–60
[a3]	S. Nitzan, J. Paroush, "Optimal decision rules in uncertain dichotomous choice situations" Internat. Economic Review , 23 (1982) pp. 289–297
[a4]	L.S. Shapley, B. Grofman, "Optimizing group judgmental accuracy in the presence of interdependencies" Public Choice , 43 (1984) pp. 329–343