Condorcet Jury Theorem: The Dependent Case

Bezalel Peleg and Shmuel Zamir

We provide an extension of the Condorcet Theorem. Our model includes both the Nitzan-Paroush framework of “unequal competencies” and Ladha’s model of “correlated voting by the jurors”. We assume that the jurors behave “informatively”, that is, they do not make a strategic use of their information in voting. Formally, we consider a sequence of binary random variables X = (X1,X2, ...,Xn, ...) with range in {0,1} and a joint probability distribution P. The pair (X,P) is said to satisfy the Condorcet Jury Theorem (CJT) if limn→∞P(∑Xi>n/2)=1. For a general (dependent) distribution P we provide necessary as well as sufficient conditions for the CJT. Let pi = E(Xi), pn = (p1 + p2, ...+ pn)/n and Xn = (X1 +X2, ...+Xn)/n. A consequence of our results is that the CJT is satisfied if lim√n(pn-1/2)=∞ and ∑i∑j≠iCov(Xi,Xj) ≤ 0 for n > N0. The importance of this result is that it establishes the validity of the CJT for a domain which strictly (and naturally) includes the domain of independent jurors. Given (X,P), let p = liminf pn, and p= limsuppn. Let y= liminf E(Xn - pn)2, y*= liminf E|Xn - pn| and y*= limsup E|Xn - pn|. Necessary conditions for the CJT are that p ≥1/2 + 1/2y∗,p ≥ 1/2 + y, and also p ≥ 1/2 + y∗. We exhibit a large family of distributions P with liminf 1/n(n-1) ∑i∑j≠iCov(Xi,Xj) > 0 which satisfy the CJT. We do that by ‘interlacing’ carefully selected pairs (X,P) and (X′,P′). We then proceed to project the distributions P on the planes (p,y∗) and (p,y), and determine all feasible points in each of these planes. Quite surprisingly, many important results on the possibility of the CJT are obtained by analyzing various regions of the feasible set in these planes.

March, 2008
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