Condorcet Jury Theorem: The Dependent Case

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• Bezalel Peleg
• Shmuel Zamir
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Abstract

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 = (X 1,X 2, ...,X n, ...) 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 lim n→∞ P(∑X i>n/2)=1. For a general (dependent) distribution P we provide necessary as well as sufficient conditions for the CJT. Let p i = E(X i), p n = (p 1 + p 2, ...+ p n)/n and X n = (X 1 +X 2, ...+X n)/n. A consequence of our results is that the CJT is satisfied if lim√n( pn -1/2)=∞ and ∑ i∑ j≠i Cov(X i,X j) ≤ 0 for n > N 0. 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 p n, and p = limsup pn . Let y = liminf E( Xn - p n) 2, y *= liminf E| Xn - p n| and y *= limsup E| Xn - p n|. Necessary conditions for the CJT are that p ≥1/2 + 1/2 y∗ ,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≠i Cov(X i,X j) > 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.

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Bibliographic Info

Paper provided by The Center for the Study of Rationality, Hebrew University, Jerusalem in its series Discussion Paper Series with number dp477.

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Length: 22 pages
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Handle: RePEc:huj:dispap:dp477

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References

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1. Myerson, Roger B., 1998. "Extended Poisson Games and the Condorcet Jury Theorem," Games and Economic Behavior, Elsevier, vol. 25(1), pages 111-131, October.
2. Laslier, Jean-François & Weibull, Jörgen, 2008. "Commitee decisions: optimality and equilibrium," Working Paper Series in Economics and Finance 692, Stockholm School of Economics, revised 11 Mar 2008.
3. Ladha, Krishna K., 1995. "Information pooling through majority-rule voting: Condorcet's jury theorem with correlated votes," Journal of Economic Behavior & Organization, Elsevier, vol. 26(3), pages 353-372, May.
4. Roger B. Myerson, 1994. "Population Uncertainty and Poisson Games," Discussion Papers 1102, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
5. Wit, Jorgen, 1998. "Rational Choice and the Condorcet Jury Theorem," Games and Economic Behavior, Elsevier, vol. 22(2), pages 364-376, February.
6. Daniel Berend & Jacob Paroush, 1998. "When is Condorcet's Jury Theorem valid?," Social Choice and Welfare, Springer, vol. 15(4), pages 481-488.
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Cited by:
1. Alexander Zaigraev & Serguei Kaniovski, 2012. "Bounds on the competence of a homogeneous jury," Theory and Decision, Springer, vol. 72(1), pages 89-112, January.

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