Abstract
We investigate necessary and sufficient conditions for the existence of Bayesian-Nash equilibria that satisfy the Condorcet Jury Theorem (CJT). In the Bayesian game G n among n jurors, we allow for arbitrary distribution on the types of jurors. In particular, any kind of dependency is possible. If each juror i has a “constant strategy”, σ i (that is, a strategy that is independent of the size n ≥ i of the jury), such that σ = (σ 1, σ 2, . . . , σ n . . .) satisfies the CJT, then by McLennan (Am Political Sci Rev 92:413–419, 1998) there exists a Bayesian-Nash equilibrium that also satisfies the CJT. We translate the CJT condition on sequences of constant strategies into the following problem:
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(**) For a given sequence of binary random variables X = (X 1, X 2, . . . , X n, . . .) with joint distribution P, does the distribution P satisfy the asymptotic part of the CJT?
We provide sufficient conditions and two general (distinct) necessary conditions for (**). We give a complete solution to this problem when X is a sequence of exchangeable binary random variables.
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Peleg, B., Zamir, S. Extending the Condorcet Jury Theorem to a general dependent jury. Soc Choice Welf 39, 91–125 (2012). https://doi.org/10.1007/s00355-011-0546-1
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DOI: https://doi.org/10.1007/s00355-011-0546-1