Extending the Condorcet Jury Theorem to a general dependent jury

We investigate necessary and sufficient conditions for the existence of Bayesian-Nash equilibria that satisfy theCondorcet Jury Theorem (CJT ). In theBayesian game Gn 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: (**) For a given sequence of binary random variables X = (X1, X2, . . . , Xn, . . .) 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.