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 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: (**) 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.