Extending the Condorcet Jury Theorem to a general dependant jury

Economics

Further details

We investigate sufficient conditions for the existence of Bayesian-Nash equilibria that satisfy the Condorcet Jury Theorem (CJT).  In the Bayesian 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".  si (that is, a strategy that is independent of the size n ≥  i  of the jury), such that  sn = (s 1,s 2, . . . ,s n . . .) satisfies the CJT, then by LcLennan (1998) there exists a Bayesian-Nash equilibrium which also satisfies the CJT. We translate the CJT condition on the sequence 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.