Consequences of the Condorcet Jury Theorem for beneficial information aggregation by rational agents

Naive Condorcet Jury Theorems automatically have "sophisticated" versions as corollaries. A Condorcet Jury Theorem is a result, pertaining to on election in which the agents have common preferences but diverse information, asserting that the outcome is better, on average, than the one that would be chosen by any particular individual. Sometimes there is the additional assertion that, as the population grows, the probability of an incorrect decision goes to zero. As a consequence of simple properties of common interest games, whenever "sincere" voting leads to the conclusions of the theorem, there are Nash equilibria with these properties. In symmetric environments the equilibria may be taken to be symmetric.