Agreement Implies Accuracy for Substitutable Signals

Inspired by Aumann's agreement theorem, Scott Aaronson studied the amount of communication necessary for two Bayesian experts to approximately agree on the expectation of a random variable. Aaronson showed that, remarkably, the number of bits does not depend on the amount of information available to each expert. However, in general the agreed-upon estimate may be inaccurate: far from the estimate they would settle on if they were to share all of their information. We show that if the experts' signals are substitutes -- meaning the experts' information has diminishing marginal returns -- then it is the case that if the experts are close to agreement then they are close to the truth. We prove this result for a broad class of agreement and accuracy measures that includes squared distance and KL divergence. Additionally, we show that although these measures capture fundamentally different kinds of agreement, Aaronson's agreement result generalizes to them as well.