Bayesian Persuasion With Complementary Information

A decision maker chooses between the status quo and an alternative action under uncertainty. The uncertainty is two-dimensional, and its realization in each dimension is independent. For each dimension, there is an expert who provides information only on that dimension. Both experts strictly prefer the alternative action and simultaneously provide information to the decision maker. We compare the experts' highest equilibrium payoff in this game with the payoff in an auxiliary game where they collude and can choose any information structure. We show that the experts strictly gain from collusion if and only if one of the experts can unilaterally persuade the decision maker to take the alternative action with positive probability.