Universally Robust Information Aggregation for Binary Decisions

We study an information aggregation setting in which a decision maker makes an informed binary decision by merging together information from several symmetric agents. Each agent provides the decision maker with a recommendation, which depends on her information about the hidden state of nature. While the decision maker has a prior distribution over the hidden state and knows the marginal distribution of each agent's recommendation, the correlation between the recommendations is chosen adversarially. The decision maker's goal is to choose an information aggregation rule that is robustly optimal. We prove that for a sufficiently large number of agents, for the three standard robustness paradigms - minimax, regret and approximation ratio - the robustly-optimal aggregation rule is identical. Specifically, the optimal aggregation rule is the random dictator rule, which chooses an agent uniformly at random and adopts her recommendation. For a small number of agents, this result no longer holds - the random dictator rule can be suboptimal for minimizing the regret even for two agents. We further characterize the minimal regret for any number of agents through the notion of concavification, and demonstrate how to utilize this characterization in the case of two agents.