A Random Dictator Is All You Need

We study information aggregation with a decision maker aggregating binary recommendations from symmetric agents. Each agent's recommendation depends on her private information about a hidden state. While the decision maker knows the prior distribution over states and the marginal distribution of each agent's recommendation, the recommendations are adversarially-correlated. The decision maker's goal is choosing a robustly-optimal aggregation rule. We prove that for a large number of agents, for the three standard robustness paradigms - minimax, regret and approximation ratio - the unique optimal aggregation rule is random dictator. We further characterize the minimal regret for any agents' number through concavification.