A Possibility Theorem on Information Aggregation in Elections

We provide a simple condition that is both necessary and sufficient for aggregation of private information in large elections where all voters have the same preference. In some states of the world, all voters prefer A; and in other states, all voters prefer B. Each voter draws a private signal independently from a distribution conditional on the state. According to our condition, there should be a hyperplane in the simplex over signals that separates the conditional distributions in states where A is preferred from those in states where B is preferred. If this condition is satisfied, information is aggregated in an equilibrium sequence: even under incomplete information, the preferred outcome obtains almost surely in each state. If the hyperplane condition is violated, there exists no feasible strategy profile that aggregates information. Therefore, information aggregation holds only for special environments.