On a three-alternative Condorcet jury theorem

We investigate whether the simple plurality rule aggregates information efficiently in a large election with three alternatives. The environment is the same as in the Condorcet Jury Theorem (Condorcet (1785)). Voters have common preferences that depend on the unknown state of nature, and they receive imprecise private signals about the state of nature prior tovoting. With two alternatives and strategic voters, the simple plurality rule aggregates information efficiently in elections with two alternatives (e.g., Myerson (1998)). We show that there always exists an efficient equilibrium under the simple plurality rule when there are three alternatives as well. We characterize the set of inefficient equilibria with two alternatives and the condition under which they exist. There is only one type of inefficient equilibrium with two alternatives. In this equilibrium, voters vote unresponsively because they all vote for the same alternative. Under the same condition, the same type of equilibrium exists with three alternatives. However, we show that the number and types of coordination failures increase with three alternatives, and that this leads to the existence of other types of inefficient equilibria as well, including those in which voters vote informatively.