How Many Representatives Do We Need? The Optimal Size of a Congress Voting on Binary Issues

Aggregating opinions of a collection of agents is a question of interest to a broad array of researchers, ranging from ensemble-learning theorists to political scientists designing democratic institutions. This work investigates the optimal number of agents needed to decide on a binary issue under majority rule. We take an epistemic view where the issue at hand has a ground truth ``correct'' outcome and each one of n voters votes correctly with a fixed probability, known as their competence level or competence. These competencies come from a fixed distribution D. Observing the competencies, we must choose a specific group that will represent the population. Finally, voters sample a decision (either correct or not), and the group is correct as long as more than half the chosen representatives voted correctly. Assuming that we can identify the best experts, i.e., those with the highest competence, to form an epistemic congress we find that the optimal congress size should be linear in the population size. This result is striking because it holds even when allowing the top representatives to become arbitrarily accurate, choosing the correct outcome with probabilities approaching 1. We then analyze real-world data, observing that the actual sizes of representative bodies are much smaller than the optimal ones our theoretical results suggest. We conclude by examining under what conditions congresses of sub-optimal sizes would still outperform direct democracy, in which all voters vote. We find that a small congress would beat direct democracy if the rate at which the societal bias towards the ground truth decreases with the population size fast enough, and we quantify the speed needed for constant and polynomial congress sizes.