Weighting Experts with Inaccurate Judges

We consider the problem of aggregating binary votes from an ensemble of experts to reveal an underlying binary ground truth where each expert votes correctly with some independent probability. We focus on settings where the number of agents is too small for asymptotic results to apply, many experts may vote correctly with low probability, and there is no central authority who knows the experts' competences, or their probabilities of voting correctly. Our approach is to designate a second type of agent -- a judge -- to weight the experts to improve overall accuracy. The catch is that the judge has imperfect competence just like the experts. We demonstrate that having a single minimally competent judge is often better than having none at all. Using an ensemble of judges to weight the experts can provide a better weighting than any single judge; even the optimal weighting under the right conditions. As our results show, the ability of the judge(s) to distinguish between competent and incompetent experts is paramount. Lastly, given a fixed set of agents with unknown competences drawn i.i.d. from a common distribution, we show how the optimal split of the agents between judges and experts depends on the distribution.