Optimizing Voting Order on Sequential Juries: A Median Voter Theorem and Beyond

We consider an odd-sized "jury", which votes sequentially between two states of Nature (say A and B, or Innocent and Guilty) with the majority opinion determining the verdict. Jurors have private information in the form of a signal in [-1,+1], with higher signals indicating A more likely. Each juror has an ability in [0,1], which is proportional to the probability of A given a positive signal, an analog of Condorcet's p for binary signals. We assume that jurors vote honestly for the alternative they view more likely, given their signal and prior voting, because they are experts who want to enhance their reputation (after their vote and actual state of Nature is revealed). For a fixed set of jury abilities, the reliability of the verdict depends on the voting order. For a jury of size three, the optimal ordering is always as follows: middle ability first, then highest ability, then lowest. For sufficiently heterogeneous juries, sequential voting is more reliable than simultaneous voting and is in fact optimal (allowing for non-honest voting). When average ability is fixed, verdict reliability is increasing in heterogeneity. For medium-sized juries, we find through simulation that the median ability juror should still vote first and the remaining ones should have increasing and then decreasing abilities.