Optimizing Voting Order on Sequential Juries: A Sealed Card Model

The celebrated Jury Theorem of Condorcet (1785) considered an odd size jury in which jurors vote simultaneously after receiving independent signals, correct with the same probability p > 1/2, concerning which of two binary possibilities (maybe innocent or guilty) is true. The reliability of the majority verdict, the probability it is correct goes to 1 as the jury size tends to infinity. Recently, Alpern and Chen (2017a) considered a variation of the Condorcet model in which jurors vote sequentially (rather than simultaneously), with knowledge of prior voting. In addition, jurors are heterogeneous in a quality called ability, which affects the usefulness of their signals for predicting the true possibility. Their notion of ability was abstract, modeling for example eyesight quality for jurors making line calls in tennis. For a jury of size three of given distinct abilities, they found that reliability was order dependent and maximized when jurors vote as follows: first the juror of median ability, then of high ability, then of low ability. This paper makes the notion of ability concrete, with a different signal distribution than Alpern-Chen. A (sealed) card is removed from a deck of m red and m black cards. Each juror draws a number of cards equal to his integer-valued ability and votes for the color of the sealed card, based on his draw and prior voting. We find that the Alpern-Chen (median-high-low) ordering has the highest reliability in most cases, and it generally is also better than simultaneous voting. In the cases where the latter is better, the abilities of the jurors tend to be similar. An analog of this ordering has high reliability for larger juries.