The following post was co-written with Adrian Vermeule.

In our previous post, we set out some puzzles about how judges should take into account the votes of other judges. Here, we sketch out some tentative answers.

Imagine that each Justice reads the statute and the briefs, and reaches a preliminary conclusion about the meaning of the statue. The Justice also has a level of confidence (high, low, middling) about her own interpretation. For example, a Justice might believe that the meaning is X with a probability of 0.99, 0.9, 0.6, 0.5, or 0.1 (which is the same as saying that she believes the meaning is Y with a probability of 0.01, 0.1, 0.4, 0.5, or 0.9). A Justice with a high confidence level believes that the statute is clear; a Justice with a confidence level in the neighborhood of 0.5 believes that the statute is ambiguous.

A Justice should update her prior in light of the information that she receives from the other Justices. If voting is sequential, then each Justice should update based on the Justices who voted before her. If voting is simultaneous, then such updating is not possible immediately—but let us suppose that Justices can change their votes in a second round of voting. Each Justice should take into account not only the number of votes for each meaning, but the confidence level of the Justices who cast those votes. A Justice who votes for X with confidence level 0.51 is not as informative as a Justice who votes for X with confidence level 0.9. It may, however, be difficult to gauge confidence level—it is certainly more difficult to gauge confidence level than the vote itself. We can imagine that Justices may try to reveal their confidence level when they cast their vote (“I’m really not sure, but for the moment X seems more plausible to me”). This may not always be the case, but let’s assume it is. Now let’s go to the questions from last post.

(1) If the initial vote reveals a 5-4 split in favor of meaning X, and all justices (sincerely) claim to be confident, then they should all certainly update their views. How they update their views is complicated. If you are confident enough, then you should presumptively not update your views; but if enough people are arrayed against you, and they are confident as well, then you should. But in this example, it certainly seems that each Justice should decide that the statute is ambiguous (by which we mean, she believes that the meaning is X with probability only slightly higher than 0.5).

(2) The second case involves a 5-4 split, where the 5 believes that the statute clearly means X, and the 4 say that the statute is ambiguous. Let’s suppose that this means that the 5 attach probability 0.9 that the statute means X, and the 4 attach probability 0.5 that the statute means X. Given these precise numbers, the 4 should update their views and agree with the 5 (while the 5 should update their views only slightly); but different numbers would yield different results.

(3) This case—where 4 Justices believe the statute clearly means X, 4 Justices believe the statutes clearly means Y, and 1 Justice believes that the statute is ambiguous—raises a question about the difference between the “collective” or aggregate meaning of an opinion, on one hand, and the views of the individual Justices on the other. For an even clearer example, suppose that all Justices believe that the statute is ambiguous but incline toward X (say, confidence level 0.55). If they then observe each other’s vote and confidence level, and also believe that each Justice’s view is independently arrived at, then they should update their belief and conclude that the probability that the correct meaning is X is very high (almost certain), in virtue of the Jury Theorem. Should the Justices unanimously vote that the statute is ambiguous or that the statute is clear? We need to work this through.

We have assumed away some types of strategic behavior. A Justice might deliberately overstate her confidence level in order to influence the votes of other Justices. In response, Justices may rationally place less value on such “cheap talk” than otherwise, depending on how honest they think the other Justices are, which is something that they may learn over time through interactions with each other. Of course, strategic behavior by Justices is a more general problem, hardly unique to this setting. Consider the certiorari process, or the possibilities for strategic behavior opened up by the Doctrinal Paradox (the choice between aggregating judicial votes over discrete issues or aggregating votes over bottom-line judgments).

Another complication is that, even if entirely sincere, a Justice may conceal information by allowing herself to be influenced by other Justices. In the famous herd-voting models, if, say, the first three or four Justices happen to vote the same, then subsequent Justices will imitate them, believing that the collective view is more information than their own. A similar problem can arise with simultaneous voting. However, this is not a problem if Justices can and do credibly reveal their confidence level as well as the outcome they believe is correct.

There is much more work to be done, on all these questions. But our tentative judgment is that the potential costs of such a system, while real, do not necessarily and invariably justify throwing away relevant information — the information contained in the votes of other judges.