Bayesian Voting (Part 3)

In our 7/17 blog post, we described a voting paradox in law and presented a simple model of bayesian voting, and in our 7/18 blog post, we showed how bayesian voting might work in practice. In that post, however, we assumed judges’ votes were sincere. In this post, by contrast, we will assume that all three judges are strategic actors (see image below); that is, all three judges are going to choose extreme values, i.e. inflate or deflate their degrees of belief as the case may be, in order to achieve their preferred outcome. I am making this assumption because I want to test whether my system of bayesian voting can be successfully manipulated or gamed.

As before, we use a simple bounded range {0 to 1} in which in which a vote of 0 is a strategic vote against plaintiff P (and for defendant D) and a vote of +1 is a strategic vote for P (and against D). Since the judges are strategically voting issue by issue, the preferences of the judges in case C can be expressed as follows:

• Judge #1 believes P has standing to sue but that D is immune, so he could assign a credence of +1 to the standing issue and a credence of 0 to the immunity issue.
• Judge #2 believes P has standing and that D is not immune, he could assign a credence of +1 to the standing issue and a credence of +1 to the immunity issue.
• Judge #3 believes that P does not have standing and that D is not immune, so he could assign a credence of 0 to the standing issue and a credence of +1 to the immunity issue.

Next, let’s add up these strategic credences by issue: (+1) + (+1) + (0) = +2 for the standing issue, and (0) + (+1) + (0) = +1 for the immunity issue. As a result, plaintiff wins on the standing issue (because the credences are above the threshold of 1.5), but plaintiff loses on the immunity issue. Notice that strategic voting produces the same result as sincere voting! (See our previous blog post on sincere bayesian voting.)

What if, however, the judges attempted to vote by overall outcome, instead of by issue. Since our sliding scale starts at 0 and ends at 1, strategic judges who want the plaintiff to prevail will assign a value of 1 to their credence in the strength of plaintiff’s case. Likewise, strategic judges who want the defendant to prevail will assign their credence a value of 0. Given these assumptions, the preferences of the judges in case C can be expressed as follows:

• Judge #1 believes P has standing but that D is immune, so he could assign an “outcome credence” of 0 to his vote, meaning that he believes that D should prevail on immunity grounds, even though P has standing to sue.
• By contrast, since Judge #2 believes P has standing to sue and that D is not immune, he could assign an outcome credence of +1 to his vote, meaning that he believes P should win.
• Lastly, since Judge #3 believes that P does not have standing, even though he believes that D would not be immune had P had standing, he would assign an outcome credence of 0 to his vote, meaning that he believes that D should prevail on standing grounds.

When we sum these individual values (0 +1 + 0), we are left with a global credence of +1. Because this global credence is below the threshold of 1.5, D should win, the same result as before with issue-by-issue voting!

To sum up, by requiring judges to assign a credence reflecting their degrees of belief in their legal holdings, bayesian voting produces consistent results in case C–regardless whether the judges are voting by issue or by overall outcome. Nevertheless, in our next few blog posts, we will consider several possible objections to bayesian voting …

Image Credit: Shonda Cobb