## Bayesian Voting (Part 2)

In my previous blog post, I showed how the outcome of an appeal can depend on the type of voting rule appellate courts use to decide cases, and I mentioned a possible solution to this paradox: bayesian voting. In brief, bayesian voting would not change the way the parties make their arguments. The party with the burden of persuasion on a given legal issue would continue to write legal memoranda, submit legal briefs, and present arguments to the court, and the opposing party would also have the opportunity to do the same things, but bayesian voting would change the way appellate judges decide cases.

Specifically, instead of voting up or down on the outcome of an issue, judges using bayesian voting would have to disclose how strongly or weakly they believed in each side’s arguments. For this method to work, however, judges would have to vote sincerely (a big if, as we shall see in our next blog post), and they would have to use the same numerical scale. By way of example, see the image below, depicting a sliding scale starting at 0 (meaning complete disbelief in the arguments made by a party) and going up to 1 (meaning complete belief in a party’s arguments). So long as the judges use the same scale, it’s okay if each judge uses his own criteria to evaluate the strength of each party’s arguments. After all, the assignment of probabilities is a subjective activity; bayesian voting just makes this fact explicit.

Under bayesian voting, each judge would assign a numerical score reflecting his degree of belief in what the proper outcome of the case should be. (Strictly speaking, each judge would score the overall persuasiveness of the arguments of the party with the burden of persuasion on each issue presented by the case.) To keep this method as simple as possible, judges could use the same sliding scale from 0 to 1. If a judge assigns a score below 0.5, this means that the party with the burden of persuasion has not persuaded the judge that he should prevail. (The closer the score is to 0, the less persuasive were the party’s arguments to the judge.) By contrast, If the judge assigns a score above 0.5, this means that the judge found the party’s arguments persuasive. (The closer the score is to 1, the more persuasive were the arguments to the judge.) Lastly, if the judge assigns a score of 0.5, this means the judge is undecided about which party should prevail–maybe because there are good arguments on both sides of the issue, or because both sets of arguments are equally persuasive.

Let’s assume the judges are “sincere judges,” i.e. they will cast their votes (i.e. assign points to each issue) in an honest manner, so their votes reflect their true subjective degrees of belief. (We will drop this simplifying assumption in our next blog post.) Also, returning to our previous example, let’s assume Judge #1 strongly believes that this is not even a close case. Since Judge #1 believes that P has standing but that D is immune, he assigns a very high value to both of these conclusions, say 0.9. Next, let’s assume Judge #2 also strongly believes that this is not a close case, but due to a difference in judicial temperament, he is only willing to assign a value of 0.7 to his conclusions (that P has standing and that D may not assert immunity). Lastly, let’s assume Judge #3 believes that this is truly a close case. He still believes in his position (that P lacks standing and that D is not immune) but is only willing to assign a value of 0.6 to his conclusions. Given these assumptions, we can present the judges’ scores in tabular form as follows:

 Standing Immunity Judge #1 Yes = Credence = 0.9 Yes = Credence = 0.9 Judge #2 Yes = Credence = 0.7 No = Credence = 0.7 Judge #3 No = Credence = 0.6 No = Credence = 0.6

Before proceeding, we need make sure we are comparing degrees of belief in the same way. To do this, we must take Judge #3’s no vote on the standing issue and convert it into a yes vote by subtracting 0.6 from 1, and likewise, we must take Judge #1’s yes vote on the immunity issue and convert it into a no vote by subtracting 0.9 from 1 (this way we are measuring the same thing). After making this conversion, we standardize the judges’ scores as follows:

 Standing Immunity Judge #1 Yes = Credence = 0.9 No = Credence = 0.1 Judge #2 Yes = Credence = 0.7 No = Credence = 0.7 Judge #3 Yes = Credence = 0.4 No = Credence = 0.6

Next, we sum up the scores. (Notice, we have to sum the scores issue-by-issue. My method of bayesian voting does not permit us to sum all the scores together to create a global score across both issues, since the two issues are distinct and separate legal issues.) If the sum of the scores exceeds the threshold of 1.5, then this means that the judges are collectively persuaded by the arguments of the party with the burden of persuasion on that particular issue. To see why, let’s go back to our hypothetical example. Regarding the standing issue, the sum of the scores is 2.0, well above the 1.5 threshold, but on the immunity issue, the sum of the scores is only 1.4, just below the cutoff. (What is so magical about 1.5? It is the dividing line between persuasive and non-persuasive arguments: an individual score of 0.5 means the judge is undecided about which party should prevail–maybe because there are good arguments on both sides of the issue, or because both sets of arguments are equally persuasive–so with three judges, the cutoff would 1.5.)

(Note: In the alternative, we could also sum all the scores on each column and divide by 3. If the final score is greater than 0.5, the plaintiff should prevail. If the final score is less than 0.5, the plaintiff should lose. In this particular example, the final score (rounded off) on the standing issue is (0.9 + 0.7 + 0.4) ¸ 3 = 0.667, while the final score on the immunity issue is (0.1 + 0.7 + 0.6) ¸ 3 = 4.667. In words, this means that the judges are fairly certain that the plaintiff has standing but are not fully persuaded that the defendant is not immune.)

In other words, bayesian voting produces a determinate result when the judges are voting sincerely. But what if the judges decide to vote strategically instead? We will consider the possibility of strategic voting (a very likely possibility, we might add) in our next blog post.

## About F. E. Guerra-Pujol

When I’m not blogging, I am a business law professor at the University of Central Florida.
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