Bayes 11: non-random adjudication with risk-averse moving parties

Note: This is my eleventh blog post in a month-long series on the basics of Bayesian probability and its application to law.

Suppose the litigation game is 90% sensitive and 90% specific, that is, suppose the process of litigation is able to determine correctly, at least 90% of the time, when a defendant has committed a wrongful act, and suppose further that the process will also determine correctly, again at least 90% of the time, when a defendant has not, in fact, committed a wrongful act. The intuition behind this assumption (non-random adjudication) is that reliable legal procedures will tend to produce just and fair results. See, for example, Henry M. Hart and Albert M. Sacks, The Legal Process (William N. Eskridge and Philip P. Frickey eds, Foundation Press, 1994). [1] Simply put, such a litigation game appears to be a highly accurate one, since it will correctly determine with 90% probability, or nine times out of 10, whether the defendant has or has not committed a wrongful act, an essential precondition before liability may justly be imposed.

Nevertheless, my model of non-random adjudication still suffers from a 10% error rate. Given this error rate, we must refer to Bayes’ rule to determine the posterior probability that liability will nevertheless be incorrectly imposed on an innocent defendant, that is to say, the probability that a defendant who has not committed a wrongful act will be incorrectly classified as a wrongful or guilty defendant. In other words, we must find the prior probability that any given defendant, selected at random, has in fact committed a wrongful act. What is this prior probability?

First, let the term ‘guilt’ stand for a guilty defendant, let ‘innocent’ represent an innocent defendant, and let the + symbol indicate the event of a positive litigation outcome for the plaintiff or prosecutor, as the case may be. That is, from the plaintiff or prosecutor’s perspective, a positive outcome, or +, occurs when liability is eventually imposed on the defendant. I will now proceed to find the values for Pr(+|guilty), Pr(+|innocent), Pr(guilty), Pr(innocent), and Pr(+). To begin with, Pr(+|guilty) is the probability that a guilty defendant will be found guilty at the end of a litigation game. Since I have assumed that the litigation game is 90% sensitive, the value for Pr(+|guilty) is equal to 0.9. By the same token, Pr(+|innocent), the probability that a particular litigation game will produce a false positive (i.e., the probability that liability will be imposed on an innocent defendant) is equal to 0.1. This value is 0.1 since, given my initial assumptions, the litigation game produces false positives only 10% of the time.

Now suppose that plaintiffs and prosecutors are risk-averse or virtuous parties, that is, assume that plaintiffs and prosecutors alike are willing to play the litigation game only when they are at least 90% certain that the named defendant has, in fact, committed an unlawful wrongful act. [2] Accordingly, given these stringent assumptions (i.e., risk-averse moving parties and non-random adjudication), the prior probability that a given defendant is guilty is 90%, or stated formally, letting A stand for the prior probability of being guilty, then Pr(A) = Pr(guilty) = 0.9. Summing up, Pr(A) or Pr(guilty) is the prior probability, in the absence of any additional information, that a particular defendant has committed a wrongful act. As stated above, this term is equal to 0.9 since I have assumed that 90% of all named defendants are guilty. In similar fashion, we can determine Pr(B) or Pr(innocent), the prior probability that a particular defendant has not committed any wrongful act. This is simply 1 – Pr(guilty) or 0.1, since 1 – 0.9 = 0.1.

Lastly, Pr(+) refers to the prior probability of a positive litigation outcome—again, ‘positive’ from the plaintiff’s or prosecutor’s perspective—in the absence of any information about the defendant’s guilt or innocence. This value is found by adding the probability that a true positive result will occur (0.9 × 0.9 = 0.81), plus the probability that a false positive will happen (0.1 × 0.1 = 0.01), and is thus equal to 0.81 plus 0.01 = 0.82. Stated formally, Pr(+) = [Pr(+|guilty) × Pr(guilty)] plus [Pr(+|innocent) × Pr(innocent)]. That is, the prior probability of a positive litigation outcome, Pr(+), is the sum of true positives and false positives and, given my assumptions above, is equal to 0.82 or 82%.

Lastly, Pr(+) refers to the prior probability of a positive litigation outcome—again, ‘positive’ from the plaintiff’s or prosecutor’s perspective—in the absence of any information about the defendant’s guilt or innocence. This value is found by adding the probability that a true positive result will occur (0.9 × 0.9 = 0.81), plus the probability that a false positive will happen (0.1 × 0.1 = 0.01), and is thus equal to 0.81 plus 0.01 = 0.82. Stated formally, Pr(+) =[Pr(+|guilty) × Pr(guilty)] plus [Pr(+|innocent) × Pr(innocent)]. That is, the prior probability of a positive litigation outcome, Pr(+), is the sum of true positives and false positives and, given my assumptions above, is equal to 0.82 or 82%.

Having translated all the relevant terms of Bayes’ theorem, I will now restate my Bayesian model of litigation and find the posterior probability, Pr(guilty|+), that civil or criminal liability will incorrectly imposed on a guilty defendant (i.e., the probability that a defendant who has not committed a wrongful act will nevertheless be incorrectly classified as a wrongful or guilty defendant):

Pr(guilty|+) = [Pr(+|guilty) × Pr(guilty)] ÷ Pr(+)

Pr(guilty|+) = [Pr(+|guilty) × Pr(guilty)] ÷ ([Pr(+|guilty) × Pr(guilty)] + [Pr(+|innocent) × Pr(innocent)])

Pr(guilty|+) = (0.9 × 0.9) ÷ [(0.9)(0.9) + (0.1)(0.1)]

Pr(guilty|+) = 0.81 ÷ 0.82 = 0.988

In plain English, given my rosy assumptions above, the outcome of any particular litigation game will be highly accurate. Specifically, the probability that a defendant who is found liable for a wrongful act is actually guilty of committing such wrongful act is close to 99%, a value that appears to vindicate Hart & Sacks’ optimistic vision of legal process, though there is still a 1% probability that an innocent defendant will nonetheless be found liable. But what happens when the litigation game is played by strategic plaintiffs or zealous prosecutors? That is, what happens when plaintiffs file a greater proportion of frivolous claims (relative to the optimal level of frivolous claims) or when prosecutors routinely ‘overcharge’ criminal defendants with extraneous or vague offenses (e.g., conspiracy)? I will consider this possibility in my next post.

[1] Of course, the existence of reliable adjudication procedures in which liability is imposed only on guilty defendants is not a sufficient condition for justice. When a defendant has broken an unjust or unfair law (licensure requirements and racial segregation laws quickly come to mind), justice would be better served by an unreliable adjudication procedure (i.e., by not enforcing the unjust or unfair law in the first place). But putting aside the underlying meaning of justice, the model of litigation I am presenting here is a highly accurate one, since it will correctly determine with 90% probability, or nine times out of 10, whether the defendant has or has not committed a wrongful act.

[2] This risk-averse conduct is considered ‘virtuous’ in my model since such moving parties are less willing than their risk-loving colleagues to gamble on the outcome of litigation, or expressed in legal language, virtuous civil plaintiffs rarely file frivolous claims and virtuous criminal prosecutors rarely abuse their discretion. The reader may rest assured, however, that I will relax these unrealistic assumptions in future blog posts.

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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