## Bayes 13: random adjudication with risk-averse moving parties

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

Suppose that litigation is a crapshoot (to quote my mentor and favorite law school professor John Langbein); that is, what if litigation outcomes are only 50% sensitive and 50% specific? In other words, what if litigation games are completely random? Under this scenario, the process of adjudication is no better than a coin toss. Although this assumption may appear fanciful, as I explained in a previous post (see “Bayes 10“), the randomness of adjudication might be a function of the level of the complexity or the level of ambiguity of the applicable legal doctrines (e.g., assumption of risk) or procedural rules (e.g., res judicata). Simply put (pun intended), the more complex or ambiguous the applicable law is, the more random or arbitrary the outcome of litigation will be.

In summary, random adjudication produces purely random results, no better than a coin toss, since it will correctly determine with one-half probability, or p = 0.5, whether the defendant has or has not committed a wrongful act. Given this inherent randomness, along with the presence of virtuous or risk-averse moving parties, we now turn to Bayes’ rule to determine the posterior probability that liability will be incorrectly imposed on an innocent defendant (i.e., the probability that a defendant who has not committed a wrongful act will be incorrectly classified as a wrongful or guilty defendant). Again, let ‘guilt’ stand for a guilty defendant, ‘innocent’ an innocent defendant, and the symbol + the event of a positive litigation outcome for the moving party (plaintiff or prosecutor). Next, we find the values for Pr(guilt), Pr(innocent), Pr(+|guilt), Pr(+|innocent), and Pr(+).

First, assuming that plaintiffs and prosecutors are virtuous or risk-averse actors and thus are willing to play the litigation game only when they are at least 90% certain that the named defendant is guilty, then Pr(guilty), the prior probability in the absence of other information that a particular defendant has committed a wrongful act, will be equal to 0.9, or stated formally, Pr(guilty) = 0.9. Likewise, Pr(innocent), the prior probability in the absence of other information that a particular defendant has not committed a wrongful act, is simply 1 – Pr(guilty) or 0.1, since 1 – 0.9 = 0.1.

Next, Pr(+|guilty), the probability that liability will be imposed on a defendant who is actually guilty, is 0.5 since the litigation game in this variation of our model purely random (i.e., 50% sensitive). Similarly, Pr(+|innocent), the probability that liability will be imposed on an innocent defendant, is also 0.5 since, given our revised assumptions, the litigation game will produce a false positive half of the time the game is played.

Lastly, recall that Pr(+) is the sum of true positives and false positives, that is, the prior probability of a positive litigation outcome, positive from the plaintiff’s or prosecutor’s perspective, in the absence of any information about the defendant’s guilt or innocence. Specifically, given our assumptions above, this value is equal to 0.5, that is, 0.5 × 0.9 = 0.45 (true positives) plus 0.5 × 0.1 = 0.05 (false positives). Thus, the prior probability of a positive litigation outcome, Pr(+), absent any information about the defendant’s guilt or innocence, is equal to 50%.

Thus, given random adjudication and virtuous or risk-averse plaintiffs, we apply Bayes’ theorem as follows:

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

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

Pr(guilty|+) = (0.5 × 0.9) ÷ [(0.5)(0.9) + (0.5)(0.1)]

Pr(guilty|+) = 0.45 ÷ 0.50

Pr(guilty|+) = 0.9 (!)

This result is perhaps the most surprising one thus far in my month-long series on Bayes and the law. Even when the litigation game is a purely random process, no better than a coin toss, the outcome of any individual litigation game will still be highly reliable, given the presence of virtuous moving parties. Specifically, under this scenario there is a 90% probability that a defendant who is found liable for a wrongful act is, in fact, actually guilty. Although this value is less than the corresponding values for Pr(guilty|+) in the previous two permutations of the model (see “Bayes 11” and “Bayes 12“), this difference is marginal at best, considering the enormous qualitative differences between non-random adjudication and a purely random legal system.

The present permutation of the model, however, assumes the presence of virtuous plaintiffs and prosecutors. What happens when the litigation game is purely random and the moving parties are less-than-virtuous? I will explore this intriguing possibility in my next post on Monday …