## Bayes 14: random adjudication with risk-loving moving parties

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

Happy Monday! Let’s now suppose that litigation is still a crapshoot but that plaintiffs and prosecutors are risk-loving or ‘less-than-virtuous’; that is, let’s assume that the moving parties are more willing to gamble than their virtuous colleagues. Specifically, I will assume that the litigation game is 50% sensitive and 50% specific and that plaintiffs and prosecutors are willing to play the litigation game even when they are only 60% certain that the named defendant has committed a wrongful act. Although these assumptions do not appear to be plausible, this permutation of my model, however implausible, may nevertheless provide an instructive counter-factual or hypothetical illustration of my Bayesian approach to litigation.

Given these revised assumptions (i.e., random results and risk-loving or ‘less-than-virtuous’ actors), we can once again turn to Bayes’ theorem 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), and once again, ‘guilt’ stands for a guilty defendant, ‘innocent’ indicates an innocent defendant, and the symbol + represents the event of a positive litigation outcome for the plaintiff or prosecutor.

As such, in the absence of any additional information or evidence, Pr(guilty), the prior probability that a particular defendant has committed a wrongful act, is equal to 0.6, while Pr(innocent), the prior probability that a particular defendant has not committed a wrongful act, is 0.4 (i.e., 1 – Pr(guilty), or 1 – 0.6). Next, Pr(+|guilty), the probability that liability will be imposed on a defendant who is actually guilty, and Pr(+|innocent), the probability that liability will be imposed on an innocent defendant, are both equal to 0.5 since, given our assumptions, this version of the litigation game is purely random. Lastly, Pr(+), the sum of true positives and false positives, is also 0.5 since, given our assumptions above, 0.5 × 0.6 = 0.3 (true positives) and 0.5 × 0.4 = 0.2 (false positives), or put another way, the prior probability of a positive litigation outcome (again, from the plaintiff’s or prosecutor’s perspective), absent any information about the defendant’s guilt or innocence, is equal to 50%.

Therefore, given random adjudication and risk-loving plaintiffs, we now 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 × 0.6) ÷ [(0.5)(0.6) + (0.5)(0.4)]

Pr(guilty|+) = 0.3 ÷ by 0.5

Pr(guilty|+) = 0.6

What is most surprising about this result is the ability of the litigation process to produce reliable results more than half the time, even when the underlying litigation game itself is purely random and even when the actors are less than virtuous. Specifically, the probability that the outcome of any individual litigation game will be accurate is 60%, even though the underlying litigation game is purely random, no more reliable than a coin toss. One way of explaining this potential paradox is to take another look at the Pr(guilty) term: the prior probability in the absence of additional information that a defendant selected at random is guilty (i.e., the prior probability that a particular defendant has committed a wrongful act). This prior probability term exerts a decisive influence in the fourth permutation of our model precisely because the outcome of litigation is purely random. That is, when litigation is a crap shoot, or to be more precise, when litigation is a coin toss, both the prior and posterior probabilities of the defendant’s guilt are the same. Here, since Pr(guilt) = 0.6, then Pr(+|guilty) = 0.6.

I will make some concluding observations in my next post …