## Bayes 12: non-random adjudication with risk-loving moving parties

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

Suppose the litigation game is still highly sensitive and specific as before (i.e., 90% sensitive and 90% specific), but that plaintiffs and prosecutors are risk-loving or less-than-virtuous actors. Specifically, assume that the moving parties are willing to play the litigation game even when they are only 60% certain (instead of 90% certain, as we assumed earlier) that the named defendant has committed a wrongful act. (Such behavior is ‘less-than-virtuous’ in my model because the moving party is less concerned with the defendant’s actual guilt than a risk-averse or virtuous moving party is.) The intuition behind this revised assumption is that, in reality, the litigation game might be played by litigants who are engaged in rent-seeking and self-serving behavior. [1]

Thus, with risk-loving moving parties, the prior probability, Pr(guilty), that a given defendant is guilty is now only 60%, while the prior probability, Pr(innocent), that a particular defendant has not committed a wrongful act is 1 – Pr(guilty), or 1 – 0.6 = 0.4. Stated formally as Pr(guilty) = 0.6 and Pr(innocent) = 0.4.

Next, we find the probability that a guilty defendant will be found guilty, or Pr(+|guilty). In this variation of our model, the value for Pr(+|guilty) is equal to 0.90 since we continue to assume the litigation game is 90% sensitive, while 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), remains 0.1. Lastly, recall that Pr(+) is the probability that a true positive result will occur (in this case, 0.9 × 0.6 = 0.54), plus the probability that a false positive will happen (0.1 × 0.4 = 0.04), and is thus equal to 0.54 plus 0.04 = 0.58. Stated formally, Pr(+) =[Pr(+|guilty) × Pr(guilty)] plus [Pr(+|innocent) × Pr(innocent)] = 0.54 plus 0.4 = 0.58.

Given these revised assumptions–non-random adjudication and less-than-virtuous plaintiffs–we now find the posterior probability that liability will be correctly imposed on a guilty or wrongful defendant 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|+) = (0.9 × 0.6) ÷ [(0.9)(0.6) + (0.1)(0.4)]

Pr(guilty|+) = 0.54 ÷ 0.58 = 0.931

In other words, despite the presence of risk-loving moving parties, the outcome of any particular litigation game will still be highly reliable. Specifically, although there is a 7% chance that an innocent defendant will be found liable, the posterior probability that a defendant who is found liable for a wrongful act is actually guilty is still 93%, a value that, once again, appears to affirm the Hart and Sacks’ rosy vision of the legal system. But what happens when litigation is a crapshoot, that is, stated formally, what happens when the litigation game is only 50% sensitive and 50% specific? I will consider that possibility in my next post …

[1] In principle, a more hard-core ‘risk-loving’ moving party might be willing to gamble on the litigation game even when he is only 50% certain of the outcome. Nevertheless, I will assume that a risk-loving moving party requires a 60% probability of a positive litigation outcome simply because he must expend resources to play the litigation game. Put another way, since the litigation game is not costless, and thus, broadly speaking, the higher the cost of playing the litigation game (relative to the resources of the moving party), the more risk-averse an otherwise risk-loving moving party will be.