In my last post, I provided a graph suggesting public defenders have above average win-rates. Most people find this surprising. Actually, this fits neatly into a model of the LSM where defense attorneys are profit maximizers and public defenders are sentence minimizers. Profit maximization *does not imply* sentence minimization. Instead, defense attorneys focus on “Win-Stay” and “Lose-Stay” outcomes. To see what I mean, consider Bayes’ Rule

All of this means that the probability of A conditional on B equals the probability of B conditional on A multiplied by the probability you assign to A, over the probability of B conditional on A multiplied by the probability you assign to A plus the probability of B conditional on not-A multiplied by the probability you assign to not-A. If the above isn’t clear, check out Bryan Caplan’s excellent lecture notes or this post at Econlog.

Here’s an example relevant to the defense attorney profit maximization problem:

P(A|B) = P(Attorney is Optimal|Bad Case Outcome)

P(B|A)= P(Bad Case Outcome|Attorney is Optimal)

P(A) = Probability Attorney is Optimal

P(~A)= Probability Attorney is not Optimal

The profit maximizing attorney wants to persuade clients with bad outcomes that their attorney was still the correct choice. This way, the attorney still has access to that client’s network (and of course for future cases with the same client). In order to do this, attorneys should focus on increasing their clients’ subjectively held belief that they are high quality and increasing the clients’ belief that bad outcomes with high quality attorneys are common. For simplicity, let’s assume that the attorney’s clients will stay with them or recommend them to others if the attorney wins their case.

These incentives create a potential agent-principal problem in the attorney-defendant relationship. If an hour of signaling “I have a great win-rate” does more to increase the probability of Lose-Stay outcomes than an hour of work increasing the probability of winning, the attorney will invest too little (from the defendant’s perspective) in actually winning.

Public defenders, as sentence minimizers, don’t have this problem. Basically – and this can be seen in the data – the average public defender is a better agent than the average private defense attorney. Of course public defenders have obvious weaknesses – essentially zero budget for non-procedural trial inputs, for example. But with respect to procedural inputs, they should behave as if they have been given infinitely large budgets.