Game Theory and the Art of Law

Game tree for a trial, follow instructions below to understand. Right click and press "open image in new tab" to see the full image and details.

Recently, my son Hunter and I were discussing the process of predicting case results using a set of variables and probability analysis. Hunter is currently working on marketing and data analysis for me this summer. When a legal colleague of mine, Travis Jameson, wrote on the subject of probability analysis in injury cases I showed the post to Hunter because I figured he would be interested in the content as well. Hunter is a rising Junior at Washington University in St. Louis studying Business Economics (“Economics & Strategy” is the formal title) and Marketing in the Olin Business school. What follows is Hunter’s analysis and response to Mr. Jameson’s probability analysis:

Mr. Jameson is addressing the process of probability analysis in injury cases and its usefulness in choosing the right avenue to pursue for the best results for his clients in his article. In other words, Jameson proposed a model for determining the expected value of a case if it goes to trial (not a universal model, merely a methodology to analyze case values). This is an interesting approach but the base analysis oversimplifies the way these cases work from a game theoretic perspective.

Jameson’s base model assumes the insurance company is an uncomplicated player, which is to say that it does not make any relevant decisions in the trial and merely takes what the odds give them. The reality of the analysis is more complicated than this. Let’s take a very basic model of mixed strategy, sequential decision game theory and apply it to a trial setting.

Please note that any facts and figures are purely made up (the actual numbers are irrelevant but I tried to stay within the bounds of feasibility). The purpose of this example is to demonstrate how deep the rabbit hole goes once we step into the realm of probability analysis in trials. Additionally, we assume that the insurance company knows whether a case will be good or bad at trial due to their vast amount of data available to make that claim with a high degree of accuracy.

Again, to see the full game tree, right click on the image above and press "open image in new tab", the diagram makes the steps more clear.

To properly create a simple model for a case you have to consider a few assumptions and rules for the “game” as follows:

1. The plaintiff attorney is depicted as the letter A, Insurance Company as letter I and Nature (an independent actor representing the probability of an event) is depicted as N.
2. The plaintiff attorney moves first in the game and has the option to accept an offer of $50k or go to trial.
3. The game is a zero-sum game, in other words any gain for the plaintiff attorney is exactly equal to the loss of the insurance company.
4. Next “Nature”, which is the probability of an event occurring in game theory, takes its move. The nature node determines whether evidence is bad or good. For simplicity we’ll assign the probability of good or bad to both be .5 (p = .5, 1 – p = .5).
5. All parties involved in the game are aware of the probabilities of any Nature node branch.
6. Both you and the insurance company understand if you have bad or good evidence after the first nature node has moved (discovery and experience owing to this understanding).
7. Next, branching off of the good or bad results the insurance company can make its move. Here it has the option to offer a higher settlement ($60k) or pursue a trial.
8. The plaintiff attorney can decide to accept the higher settlement or reject the higher settlement in favor of a trial.
9. Nature again has a move here to determine the probability of settlement. If nature has determined you have good evidence, your probability (q) of acquiring a good settlement is .8 (q = .8 or 80% and 1 – q = .2 or 20%).
10. If nature has determined you have bad evidence, these probabilities switch giving a q = .2 for a good settlement and 1 – q = .8 for a bad settlement.
11. A good trial result gives a payoff of $100k and a bad trial result gives a payoff of $25k.
12. Payoffs are listed (Plaintiff, Insurance Company)
13. Now we can go about solving the game. The expected result of the Good evidence branch is a trial with Expected Payoffs (85, -85).
14. The expected result of the Bad evidence branch is (40, -40).
15. The optimum choice routes for any party that can make decisions (A and I) are depicted as green arrows.
16. The overall Nash Equilibrium (the most rational result) of the game is for the plaintiff attorney to pursue trial for an expected payoff set of (62.5, -62.5).

This model can adapt based off of other factors. For example, a plaintiff and their attorney may incur costs for pursuing trial that reduce their payoffs in that event which makes settling more favorable. Alternatively, the plaintiff and their attorney may receive a premium utility from winning the case. In other words a client that goes to trial may enjoy the possibility of winning so much that they value winning as an additional pay off to the value of any case result.

It should be noted that this game tree is the same as what game theorists would describe formally as an adverse selection problem and colloquially as the lemon problem (after lemon cars). Adverse selection games occur when the less informed party (the plaintiff attorney) moves first. We model the plaintiff attorney as less informed because the insurance company has access to enough data to make a reliable prediction on case quality.

My point with this analysis isn’t to call out Mr. Jameson’s thought process, but I believe it is very important (if we decide to go this route for trial analysis) to understand how the full game plays out in reality. This sort of subject is terribly complicated and even with my more complicated model it doesn’t adequately explain every variable that goes into a case. As such without a great deal of resources it is difficult to determine where you stand in a case before going to trial with a high degree of certainty. Statisticians usually use a significance level of alpha = .05 as a criterion, which means they are 95% certain of a factor or model’s significance, to determine whether or not a model is useful. I would be hard pressed to say any model we could produce with in-firm data could provide that level of accuracy, hence the value of acquiring experience trying cases.

This is why I like to think of law as both an art and a science. While models and data are all very helpful, there’s always the unexplained portion of the equation that, in law, can only be estimated through experience and subjective analysis.