Managing uncertainty in mediation decision making is a basic negotiation skill. Unfortunately, there is not a lot of training in law school or in continuing legal education that focuses on the challenges of decision making in the face of uncertainty. I hope this article will fill some of the void.

When analyzing uncertainty, statisticians and actuaries rely on the mathematics of probability. Lay people rely on intuition and gut checks. This holds true for lawyers as well as for clients. However, the research demonstrates that human brains are terrible at guesstimating likely outcomes. Unless you engage your algorithmic and reflective brain in conscious probability analysis, your opinions will be wrong. Period. Fortunately, there are some down-and-dirty tools that are easy to learn and apply. Use these to avoid decisional errors cause by cognitive biases and distortions.

The foundational tool for managing uncertainty in mediation decision making is a mathematical construct known as Bayes Theorem. I know, you went to law school to avoid doing math. Get over it! Bayes Theorem is not hard to understand, plus you can wow your colleagues by casually mentioning how you are doing some Bayesian analysis of your case.

Bayes Theorem is named after an 18^{th} century pastor, Thomas Bayes. In its simplest form, Bayes Theorem allows us to quantify the probabilities of future uncertain events. Thus, it is a useful solution for trial lawyers wanting to quantify risk for clients.

“What’s the likelihood that we will prevail at trial?” asks Client.

Lawyer ponders, then looks slightly uncomfortable. “Well, these matters are always difficult to assess. A judge or jury can really do just about anything. But, I think we have a pretty good chance at prevailing. Of course, things can always go sideways for us so there are no guarantees.”

Sound familiar?

Here’s another approach that uses Bayes Theorem.

“What’s the likelihood that we will prevail at trial?” asks Client.

The answer is that there are a number of outcomes, from ringing the bell to skulking out of the courtroom in complete defeat. Each outcome is a value, measured in dollars. Each outcome has some probability of occurring. Through the simple application of Bayes Theorem, we can derive some information that might be more useful than “I think we have a pretty good chance at prevailing.”

Consider the following situation:

- There is a 15% chance of a defense verdict ($0).
- There is a 50% chance of a $25 verdict.
- There is a 20% chance of a $50 verdict.
- There is a 10% chance of a $75 verdict.
- There is a 5% chance of a $100 verdict

Now, we have five outcomes with 5 probability values. The sum of the probabilities cannot be great than 1. Bayes Theorem simply tells us to multiply the assigned probability value with its outcome.

Probability of 1 |
X | Value of 1^{st} Outcome |

Probability of 2nd outcome | X | Value of 2^{nd} Outcome |

Probability of 3rd outcome | X | Value of 3rd Outcome |

Probability of 4^{th} outcome |
X | Value of 4^{th} Outcome |

Probability of 5th outcome | X | Value of 5th Outcome |

Putting in the numbers looks like this:

$0 | X | 15% | $0 |

$25 | X | 50% | $12.50 |

$50 | X | 20% | $10 |

$75 | X | 10% | $7.50 |

$100 | X | 5% | $5 |

100% | $35 |

When we add up the third column, we get 100%, the sum of all of the probabilities. When we add up the fourth column, we get $35, which is called the Expected Value. The Expected Value is the average result if the case were tried a hundred times. Observe that the Expected Value is not a prediction. We cannot predict any specific outcome with certainty. However, the Expected Value helps us assess whether an offer or demand is within some range of reasonableness.

A more visual way of doing this analysis is with a decision tree. Here’s what the problem looks like in that format.

Each branch of the decision tree represents one of the outcomes. You assign the probabilities and values; the program does the arithmetic.

The true value of this tool is the ability to change the probabilities and see how the individual outcomes and Expected Values change. Thus, you would create your first assessment based on your current thinking about the case. Then, you would do an assessment assuming you are woefully overconfident. You can see how your probability assessments, whether optimistic or bleak change the Expected Value of the case.

If you want to be even more sophisticated, you can plot the probabilities and outcomes of your best and worst assumptions into an area chart. Then you get a picture like this for the defendant:

One like this for the plaintiff:

And if you overlap them, you get this picture:

The overlap area is what I call the Zone of Probable Settlement. It is nothing more than a visual depiction of how the plaintiff and defense probabilities and outcomes overlap each other.

You can do this analysis with sophisticated software like TreeAge or you can do it on a legal pad in front of your client.

The benefits of Bayesian analysis in managing uncertainty in mediation decision making are:

- You can quantify hunches, guesses, and intuitions into probabilities
- You can test different hypothetical sets of probable outcomes
- You can create a visual analysis that clients find easy to understand
- You can test the other side’s probabilities
- You can create area charts to reveal zones of probable settlement

However, there some limitations:

- This analytical method is not predictive. It will not tell you what is actually going to occur.
- You have to run more than one model of probabilities and outcomes to gain any useful information.
- The Expected Value is not the settlement value of the case because the Expected Value changes with different probability and outcome values.

Even with the limitations, the information you obtain from the analysis gives you a much better feel for the uncertainties of the case. Your client will benefit from a thoughtful analysis of the risks and outcomes. You will both make better decisions in your mediation.

P.S. I teach this at Pepperdine and would be happy to present a workshop on this type of decision making to your bar association, law firm, or corporate legal department.