I hope you enjoyed playing Resolution Roulette, which is often called the Monty Hall Problem after the TV game show host. Did you win? More importantly, did you process the rules and information to make the correct decision? Yes, the silly game illustrates a point.
When given the opportunity did you switch and take the other door? The overwhelming number of people refuse to switch because they believe: (1) that there is an equal probability that the prize is behind the two doors; and (2) because it is a 50-50 proposition, they keep their initial selection to avoid the potential angst and regret of switching from a winner. However, most people are wrong — contestants that switch doors double the likelihood of winning.
Wait, what? If your reaction is that there cannot be any advantage to switching, you are not alone. The Monty Hall problem has confounded people for years. In 1990 when Marilyn vos Savant (who purportedly possessed one of the highest IQs in the world) noted in Parade Magazine that there was an advantage to switching doors, thousands of people – including prominent mathematicians – insisted that she was wrong. Books and articles have analyzed why the Monty Hall problem is so difficult to solve. Not only is the problem difficult to understand, even after hearing explanations, many people remain skeptical that switching is, in fact, the correct course of action. In fact, one study concludes that pigeons more quickly realize that switching is advantageous than humans. Yes, pigeons.
Before chiming in on the problem’s relevance to mediation, let me attempt to convince you that you should, in fact, switch doors. Ms. vos Savant’s first attempt to pacify her outspoken critics was to change the rules. Let’s assume that instead of three there are now one million doors and you select door number 54,782 because, of course, that is your favorite number. Monty then reveals 999,998 goats (or in our simulation years of expensive litigation). There are now only two doors left and one has the beneficial settlement — do you switch? At this point most people begin to realize that their initial selection was a one in a million shot. Did they really get lucky and pick the prize? Do they really now have a 50-50 shot of being right after Monty revealed all those doors with bad outcomes? Studies show that when the number of doors is increased, people intuitively understand that they should switch from their initial selection.
Thankfully, the Monty Hall problem presents a finite set of possibilities for the distribution of the two goats (“G”) and one car (“C”) (again in our Resolution Roulette game the car is a settlement and the goats are protracted litigation). I depict the possibilities as follows with each respective door separated by a comma:
G, C, G
G, G, C
C, G, G
Under the rules of the game, after you select a door Monty must reveal a goat. That’s why after you selected a door, we revealed a door with one of the unfavorable outcomes of protracted litigation. Let’s assume you select Door Number 1. Placing an X to indicate when Monty is forced to reveal a goat reveals the following possibilities:
G, C, X
G, X, C
C, X, G or C, G, X
This demonstrates two crucial facts. First, in selecting Door Number 1, there was a 1 in 3 chance that you initially correctly selected the prize and a 2 in 3 chance that you did not. Second, under the rules of the game, in both situations where your original selection was incorrect, after Monty reveals a goat, the remaining door must now conceal the car. In other words, Monty is being forced by the rules to reveal the favorable settlement. In 2 out of 3 options, when you switch you will win the prize. Switching doubles the likelihood of winning!
A further illustration might help. What if Monty no longer must reveal a door with protracted litigation? Let’s assume that when Monty reveals one of the unselected doors he now can reveal the beneficial settlement, and if he does, you lose and the game ends. As before, if he reveals a losing door, he offers you the opportunity to switch. Under this new, modified scenario, there is no benefit to switching because there now is a 50-50 likelihood that the prize is behind either door. In other words, only in the modified approach does revealing a door with an unfavorable outcome actually mean that the odds for your original selection have improved, from 33% to 50%. This is not true in our Resolution Roulette game. A formula called Bayes’ Theorem allows you to calculate how certain events impact the probability of other events, or contingent probability.
I am not saying that mediators and lawyers need to have ever heard of Bayes’ Theorem – let alone apply it – to settle cases. However, a general understanding of probability cannot hurt in assessing the likelihood of future events, which, after all, is the lynchpin for valuing a case. Experts in mathematics, psychology and philosophy use the Monty Hall problem to demonstrate our cognitive limitations. One book highlights man’s poor choices, coining the phrase Bozo Sapiens. The Monty Hall problem underscores that humans are hard-wired to misunderstand seemingly simple probability calculations. And if mankind is really Bozo Sapien, a sub-species certainly must be the number fearing, Homo Advocatus. Lawyers, like most people, are often prone to miscalculations in situations where their gut strongly points to an incorrect conclusion. The pigeon experiment discussed above also demonstrates that humans are likely to have an emotional attachment to initial assumptions that causes them to miscalculate certain probabilities.
The purpose of this discussion is not to exhaustively address probability theory, but to suggest that thinking about it can help lawyers value cases and find common ground. I believe that the Monty Hall problem is relevant to settlement negotiations for several reasons, which can be used by a skillful mediator:
Despite all of our limitations, mediation is typically successful and most cases settle. Why?
One reason is that a mediator’s job is not to force the parties to agree on the “right” settlement amount. Parties frequently settle to avoid the risk that a judge or jury might misunderstand the law, evidence or probability. The reason the parties ultimately agree on an acceptable resolution is irrelevant.
To achieve a settlement, a mediator must be a good listener, be trusted as a neutral, understand the nature of the legal dispute, and understand how the parties value their case. A mediator that can evaluate assumptions and calculations that drive the party’s valuation of the case from both a legal and probability perspective is invaluable.
A good mediator must be flexible and convince the parties that they must be as well. Sometimes good-faith calculations made months ago to obtain settlement authority from management may become outdated because they fail to take into consideration new factors. A good mediator can convince parties that it is in their best interests to reevaluate the “lines in the sand” that they have drawn.
I have broad interests and experiences that allow me to assist in the resolution of a wide array of disputes. Not all cases require an understanding of statistics. Most cases present legal disputes and disagreements over the material facts. That said, statistics are often disputed in courts around the country. I would be happy to assist you in resolving these and other such cases.
If you have not yet found a preferred mediator or the other side will not accept your first choice(s), give me a chance and Let’s Make a Deal!
And if you have made it through this entire article and want to use me as a mediator, provide the following code: 60614-78731 to get a discount on your first mediation.
A more detailed version of this article with citations can be found by clicking here.
1 Rosenhouse, Jason, The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser (Oxford University Press 2009) provides a thorough analysis of the problem.
2 We could include another scenario of C, G, G with the two goats reversed (i.e., we could number the goats as Goat 1 and Goat 2). However, both C, G, G scenarios would collectively have a 33% likelihood because initially there is a 1 in 3 chance that the car is behind each door.
3 Kaplan, Michael and Kaplan, Ellen, Bozo Sapiens: Why to Err is Human (2009 Bloomsbury Press).
4 See e.g., Schneps, Leila and Colmez, Coralie, Math on Trial: How Numbers Get Used and Abused in the Courtroom (Basic Books 2013).
5 Complex litigation usually becomes a battle of the experts. In Federal Court, district court judges are tasked with being the “gatekeeper” in an effort to keep “junk science” out of the courtroom. Daubert v. Merrell Dow Pharms., Inc., 509 U.S. 579 (1993). In products liability cases, plaintiffs often attempt to present experts that opine that the defendant’s drugs caused the plaintiff’s harm, sometimes by relying on statistical analysis. Defendants typically seek to bar such testimony claiming the data is not “statistically significant.” Compare, In re: Zoloft, 2015 WL 7776911 (barring plaintiff’s expert and noting he failed to demonstrate that the concept of “statistical significance” had been abandoned) with Milward v. Acuity Specialty Prods. Grp., Inc., 639 F.3d 11, 25 (1st Cir. 2011)(overturning district court and determining that the “lack of statistical significance” is not the proper basis for excluding an expert’s opinion). That courts take different approaches concerning probability in the courtroom underscores the risks of litigation and is something about which a good mediator can discuss with the parties and their lawyers.