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Decision Analysis

Occasionally in order to avoid an impasse, it is helpful for the parties’ case evaluation if they can calculate, based on their best guesses about the key variables, what the average result would be if the case were tried ten times.  “Decision analysis” is a powerful tool for making that calculation.  I have found it useful in air passenger, employment, personal injury, real estate and securities cases. Decision analysis will not magically convert best guesses into sure outcomes, but it can be a significant aid in settlement negotiations.

Originally known as “decision tree analysis” because of the use of schematic trees, the analysis produces what economists call the “expected value” of trying a case.  Simply put, the expected value is the average (mean) result from trying the case 100 times.

Decision analysis is not needed in simple cases.  Suppose a plaintiff had a 65% chance of winning a breach of contract claim, where there were stipulated damages of $100,000.  Most of us would intuitively understand that the case had a value for the plaintiff of about $65,000.  We get that result from multiplying the odds of winning times the damage figure.

In a more complicated situation, though, the analysis is much trickier.  Suppose in a negligence case a plaintiff has a 90% chance of defeating the defendant’s motion for summary judgment.  Suppose further there is a 70% chance the plaintiff will make a good witness.  If so, suppose the plaintiff has a 75% chance of winning at trial on the negligence issue, an 80% chance of winning on the causation issue, a 30% chance of a $300,000 damage award assuming liability, a 60% chance of a $500,000 damage award, a 10% chance of $1,000,000 damage award, and finally expected unrecoverable costs of $10,000 after the summary judgment motion to take the case through trial.  Suppose however that if the plaintiff is a bad witness, the chance of winning on negligence drops to 50%, the chance of a low damage award increases to 40% and there is no chance of a high damage award.  Now, probably very few people could intuitively value the case.  With a decision analysis tree, it is possible to calculate the expected value – the expected average result from trying the case 100 times.  Given these assumptions, the expected value is about $222,000.  Here’s what the tree would look like for this negligence example: 

To help understand how a decision tree works, here’s a diagram of a simple tree:

This tree has only two “chance nodes” (the round circles) – one for liability and one for damages.  Starting at the left of the tree, there are two possible results at the liability chance node.  The tree shows 70% odds of a finding of liability and 30% odds there will be no liability.  Following out the “yes” liability branch, the tree then shows 20% odds of a $70,000 award, 60% odds of a $40,000 award and 20% odds of a $10,000 award.  (The odds at a chance node must always add to 100.)  Since the claimant would lose if there was no liability, there is no chance node for damages at the end of the “no” liability branch.

To calculate the expected value, you start at the right hand side of the tree.  For each set of branches off of a chance node, you multiply the odds times the damage figure, and add up the numbers.  Here, 20% of $70,000 is $14,000, 60% of $40,000 is $24,000 and 20% of $10,000 is $2,000.  Adding up the three numbers gives a sum of $40,000.  This process is then repeated for the next chance node to the left.  You take this $40,000 figure and multiply it by the 70% odds of a “yes” result on liability, which gives $28,000.  For the lower “No” branch, you multiply the 30% odds times the $0 result, which of course gives $0.  Adding the $28,000 and $0 figures gives the final, expected value of $28,000.  Again, this decision analysis tree shows that, based on the assumed odds and values for the possible damage awards, on average if the case were tried 100 times the average result would be an award of $28,000.

It is important to note that the expected value figure may not be a result that could be obtained in any one trial.  To illustrate, suppose you were asked to bid on the right to guess whether the flip of a one dollar coin would come up heads or tails.  Knowing that you have a 50% chance of guessing correctly, you know that the expected value of the right to guess is fifty cents.  While the expected value is fifty cents that is not a result you would get in any one toss since you either win the entire dollar or lose it all.

For a more complicated decision analysis tree example, here is a tree that might be used in analyzing a claim by a customer of a securities brokerage firm that the securities recommended by the broker were not suitable given the customer’s financial situation and investment objectives:

Frequently these suitability cases turn on how sympathetic the claimant is, so for this tree that issue is the first chance node.  The “Yes” and “No” branches that lead away from that first chance node are identical here in structure.  Each branch has a chance node for the suitability issue and then chance nodes for possible damage awards.  I have assumed here that there is a 70% chance the arbitrators will find the claimant sympathetic, and if so, a 70% chance the arbitrators will then find the investments were not suitable.  Assuming a finding the investments were not suitable, I have assumed that there is a 30% chance of an award of $300,000 (a Net Out of Pocket loss figure less a deduction for a failure to mitigate), a 65% chance of a full NOP award of $500,000, and a 5% chance of an award of the full NOP plus another $500,000 in punitive damages.  On the other hand, in the tree I have assumed that if the arbitrators find the claimant is not sympathetic, then the odds of a finding the investments not suitable drops to 30%, the odds of a lower end “NOP-Mitigate” award increase to 60%, the odds of a full NOP award decrease to 40% and there is no chance for an award of the full NOP plus punitive damages.  (This tree is from the claimant’s perspective, and is simplified by the omission of unrecoverable costs in a loss.  A tree from the respondent’s perspective could look nearly identical in structure, with the inclusion of expected defense fees and costs and usually with different assumed odds and damage figures.)  Given these assumptions, this decision analysis tree shows the expected value of the suitability claim is $262,050.

One can construct decision analysis trees with many more branches.  One could have a chance node for each element in the claimant’s case.  The federal government apparently uses trees with tens of thousands of branches.  For the purposes of a mediation, though, less branches probably are better than more.  The more branches are used, the more difficult it is to explain, track and print the tree.

Having calculated the expected value of a claim, it is important to spend a few more minutes performing what is known as a “sensitivity analysis”.  This is done by changing one or more variables – odds and/or damage figures – and seeing what the impact would be upon the expected value figure.  The reasons for doing the sensitivity analysis include making sure your tree comports with reality and identifying what the most important variables are.  For example, suppose that in the above hypothetical suitability claim the claimant’s lawyer thought that the most the claimant could get would be an award of $1,000,000 representing the NOP loss  plus punitive damages, but the claimant believed that the high award reasonably could be $2,000,000.  Suppose further that based on that belief, the claimant was holding out for a much higher settlement than the $262,050 expected value.  If the tree were recalculated with this higher top end award figure, it would raise the expected value only to $286,550.  What this analysis shows is that because here the odds of a full NOP plus punitive damages award are so attenuated (70% times 70% times 5%), the expected value figure is not very sensitive to even a dramatic increase in the potential punitive damages award.

There is one more important product of the decision analysis to review.   That is the “risk profile” shown here:

Based upon the assumptions used here, the Risk Profile chart shows that while arbitrating the case 100 times on average would result in a $262,050 award, in 42 out of those 100 times the claimant would lose.  Clearly, this point must be understood by the claimant before proceeding to arbitration.

The expected value figure produced by decision analysis cannot be used simply as a party’s “bottom line.”  For example, there may be a need to adjust the expected value figure for a party’s risk tolerance.  In the hypothetical suitability claim here, there must be a discussion regarding the claimant’s tolerance for risk.  If the claimant were the proverbial senior citizen who lost most of his or her irreplaceable retirement funds, even though the claim would be very solid it would be a huge disaster to lose at arbitration.  This claimant probably should take a downward adjustment in the $262,050 expected value figure in order to come away from the mediation with a known recovery.  If the claimant were a billionaire for whom a total loss at arbitration would be inconsequential, no risk tolerance adjustment would be appropriate.  The same risk tolerance issues may be faced by defendants.  For example, suppose in the suitability hypothetical there were pending ten additional, similar claims by other claimants.  The broker dealer might want to pay a settlement figure higher than the calculated expected value of the instant case as an “insurance policy” so as to avoid an increased exposure in the remaining cases caused by losing the first one.  Another example might be where a corporate defendant is faced with a claim that potentially would bankrupt the company if there was a large verdict.  There are other reasons why a party might want to vary from the expected value for settlement purposes, including the physical and emotional impact of going through a trial or arbitration and the possibility of unwanted and/or adverse publicity.

Although decision analysis can seem a little daunting at first, the mechanics of performing the analysis are relatively simple.  You can do these analyses by paper and pen, but it takes a while, you don’t get a risk profile, and it is cumbersome to change the variables.  I use a decision analysis computer program that makes it easy and quick to construct a tree, plug in the odds and values and get the calculated expected values and risk profile.  Performing the analysis takes about a half-hour to forty-five minutes.  Where possible, I hook my laptop to another monitor so that party and counsel easily can see the construction of the tree without having to squeeze behind me.  I use my portable printer to print out copies of the trees.

Why take the time to do a decision analysis in a mediation?  Sometimes there is an obvious discrepancy between the views of a party and the party’s counsel regarding the settlement value.  Simply walking through the structure of a tree can help clarify the case evaluation analysis.  Then, discussing and assigning odds and damage figures and running the analysis can be very helpful in getting everyone on that side on the same page.  In an evaluative mode, if my opinions are different from a party’s and counsel’s, I will use my own figures for the odds and reasonable damage awards and run the analysis.  Whether the odds and damage figures come from me or the party, the decision analysis is a much more powerful tool than best, worst and most probable charts which do not factor in the probability of these different potential results.  The use of decision analysis can also help avoid the natural tendency to overweight a large but only remotely possible outcome.  Decision analysis can also help deal with emotional bias.

There are some potential downsides to using decision analysis in a mediation.  There can be undue time consumption.  A party may perceive the analysis as a parlor trick designed to make them believe best guesses have been converted into certainties.  The mediator must not let the analysis cause excessive focus on position based bargaining as opposed to interest based bargaining.  There is also a risk that a party may seize upon an expected value figure and give little or no consideration to other possible odds and damage figures or advisable risk tolerance adjustments.

If I think decision analysis will be useful in avoiding an impasse, I will ask a party’s permission to use it.  It definitely is not a tool for most mediations, but in the right time and place it can help greatly in bringing the parties together on a satisfactory agreement.


Robert H. Logan

As a mediator since 2005, Bob Logan has experience mediating securities, mortgage banking, title insurance, Montreal Convention (cargo and passenger), employment, personal injury, premises liability and a variety of other cases. In addition to the knowledge and experience gained as lead or supervising counsel in hundreds of mediations, Bob has had extensive formal… MORE >

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