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# Decision Analysis with Decision Tree

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Jack Nimble, an employee of Daniel Construction Company, claims to have injured his back as a result of a fall while repairing the roof at one of the Eastview apartment buildings. He filed a lawsuit against Dave Gilmour, the owner of Eastview apartments, asking for damages of \$1,500,000. Jack claims that the roof had rotten sections and that his fall could have been prevented if Dave Gilmour had told Daniel Construction Company about the problem. Dave Gilmour notified his insurance company, Allstate Insurance, of the lawsuit. Allstate insurance must defend Dave Gilmour and decide what action to take regarding the lawsuit.

Some depositions and a series of discussions took place between both sides. As a result, Jack Nimble offered to accept a settlement of \$750,000. Thus, one option is for Allstate Insurance to pay Jack \$750,000 to settle the claim. Allstate is also considering making Jack Nimble a counteroffer of \$400,000 in the hope that he will accept a lesser amount to avoid the time and cost of going to trial. Allstate's preliminary investigation shows that Jack Nimble's case is strong; Allstate is concerned that Jack Nimble may reject their counteroffer and request a jury trial. Allstate's lawyers spent some time exploring Jack's likely reaction if they make a counteroffer of \$400,000.

The attorneys concluded that it is adequate to consider three possible outcomes to represent Jack Nimble's possible reaction to a counteroffer of \$400,000.
(1) Jack Nimble will accept the counteroffer of \$400,000 and the case will be closed;
(2) Jack Nimble will reject the counteroffer and elect to have a jury decide the settlement amount; or
(3) Jack Nimble will make a counteroffer to Allstate Insurance of \$600,000.

If Jack Nimble does make a counteroffer, Allstate Insurance decided that they will not make additional counteroffers. Allstate Insurance will either accept Jack Nimble's counteroffer of \$600,000 or go to trial.

If the case goes to a jury trial, Allstate Insurance considers three outcomes possible:
(1) The jury may reject Jack Nimble's claim and Allstate Insurance will not be required to pay any damages;
(2) the jury will find in favor of Jack Nimble and award him \$750,000 in damages; or
(3) the jury will conclude that Jack Nimble has a strong case and award him the full amount of \$1,500,000.

Key considerations as Allstate Insurance develops its strategy for disposing of the case are the probabilities associated with Jack Nimble's response to an Allstate Insurance counteroffer of \$400,000 and the probabilities associated with the three possible trial outcomes. Allstate Insurance's attorneys believe the probability that Jack Nimble will accept a counteroffer of \$400,000 is 0.10, the probability that Jack Nimble will reject a counteroffer of \$400,000 is 0.40, and the probability that Jack Nimble will, himself, make a counteroffer to Allstate Insurance of \$600,000 is 0.50. If the case goes to court, the Allstate Insurance's attorneys believe that the probability the jury will award Jack Nimble damages of \$1,500,000 is 0.30, the probability that the jury will award Jack Nimble damages of \$750,000 is 0.50, and the probability that the jury will award Jack Nimble nothing is 0.20.

Perform an analysis of the problem facing Allstate Insurance and prepare a report that summarizes findings and recommendations. Include the following:

1. Decision tree in Excel workbook

2. Recommendation regarding whether Allstate Insurance should accept Jack Nimble's initial offer to settle the claim for \$750,000.

3. Decision strategy that Allstate Insurance should follow if they decide to make Jack Nimble a counteroffer of \$400,000.

4. Risk Profile for recommended strategy.

https://brainmass.com/math/probability/decision-analysis-with-decision-tree-132301

#### Solution Preview

Jack is asking for \$1,500,000. He would accept a \$750,000 settlement. Allstate has three options:

(1) Settle for \$750,000. If they choose this option, they would definitely have to pay \$750,000.
(2) Go to trial. In this case, they could pay \$1,500,000, \$750,000, or \$0. The expected value that they would have to pay is \$825,000.
(3) Make a counteroffer. If Allstate makes a counteroffer, Jack might accept, in which case they would have to pay \$400,000. If Jack rejects the offer, they might go to trial, in which case, they might have to pay \$1,500,000, \$750,000, or \$0. Finally, Jack might make a counter-counteroffer of \$600,000. Allstate could accept the offer and pay \$600,000, or they could go to trial. The expected value of making a counteroffer is \$1,082,500.

The calculations for finding the expected values are in the attached Excel spreadsheet. Because the expected value for option 1 is the lowest, Allstate should accept Jack's offer of ...

\$2.49