Used with permission from Making Hard Decisions with DecisionTools by Robert T. Clemen and Terence Reilly, copyright 2000 Duxbury Thomson Learning.
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In 1984, Pennzoil and Getty Oil agreed to a merger. But before it was finalised, Texaco offered Getty a substantially better price, and Getty Oil reneged on the Pennzoil deal and sold to Texaco. Pennzoil immediately sued Texaco, alleging Texaco had illegally interfered in the negotiations. Pennzoil won the case in 1985 and was awarded $11.1 billion, the largest judgment ever in the U.S. at the time. A Texas appeals court reduced the judgment by $2 billion, but interest and penalties drove it back up to $10.3 billion. Texaco said they would appeal the case to the Supreme Court, and would file for bankruptcy if forced to pay. In April 1987, just before Pennzoil began to file liens against Texaco's assets, Texaco offered to pay Pennzoil $2 billion to settle the entire case. Pennzoil wanted between $3 and $5 billion. What should Pennzoil do? Take the $2 billion and run? Or try for $5 billion?

If Pennzoil takes the $2 billion, they risk getting a lot less money than they feel they are entitled to. If they refuse the $2 billion and make a counteroffer of $5 billion, Texaco might agree, or they might counter with $3 billion, or simply pursue further appeals. Although this situation is simplified for illustration purposes (Pennzoil could counteroffer a lot of values besides $5 billion, and Texaco could then counter with other values; the possibility of Texaco filing for bankruptcy is not included; etc.), it captures the essential features of a very realworld business problem.
At the root (left) of the tree, a green square decision node represents the choice Pennzoil has to accept or reject Texaco's $2 billion settlement offer. If they counteroffer $5 billion, there are three possible reactions Texaco could have  represented by a red chance node with three branches. Texaco could accept the $5 billion offer, thinking it's better than possibly being forced to pay more later on, or go bankrupt. Texaco could refuse the counteroffer and let the courts decide the final settlement. Or Texaco could counteroffer $3 billion. Each of these possible outcomes has a different probability of occurring: 17% for accepting the $5 billion offer; 50% for refusing it and offering no alternative; and 33% for refusing it and counteroffering $3 billion. These probabilities may be entered directly in the spreadsheet and may be gleaned from expert opinion. In this case, Pennzoil chairman Hugh Liedtke and his advisors may have decided that, given the tough negotiating stance of himself and Texaco CEO James Kinnear, there is an even (50%) chance that Texaco would refuse to negotiate further. Furthermore, Liedtke might figure that a counteroffer of $3 billion is about twice as likely as Texaco accepting the $5 billion. Thus, because there is a 50% chance of refusal, there must be a 33% chance of a Texaco counteroffer and a 17% chance of Texaco accepting $5 billion.
If Texaco refuses to negotiate, or if Texaco makes a counteroffer that Pennzoil refuses, then it goes to the courts to decide. What are the possible outcomes then, and the probabilities of each? Liedtke admitted that Texaco could win the case, leaving Pennzoil with nothing but lawyer bills. Thus, there is a significant possibility that the outcome would be zero. Given the strength of Pennzoil's case so far, there is a good chance that the court will uphold the judgment as it stands. Finally, the possibility exists that the judgment could be reduced somewhat (to $5 billion in this model). Let's assume that Liedtke and his advisors agree that there is a 20% chance that the court will award the entire $10.3 billion and a slightly larger, or 30% chance the award will be zero. Thus, there must be a 50% chance of an award of $5 billion.
Figure 1: Decision tree. 
Now that the model is set up, how do you decide what the best choices are? One way to choose among risky alternatives is to pick the one with the highest expected value (EV). PrecisionTree does this automatically. Basically, it starts at the endpoints of the branches and moves left, calculating expected values when it encounters a chance node, or choosing the branch with the highest value when it encounters a decision node. For a detailed explanation of how expected value is calculated, see pp. 115117 of Making Hard Decisions with DecisionTools. From the figure, we can see that PrecisionTree has determined the expected value of counteroffering $5 billion to be $4.63 billion, vs. $2 billion if Pennzoil accepts Texaco's initial offer. Pennzoil should refuse the initial offer and counteroffer $5 billion. If Texaco turns down the $5 billion and makes another counteroffer of $3 billion, Pennzoil should refuse again and take their chances in court. The decision tree allows Pennzoil to develop a contingent strategy. If a particular course of events occurs (Texaco's counteroffer), then there is a specific course of action to take (refuse the counteroffer). In order to make a good decision at the current time, we have to know what appropriate contingent strategies are in the future.
Even though counteroffering $5 billion and then refusing a Texaco counteroffer of $3 billion may be the best choices, what about the risks associated with these various decisions? What are the chances of different payoffs for Pennzoil occurring? For that, just click PrecisionTree's Decision Analysis button to view various risk profile graphs. PrecisionTree gives you histograms, cumulative charts, and scatterplots to display your risks. In this example, if you follow the optimal choices through the tree, there is a 58.5% chance that the eventual settlement is $5 billion, a 16.6% chance of $10.3 billion, and a 24.9% chance of nothing (see charts below). PrecisionTree can also give you a truncated version of your tree showing only the optimal routes to take. This Policy Suggestion report is also shown below.
Figure 2: Risk Profile Histogram. 
Figure 3: Risk Profile Cumulative. 
Figure 4: Risk Profile Scatterplot. 
Figure 5: Policy Suggestion Report. Click to view larger graphic. 
Together, PrecisionTree's decisions trees and risk profiles can provide invaluable insight to the decision maker faced with any tough decision.
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