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Decision Tree and Acquisition of Commercial Properties

Need help with analysis of a sequential decision problem with decision tree;
Need to draw a decision tree for all possibilities.

Mr. Warren operated a real estate agency that specialized in finding buyers for commercial properties. Warren was approached one day by a prospective client, who had three properties that he wished to have sold. The client wished to receive the following prices:
Property Price
Allston $25,000
Belmont $50,000
Cambridge $100,000
Warren would receive a commission of 4% on any of the properties he was able to sell.

The client laid down the following conditions for an exclusive listing:
(i) Allston had to be sold first.
(ii) If Warren failed to sell Allston within a month, then the deal was off. Warren would not receive any commission or the chance to sell other properties.
(iii) If Warren could sell Allston within a month, he would get the commission for the sale and an option to sell the second property. At this stage, if Warren decided not to sell any more properties, then the contract would terminate. However, if Warren decided to sell a second property, then he had to choose between Belmont and Cambridge at this stage (but not both). Once he made the choice, he had one month to sell the property.
(iv) If Warren failed to sell the second property within a month, then the deal was off.
Warren would not receive any further commission or the chance to sell the third property.
(v) If successful, Warren had the option to terminate the contract or sell the third
property. If Warren decided to sell the third property, the terms were same as in the previous cases.

After the client left, Warren proceeded to analyze the proposal to determine whether or not to accept it. He figured his selling costs and his chances of selling each property at the price set by the client to be:
Selling Cost Probability of Sale
Allston $800 0.7
Belmont $200 0.6
Cambridge $400 0.5
He believed that the sale of a particular property would not make it more or less likely that the two remaining properties could be sold. Selling costs would have to be incurred whether or not a particular property was sold but could be avoided by deciding not to attempt to sell the property.
a. Draw a decision tree for the problem. Identify the decision nodes and chance nodes.
Demonstrate how you got the terminal values.
b. Suppose Warren successfully sold Allston first and Belmont second. Should he decide to terminate the contract? Why or why not?
c. Suppose Warren successfully sold Allston first and Cambridge second. Should he decide to terminate the contract? Why or why not?
d. Suppose Warren sold Allston successfully. Should he terminate the contract? If not, which property should he sell second?
e. Should Warren accept the contract?
f. Based on the decision tree, what is the option value of the contract (that is, if Warren accepts the contract instead of rejecting it, how much more or less can he expect to make)?

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Solution Preview

Please find the decision tree (in excel)attached. The decision tree is on the first worksheet and the answers with explanations are on the second worksheet. I have used Precision Tree to make the decision tree software. I highly recommend that software because it make it very easy and fast to make decision trees.

Answer (a): The decision tree is drawn on the previous spreadsheet. All the chance nodes are drawn in marron circle and all the decision nodes are in green squares.
The terminal values were obtained by adding/subtracting the values at each of the nodes. For example, the value at the topmost terminal node was obtained by adding 3600 to 1800. Also, the formulas for all the individual values throughout the ...

Solution Summary

The solution provides a very good decison tree for the problem below.