An oil company has been given the option to drill in one of two oil fields F1 an
ID: 158365 • Letter: A
Question
An oil company has been given the option to drill in one of two oil fields F1 and F2 but not both. The company believes that the existence of oil in one field is independent of the other with the probability of oil in F1 being 0.4 and in F2 being 0.2. A net profit of $770m is expected if oil is struck in F1 and $1950m if oil is struck in F2. The company can pay $60m to investigate either but not both of the fields if it chooses. Whether or not it takes this option it can then choose not to drill (d0) or drill Fi (di) i = 1,2. The investigation is not entirely foolproof. The DM believe that when oil is present the investigators advise drilling with probability 0.8 and when oil is not present will advise drilling with probability 0.4. The cost of accepting the option on either field is $31m. Draw the rollback tree of this problem and find the decision maximizing the company expected payoff.Explanation / Answer
Decision-tree models offer a visual tool that can represent the key elements in a model for decision making under uncertainty and help organize those elements by distinguishing between decisions (controllable variables) and random events (uncontrollable variables).
In a decision tree, we describe the choices and uncertainties facing a single decision-making agent.
This usually means a single decision maker, but it could also mean a decision-making group or a company
•Whereas we build the tree left to right, to reflect the temporal sequence in which a decision is followed by a chance event, we evaluate the tree in the reverse direction.
•At each chance node, we can calculate the expected payoff represented by the probability distribution at the node.
•This value becomes associated with the corresponding action branch of the decision node.
•Then, at the decision node, we calculate the largest expected payoff to determine the best action.
This process of making the calculations is usually referred to as rolling back the tree
•The distribution associated with a particular action is called its risk profile.
•The risk profile shows all the possible economic outcomes and provides the probability of each: It is a probability distribution for the principal output of the model.
•This form reinforces the notion that, when some of the input parameters are described in probabilistic terms, we should examine the outputs in probabilistic terms.
•After we determine the optimal decision, we can use a probability model to describe the profit outcome.
Principles for Building and Analyzing Decision Trees
1.Determine the essential decisions and uncertainties.
2.Place the decisions and uncertainties in the appropriate temporal sequence.
3.Start the tree with a decision node representing the first decision.
4.Select a representative (but not necessarily exhaustive) number of possible choices for the decision node.
5.For each choice, draw a chance node representing the first uncertain event that follows the initial decision.
6.Select a representative (but not necessarily exhaustive) number of possible states for the chance node.
7.Continue to expand the tree with additional decision nodes and chance nodes until the overall outcome can be evaluated.
Rollback Procedure for Analyzing Trees
1.Start from the last set of nodes—those leading to the ends of the paths.
2.For each chance node, calculate the expected payoff as a probability-weighted average of the values corresponding to its branches.
3.Replace each chance node by its expected value.
4.For each decision node, find the best expected value (maximum benefit or minimum cost) among the choices corresponding to its branches.
5.Replace each decision node by the best value, and note which choice is best.
6.Continue evaluating chance nodes and decision nodes, backward in sequence, until the optimal outcome at the first node is determined.
Now it is easy for you to go through the rollback tree diagram drawn below