Decision Theory Professor Ahmadi Learning Objectives Structuring the decision problem and decision trees Types of decision making environments: Decision making under uncertainty when probabilities are not known Decision making under risk when probabilities are known
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Consider the following problem with two decision alternatives (d1 & d2) and two states of nature S1 (Market Receptive) and S2 (Market Unfavorable) with the following payoff table representing profits ( $1000):
States of Nature
d1 20 6
d2 25 3
An optimistic decision maker would use the optimistic approach. All we really need to do is to choose the decision that has the largest single value in the payoff table. This largest value is 25, and hence the optimal decision is d2.
choose d2d2 25 maximum
A conservative decision maker would use the conservative approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs.
choose d1d1 6 maximum
For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. The resulting regret table is:
d1 5 0 5
d2 0 3 3 minimum
Then, select the decision with minimum regret.
Equally likely, also called Laplace, criterion finds decision alternative with highest average payoff.
Average for d1 = (20 + 6)/2 = 13
Average for d2 = (25 + 3)/2 = 14 Thus, d2 is selected
In our example let = 0.8
Payoff for d1 = 0.8*20+0.2*6=17.2
Payoff for d2 = 0.8*25+0.2*3=20.6 Thus, select d2
where: N = the number of states of nature
P(sj) = the probability of state of nature sj
Vij = the payoff corresponding to decision alternative di and state of nature sj
Refer to the previous problem. Assume the probability of the market being receptive is known to be 0.75. Use the expected monetary value criterion to determine the optimal decision.
Determine the optimal return corresponding to each state of nature.
Compute the expected value of these optimal returns.
Subtract the EV of the optimal decision from the amount determined in step (2).
Calculate the expected value for the best action for each state of nature and subtract the EV of the optimal decision.
EVPI= .75(25,000) + .25(6,000) - 19,500 = $750
For each state of nature, multiply the prior probability by its conditional probability for the indicator -- this gives the joint probabilities for the states and indicator.
Sum these joint probabilities over all states -- this gives the marginal probability for the indicator.
For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution.
Answer the following questions based on the above information.
1. Draw a complete probability tree.
2. Find the posterior probabilities of all states of nature.
3. Using the posterior probabilities, which plan would you recommend?
4. How much should one be willing to pay (maximum) for the research survey? That is, compute the expected value of sample information (EVSI).