Chapter 3 Decision Analysis

The Six Steps in Decision Making Clearly define the problem at hand List the possible alternatives Identify the possible outcomes or states of nature List the payoff or profit of each combination of alternatives and outcomes Select one of the mathematical decision theory models Apply the model and make your decision

Case of Maria Rojas Maria Rojas is considering the possibility of opening a small dress shop on Fairbanks Avenue, a few blocks from the university. She has located a good mall that attracts students. Her options are to open as mall shop, a medium-sized shop, or no shop at all. The market for a dress shop can be good, average, or bad. The probabilities is 1/3 for each market. The net profit or loss for the medium-sized and small shops for the various market conditions are given in the following table. Building no shop at all yields no loss and no gain.

Case of Maria Rojas

Types of Decision-Making Environments Type 1: Decision making under certainty Decision maker knows with certainty the consequences of every alternative or decision choice Type 2: Decision making under uncertainty The decision maker does not know the probabilities of the various outcomes Type 3: Decision making under risk The decision maker knows the probabilities of the various outcomes

Decision Making Under Uncertainty There are several criteria for making decisions under uncertainty Maximax (optimistic) Maximin (pessimistic) Criterion of realism (Hurwicz) Equally likely (Laplace) Minimax regret

Maximax Used to find the alternative that maximizes the maximum payoff Locate the maximum payoff for each alternative Select the alternative with the maximum number STATE OF NATURE ALTERNATIVE GOOD MARKET ($) AVERAGE MARKET ($) BAD MARKET MAXIMUM IN A ROW ($) Small shop 75,000 25,000 -40,000 Medium-sized shop 100,000 35,000 -60,000 Do nothing Maximax

Maximin Used to find the alternative that maximizes the minimum payoff Locate the minimum payoff for each alternative Select the alternative with the maximum number STATE OF NATURE ALTERNATIVE GOOD MARKET ($) AVERAGE MARKET ($) BAD MARKET MINIMUM IN A ROW ($) Small shop 75,000 25,000 -40,000 Medium-sized shop 100,000 35,000 -60,000 Do nothing Maximin

Criterion of Realism (Hurwicz) A weighted average compromise between optimistic and pessimistic Select a coefficient of realism  Coefficient is between 0 and 1 A value of 1 is 100% optimistic Compute the weighted averages for each alternative Select the alternative with the highest value Weighted average = (maximum in row) + (1 – )(minimum in row)

Criterion of Realism (Hurwicz) For the small shop alternative using  = 0.8 (0.8)(75,000) + (1 – 0.8)(–40,000) = 52,000 For the medium-sized shop alternative using  = 0.8 (0.8)(100,000) + (1 – 0.8)(–60,000) = 68,000 STATE OF NATURE ALTERNATIVE GOOD MARKET ($) AVERAGE MARKET ($) BAD MARKET CRITERION OF REALISM ( = 0.8)$ Small shop 75,000 25,000 -40,000 52,000 Medium-sized shop 100,000 35,000 -60,000 68,000 Do nothing Realism

Equally Likely (Laplace) Considers all the payoffs for each alternative Find the average payoff for each alternative Select the alternative with the highest average STATE OF NATURE ALTERNATIVE GOOD MARKET ($) AVERAGE MARKET ($) BAD MARKET ROW AVERAGE ($) Small shop 75,000 25,000 -40,000 20,000 Medium-sized shop 100,000 35,000 -60,000 Do nothing Equally likely

Minimax Regret Based on opportunity loss or regret, the difference between the optimal profit and actual payoff for a decision Create an opportunity loss table by determining the opportunity loss for not choosing the best alternative Opportunity loss is calculated by subtracting each payoff in the column from the best payoff in the column Find the maximum opportunity loss for each alternative and pick the alternative with the minimum number

Minimax Regret Opportunity Loss Tables STATE OF NATURE GOOD MARKET ($) AVERAGE MARKET BAD MARKET 100,000 - 75,000 35,000 - 25,000 0 – (- 40,000) 100,000 - 100,000 35,000 - 35,000 0 – (- 60,000) 100,000 - 0 35,000 - 0 Opportunity Loss Tables STATE OF NATURE ALTERNATIVE GOOD MARKET ($) AVERAGE MARKET ($) BAD MARKET Small shop 25,000 10,000 40,000 Medium-sized shop 60,000 Do nothing 100,000 35,000 Table 3.7

Minimax Regret Minimax STATE OF NATURE ALTERNATIVE GOOD MARKET ($) AVERAGE MARKET ($) BAD MARKET MAXIMUM IN A ROW Small shop 25,000 10,000 40,000 Medium-sized shop 60,000 Do nothing 100,000 35,000 Minimax Table 3.8

Decision Making Under Risk Decision making when there are several possible states of nature and we know the probabilities associated with each possible state Most popular method is to choose the alternative with the highest expected monetary value (EMV) EMV(alternative i) = (payoff of 1st state of nature) x (prob. of 1st state of nature) + (payoff of 2nd state of nature) x (prob. of 2nd state of nature) + … + (payoff of last state of nature) x (prob. of last state of nature)

EMV for Maria Rojas Each market has a probability of 1/3 Which alternative would give the highest EMV? The calculations are EMV (small shop) = (1/3)($75,000) + (1/3)($25,000) + (1/3)($-40,000) = $20,000 EMV (medium shop) = (1/3)($100,000) + (1/3)($35,000) + (1/3)($-60,000) = $25,000 EMV (do nothing) = (1/3)($0) + (1/3)($0) + (1/3)($0) = $0

EMV for Maria Rojas Largest EMV STATE OF NATURE ALTERNATIVE GOOD MARKET ($) AVERAGE MARKET ($) BAD MARKET ROW AVERAGE ($) Small shop 75,000 25,000 -40,000 20,000 Medium-sized shop 100,000 35,000 -60,000 Do nothing Probability 1/3 Largest EMV EMV (small shop) = (1/3)($75,000) + (1/3)(25,000) + (1/3)(-40,000) = $20,000 EMV (medium shop) = (1/3)($100,000) + (1/3)(35,000) + (1/3)($-60,000) = $25,000 EMV (do nothing) = (1/3)($0) + (1/3)($0) + (1/3)($0) = $0

Expected Value of Perfect Information (EVPI) EVwPI (Expected Value with Perfect Information) is the long run average return if we have perfect information before a decision is made EVwPI = (best payoff for 1st SoN)x P1st SoN + (best payoff for 2nd SoN)x P2nd SoN + … + (best payoff for nth SoN)x Pnth SoN EVPI (Expected Value of Perfect Information) places an upper bound on what you should pay for additional information EVPI = EVwPI – Maximum EMV

Expected Value of Perfect Information (EVPI) Scientific Marketing, Inc. offers analysis that will provide certainty about market conditions (favorable) Additional information will cost $25,000 Is it worth purchasing the information?

Expected Value of Perfect Information (EVPI) STATE OF NATURE ALTERNATIVE GOOD MARKET ($) AVERAGE MARKET ($) BAD MARKET ROW AVERAGE ($) Small shop 75,000 25,000 -40,000 20,000 Medium-sized shop 100,000 35,000 -60,000 Do nothing Probability 1/3 Best alternative for good state of nature is opening a medium shop with a payoff of $100,000 Best alternative for average state of nature is opening a medium shop with a payoff of $35,000 Best alternative for bad state of nature is to do nothing with a payoff of $0 EVwPI = ($100,000)(1/3) + ($35,000)(1/3) + ($0)(1/3) = $45,000 The maximum EMV without additional information is $25,000 EVPI = EVwPI – Maximum EMV = $45,000 - $25,000 = $20,000

Expected Value of Perfect Information (EVPI) Best alternative for good state of nature is opening a medium shop with a payoff of $100,000 Best alternative for average state of nature is opening a medium shop with a payoff of $35,000 Best alternative for bad state of nature is to do nothing with a payoff of $0 EVwPI = ($100,000)(1/3) + ($35,000)(1/3) + ($0)(1/3) = $45,000 The maximum EMV without additional information is $25,000 EVPI = EVwPI – Maximum EMV = $45,000 - $25,000 = $20,000 So the maximum Maria should pay for the additional information is $20,000

Expected Opportunity Loss Expected opportunity loss (EOL) is the cost of not picking the best solution First construct an opportunity loss table For each alternative, multiply the opportunity loss by the probability of that loss for each possible outcome and add these together Minimum EOL will always result in the same decision as maximum EMV Minimum EOL will always equal EVPI

Expected Opportunity Loss Opportunity loss table STATE OF NATURE ALTERNATIVE GOOD MARKET ($) AVERAGE MARKET ($) BAD MARKET MAXIMUM IN A ROW Small shop 25,000 10,000 40,000 Medium-sized shop 60,000 20,000 Do nothing 100,000 35,000 45,000 Probability 1/3 Minimum EOL EOL (small shop) = (1/3)($25,000) + (1/3)($10,000) + (1/3)($40,000) = $25,000 EOL (medium shop) = (1/3)($0) + (1/3)($0) + (1/3)($60,000) = $20,000 EOL (do nothing) = (1/3)($100,000) + (1/3)($35,000) + (1/3)($0) = $45,000

Homework 03 Prob. 3.16, 3.18, 3.19, 3.22, 3.26, 3.27