Chapter 19: Decision Analysis

Learning Objectives LO1 Make decisions under certainty by constructing a decision table. LO2 Make decisions under uncertainty using the maximax criterion, the maximum criterion, the Hurwicz criterion, and the minimax regret. LO3 Make decisions under risk by constructing decision trees, calculating expected monetary value and expected value of perfect information, and analyzing utility. LO4 Revise probabilities in light of sample information by using Bayesian analysis and calculating the expected value of sample information.

Decision-Making Scenarios Decision-making under certainty Decision-making under uncertainty Decision-making under risk LO1

Three Variables in Decision Analysis Model Many decision analysis problems can be viewed as having variables Decision Alternatives are the various choices or options available to the decision maker in any given problem situation (actions or strategies) States of nature are the occurrences of nature that can happen after a decision is made that can affect the outcome of the decision and over which the decision maker has little or no control. States of nature can be environmental, business climate, political, or any condition or state of affairs. Payoffs are the benefits or rewards (positive or negative) that result from selecting a particular decision alternative. They are often expressed in dollars, but may be stated in other units, such as market share. LO1

Construction of the Decision or Payoff Table The concepts of decision alternatives, states of nature, and payoffs can be examined jointly by using a decision table or payoff table. The table a cross tabular table with “states of nature” and “decision alternatives” as classification variables. The associated outcomes in the cells are the payoffs or benefits resulting from making certain choices and a certain state of nature occurring. Table 19.1 on the next slide illustrates the structure of the problem. LO1

Table 19.1: The Decision or Payoff Table States of Nature Decision Alternatives where: sj = state of nature dj = decision alternative Pi,j = payoff for decision i under state j LO1

Yearly Payoffs on an Investment of $10 000: Description of Problem An investor is faced with the decision of where and how to invest $10,000 under several possible states of nature States of Nature A stagnant economy A slow-growth economy A rapid-growth economy Decision Alternatives being considered Invest in the stock market Invest in the Bond market Invest in GICs Invest in a mixture of stocks and bonds The payoffs are presented in Table 19.2 on the next slide. LO1

Table 19.2: Decision Table for an Investor Stagnant Slow Growth Rapid Stocks (500) $ 700 2,200 Bonds (100) 600 900 GICs 300 500 750 Mixture (200) 650 1,300 Annual payoffs for an investment of $10,000 LO1

Rule for Decision Making Under Certainty In making decisions under certainty the states of nature are known. The decision maker needs merely to examine the payoffs under different decision alternatives and select the alternative with the highest with the largest payoff LO1

Decision Making Under Certainty The states of nature are known. The Greatest Possible Payoff The economy will grow rapidly. Invest in stocks. Stagnant Slow Growth Rapid Stocks (500) $ 700 2,200 Bonds (100) 600 900 GICs 300 500 750 Mixture (200) 650 1,300 Annual payoffs for an investment of $10,000 LO1

Criteria for Decision Making Under Uncertainty Maximax payoff: Choose the best of the best (An optimist’s choice) Maximin payoff: Choose the best of the worst (A pessimist’s choice) Hurwicz payoff: Use a weighted average of the extremes (optimist and pessimist) Minimax regret: Minimize the maximum opportunity loss LO2

Maximax Criterion Stagnant Slow Growth Rapid Maximum Stocks Bonds GICs 1. Identify the maximum payoff for each alternative. 2. Choose the alternative with the largest maximum. Stagnant Slow Growth Rapid Maximum Stocks Bonds GICs Mixture (500) $ 700 2,200 (100) 600 900 300 500 750 (200) 650 1,300 LO2

Maximin Criterion Slow Rapid Stagnant Growth Growth Minimum Stocks $ 1. Identify the minimum payoff for each alternative. 2. Choose the alternative with the largest minimum. Slow Rapid Stagnant Growth Growth Minimum Stocks $ (500) $ 700 $ 2,200 $ (500) Bonds $ (100) $ 600 $ 900 $ (100) GICs $ 300 $ 500 $ 750 $ 300 Mixture $ (200) $ 650 $ 1,300 $ (200) LO2

Hurwicz Criterion Stagnant Slow Growth Rapid Maximum Minimum Weighted 1. Identify the maximum payoff for each alternative. 2. Identify the minimum payoff for each alternative. 3. Calculate a weighted average of the maximum and the minimum using  and (1 - ) for weights. 4. The size of α is between 0 and 1 and will depend on how optimistic or pessimistic the decision-maker is. 4. Choose the alternative with the largest weighted average. Stagnant Slow Growth Rapid Maximum Minimum Weighted Average Stocks (500) $ 700 2,200 1,390 Bonds (100) 600 900 GICs 300 500 750 615 Mixture (200) 650 1,300 850  =.7  =.3 LO2

Decision Alternatives for Various Values of  Stocks Bonds GICs Mixture Max Min  1- 2,200 -500 900 -100 750 300 1,300 -200 0.0 1.0 0.1 0.9 -230 345 -50 0.2 0.8 40 100 390 0.3 0.7 310 200 435 250 0.4 0.6 580 480 400 0.5 850 525 550 1120 500 570 700 1390 600 615 1660 660 1000 1930 800 705 1150 2200 1300 LO2

Graph of Hurwicz Criterion Selections for Various Values of  LO2

Investment Example: Selected Regrets I invested in stocks, and the economy grew slowly. I have no regrets. Stagnant Slow Growth Rapid Stocks (500) $ 700 2,200 Bonds (100) 600 900 GICs 300 500 750 Mixture (200) 650 1,300 I invested in stocks. Then the economy stagnated. I regret not investing in GICs. I am $800 down from where I could have been. I invested in GICs. Then the economy grew rapidly. I am out $1,450. LO2

Investment Example: Opportunity Loss Table Stagnant Slow Growth Rapid Stocks 800 Bonds 400 100 1,300 GICs 200 1,450 Mixture 500 50 900 LO2

Investment Example: Calculating Opportunity Loss Stagnant Slow Growth Rapid Stocks (500) $ 700 2,200 Bonds (100) 600 900 GICs 300 500 750 Mixture (200) 650 1,300 Payoff Table 800 400 100 200 1,450 50 Opportunity Loss Table OLi,j = Max(column j) - Pi,j LO2

Minimax Regret Stagnant Slow Growth Rapid Maximum Stocks 800 Bonds 400 1. Identify the maximum regret for each alternative. 2. Choose the alternative with the least maximum regret. Stagnant Slow Growth Rapid Maximum Stocks 800 Bonds 400 100 1,300 GICs 200 1,450 Mixture 500 50 900 LO2

Decision Making under Risk Probabilities of the states of nature have been determined Decision making under uncertainty: probabilities of the states of nature are unknown Decision making under risk: probabilities of the states of nature are known (have been estimated) Decision Trees Expected Monetary Value of Alternatives LO3

Decision Table with States of Nature Probabilities for Investment Example LO3

Decision Tree for the Investment Example Stocks Bonds GICs Mixture Slow growth (.45) Stagnant (.25) Rapid Growth (.30) -$500 $700 $2,200 -$100 $600 $900 $300 $500 $750 -$200 $650 $1,300 Decision Node Chance LO3

Expected Monetary Value Criterion LO3

EMV Calculations for the Investment Example LO3

Decision Tree with Expected Monetary Values for the Investment Example Stocks Bonds GICs Mixture Slow growth (.45) Stagnant (.25) Rapid Growth (.30) -$500 $700 $2,200 -$100 $600 $900 $300 $500 $750 -$200 $650 $1,300 $850 $515 $525 $623.50 LO3

Decision Tree with Expected Monetary Values for the Investment Example LO3

EMV Criterion for the Investment Example 1. Calculate the expected monetary value of each alternative. 2. Choose the alternative with the largest EMV: $850 LO3

Definition of Expected Value of Perfect Information What is the value of knowing which state of nature will occur and when? What is the value of sampling information or undertaking the prediction of an event? The concept of the expected value of perfect information answers these questions and provide some insight into how much the decision maker should pay for market research. LO3

Definition of Expected Value of Perfect Information The expected value of perfect information the difference between the payoff that would occur if the decision maker knew which state of nature would occur and the expected monetary payoff from the best decision alternative when there is no information about the occurrence about the states of nature Expected Value of Perfect Information = Expected Monetary Payoff with Perfect Information – Expected Monetary Payoff with Information

Choice Criterion Under Perfect Information: Choose the Maximum Payoff for any Given State of Nature LO3

Expected Monetary Payoff with Perfect Information for the Investment Example The investment of stocks was selected under the EMV strategy because it resulted in the maximum payoff of $850. This decision was made with no information about the states of nature (Refer to slide above Perfect Information) Maximum Payoffs for each state of nature under perfect information: Stagnant Economy = $300; Slow Growth = $700; Rapid Growth = $2,200 (refer to slide above: Perfect Information Criterion ) The expected Monetary Value with perfect information = (300)(0.25) + ($700)(0.45) + ($2,200)(0.30) = $1,050 LO3

Expected Value of Perfect Information for the Investment Example = Expected Monetary Payoff with Perfect Information - Max(EMV[di]) = $1050 - $850 = $200 It would not be economically wise to spend more than $200 to obtain perfect Information about these states of nature. The cost of collecting and processing the information is very high relative to the benefits. LO3

Utility Utility is the degree of pleasure or displeasure a decision maker has in being involved in the outcome selection process given the risks and opportunities available. The degree of pleasure will depend on the individual tolerance of risk. An investor may be classified as Risk-Avoider Risk-Neutral Risk-Taker LO3

Measurement of Utility: Standard Gamble Method A person has the chance to enter a contest with a 50-50 chance of winning $100,000 If the person wins the contest, he or she wins $100,000. If the person loses, he or she receives $0. Cost of entering the game is zero dollars. The Expected value of the game is : ($100,000)*(.5)+($0)*(.5) = $50,000. But the person betting will not get this unless he or she continues to bet indefinitely on the game. Would a person take an offer of $30,000 for certain, in the condition that he or she drops out of the game. The answer to this depends on the person’s assets and whether the person is risk neutral, a risk avoider, or a risk taker. LO3

Utility Curves for Three Types of Game Players The straight line is where the expected value of the game is equal the payment offered to drop out of the game (Risk Neutral) rather than continue the gamble For the risk avoider the expectation of winning must be higher than the long run probability that makes EMV = the equivalent certainty value: the utility curve is above the Risk Neutral line The risk taker will bet on the gamble even if the chances of winning is below that required to make EMV = to the equivalent certainty value . The utility curve is below the Risk Neutral line Chance of Winning the Contest Monetary Payoff Risk-Avoider Risk Neutral Risk-Taker LO3

Risk Neutral Game Player in a Standard Gamble Game: Indifferent to Owning “a” or “b” The game player decides to take the $50,000 and not continue gambling. The amount is equal to the expected value of the game at probability of winning =0.5 a b $100 000 - $0 .5 $50 000 LO3

Risk Avoider in a Standard Gamble Game: Indifferent to Owning “a” or “b” Game player decides to take the $20,000 for certain, rather than continue to play, even though the expected value of the game is much higher ($50,000) a b $100 000 - $0 .5 $20 000 LO3

Risk Taker in a Standard Gamble Game: Indifferent to Owning “a” or “b” Game player decides not to take the offer of $70,000 to leave the game, despite the fact that the expected value of the gamble is much less ($50,000). a b $100 000 - $0 .5 $70 000 LO3

Risk Curves For Three Game Players LO3

Revising Probabilities in Light of Sample Information Bayes’ Rule Expected Value of Sample Information LO4

Interpretation X represents the gamble responses of a risk-avoider X makes decision based on a utility the segment of parabolic function above the risk-neutral line Y represents the gamble responses of a risk-taker Y makes decisions based on an exponential utility function below the risk –neutral. LO4

Interpretation Let Z represent the responses of a risk-neutral game player Z is indifferent between a certain guarantee amount, and gambling or not gambling. He remains on the EMV line The gamble is $10,000. Probability of winning is p= 0.5, EMV = $50,000. The risk curve shows that for a guarantee of $50,000 to drop gambling in the game , the risk avoider (X) will only gamble if the probability of winning is p=0.8. On the other hand the risk-taker will gamble even if the guarantee is just under$80,000, approximately $30,000 more than the EMV at p= 0.5. LO4

Decision Table for Investment Problem No Growth (.65) Rapid (.35) Bonds 500 $ 100 Stocks (200) 1,100 LO4

Expected Monetary Value Criterion for the Investment Example No Growth Rapid Expected Monetary Value 0.65 0.35 Bonds 500 $ 100 360.00 Stocks (200) 1,100 255.00 LO4

Revising Probabilities in the Light of Sample Information In this section we address the revision of prior probabilities using Bayes’ rule with sampling information in the context of the $10,000 case discussed above. The probabilities of the various states of nature are frequently not fixed or known in an exact way. Thus prior subjective probabilities (or probabilities based on our best guess) may be used initially to obtain the EMV. These probabilities can be updated by introducing information obtained from samples. The updated probabilities can be incorporated into the decision process to hopefully help make better decisions. LO4

Simplified Version of the $10,000 Investment Decision Problem: Table 19.6 LO4

Decision Tree for the Investment Example: Figure 19.5 Stocks Bonds No Growth (.65) Rapid Growth (.35) $500 $100 -$200 $1,100 EMV=$360 EMV=$255 ($360) LO4

The Success and Failure Rates of the Forecaster in Forecasting the Two States of the Economy Actual State of Economy No Growth (s 1 ) Rapid Growth 2 Forecaster Predicts No Growth (F ) .80 .30 Rapid Growth (F2 ) .20 .70 P(Fi|sj) LO4

Bayes’ Rule LO4

Revision Based on a Forecast of No Growth (F1) State of Economy Prior Probabilities Conditional Joint Revised No Growth (s 1 ) P(s ) = .65 P(F | s ) = .80  s ) = .520 .520/.625 = .832 Rapid 2 ) =.35 ) = .30 ) = .105 .105/.625 = .168 P(F1) = .625 P(sj|F1) LO4

Revision Based on a Forecast of Rapid Growth (F2) State of Economy Prior Probabilities Conditional Joint Revised No Growth (s 1 ) P(s ) = .65 P(F 2 | s ) = .20  s ) = .130 .130/.375 = .347 Rapid ) =.35 ) = .70 ) = .245 .245/.375 = .653 P(F2) = .375 P(sj|F2) LO4

Decision Tree for the Investment Example After Revision of Probabilities: Figure 19.6 Stocks Bonds No Growth (.832) Rapid Growth (.168) $500 $100 -$200 $1,100 $432.80 $18.40 No Growth (.347) Rapid Growth (.653) $238.80 $648.90 Forecast No Growth (.625) Rapid Growth (.375) $513.84 Buy LO4

Expected Value of Sample Information for the Investment Example In general, the expected value of sample information = expected monetary value with information - expected monetary value without information = $513.84 - $360 = $153.84 But what if the decision maker had to pay $100 for the forecaster’s prediction? This would reduce the value of getting perfect information from $513.84 shown in Figure 19.6 in the previous slide to $413.84. Note that this is still superior to the $360 without sample information LO4

Decision Tree Investment Example All Options Included Figure 19.7 is constructed by combining Figures 19.5 and 19.6. This is the Investment Tree for the investment information with the options of buying the information or not buying the information included . It includes a cost of buying Information ($100) and the EMV with this purchased information ($413.84) LO4

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