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BU.520.601 Decision Models Decision Analysis DecisionAnalysis Summer 2013 BU.520.601 1 Let us flip a fair coin once (there is a fee). If you win I give you $102. If I win, you give me $100. How much fee will you pay me for playing the game: $5, $2, $1, $0? You can select any other amount. Let us flip a fair coin 1000 (there is a fee). If you win a toss, I give you $102. If I win, you give me $100. How much fee will you pay for the playing the entire game? Suppose you are getting ready to go the office in a crowded metro. Carrying an umbrella is a hassle; you will carry it only when you feel necessary. Forecast for today is 70% chance of rain and the sky is overcast. Should you carry an umbrella - Yes or no? My decision would be “Yes” and it is a good decision. However, there are two possible outcomes - it will rain or not. If it does not rain, it does not mean I have made a bad decision. DecisionAnalysis BU.520.601 2 Decision Analysis (DA) • • • DA is a methodology applicable to analyze a wide variety of problems. Although DA was used in the 1950s (at Du Pont) and early 1960s (at Pillsbury), major DA development took place in mid sixties. One of the earliest application (at GE) was to analyze whether a super heater should be added to the current power reactor. DA has been considered as a technology to assist (individuals and) organizations in decision making by quantifying the considerations (even though they may be subjective) to deduce logical actions. DecisionAnalysis BU.520.601 3 Decision Analysis (DA) One can discuss many topics listed below; we will look at a few. • Problem Formulation. • Decision Making with / without Probabilities. • Risk Analysis and Sensitivity Analysis. • Decision Analysis with Sample / Perfect Information. • Multistage decision making. Tools and terminology • Basic statistics and probability • Influence diagram / payoff table / decision tree • EMV: Expected Monetary Value • EVSI / EVPI : Expected Value of Sample / Perfect Information DecisionAnalysis BU.520.601 • • • • • Bayes’ rule Decision vs. outcome Risk management Minimax / maximin / Utility theory 4 Decision analysis without probabilities Concepts covered: Payoff table. Different approaches: Maximax, maximin, minimax regret Example: There are four projects; I can select only one. The payoff table shows potential “payoff” depending upon likely economic conditions. Alternatives Economic Condition Recession Normal Boom Project A 4075 5000 6100 Project B 0 5250 12080 Project C 2500 7000 10375 Project D 1500 6000 9500 DecisionAnalysis BU.520.601 Since the payoff in project C is higher than the payoff for D for every economic condition, we say that project C is dominant. We can eliminate project D from consideration. 5 Maximax If you are an optimist, you will decide on the basis of Maximax. Alternatives Economic Condition Recession Normal Boom Project A 4075 Project B 0 5250 12080 Project C 2500 7000 10375 5000 Step 1: Pick the max value for each alternative. 6100 6100 12080 10375 Step 2:Then pick the alternative with max payoff. DecisionAnalysis BU.520.601 6 Maximin If you are a conservative you will use Maximin. Alternatives Economic Condition Recession Normal Boom 1: Pick the min value for each alternative. Project A 4075 5000 6100 4075 Project B 0 5250 12080 0 Project C 2500 7000 10375 2500 2: Then pick the alternative with max payoff. DecisionAnalysis BU.520.601 7 Minimax Regret You are neither optimist nor conservative. Alternatives Economic Condition Recession Normal Boom Project A 4075 5000 6100 Project B 0 5250 12080 Project C 2500 7000 10375 4075| 7000| 12080 Alternatives Regret Table Recession Normal Boom Project A 0 2000 5980 Project B 4075 1750 0 Project C 1575 0 1705 Step 1: Calculate the maximum for each outcome. Stet 2: Prepare “Regret Table” by subtracting each outcome cell value from its maximum. At least one number for each regret table outcome is zero and there are no negative numbers. Why? DecisionAnalysis BU.520.601 8 Minimax Regret.. Alternatives Economic Condition Recession Normal Boom Project A 4075 5000 6100 Project B 0 5250 12080 Project C 2500 7000 10375 4075| 7000| 12080 Alternatives Regret Table Recession Normal Step 3: Pick the max value Boom for each alternative. Project A 0 2000 5980 5980 Project B 4075 1750 0 4075 Project C 1575 0 1705 1705 Step 4: Pick the alternative with minimum regret. DecisionAnalysis BU.520.601 9 General comments Payoff table Alternatives Table columns show outcomes (also called state of nature). Economic Condition Recession Normal Boom Project A 4075 Project B 0 5250 12080 Project C 2500 7000 10375 5000 6100 • The maximax payoff criterion seeks the largest of the maximum payoffs among the actions. • The maximin payoff criterion seeks the largest of the minimum payoffs among the actions. • The minimax regret criterion seeks the smallest of the maximum regrets among the actions. The above three approaches we used involved Decision Making without Probabilities. DecisionAnalysis BU.520.601 10 Decision analysis with probabilities Typically, we use a tree diagram for the decision analysis. 1. A decision point is shown by a rectangle 2. Alternatives available at a decision point DB are shown as decision branches (DB). 3. At the end of each DB, there CB can be two or more chance 20% events shown by a node and 55% chance branches (CB). Decision 25% Chance events must be mutually point exclusive and exhaustive (total probability = 1). 4. At the end of each branch is an endpoint shown as a triangle where a payoff will be identified. DecisionAnalysis BU.520.601 11 Decision analysis with probabilities Decision point: Chance event : End point: DB: Decision Branch CB: Chance Branch At the chance node, we calculate the average (i.e. expected) payoff. The terminology used is DB Expected Monetary Value (EMV) If there is no chance event for a CB particular decision branch, it’s 20% EMV is equal to the payoff. 55% 25% We select the decision with the highest EMV . What if we are dealing with costs? DecisionAnalysis BU.520.601 12 A larger tree diagram DecisionAnalysis BU.520.601 13 Example 1 You bought 500 units of X @$10 each. A dealer has offered to buy these from you @$14 each ( you can make $4/unit profit). You can sell these yourself for $16 Demand: X 300 400 500 600 each ($6/unit profit) but the 0.30 0.45 0.20 0.05 demand is uncertain. The demand Pr(X) distribution is shown in the table. Obviously, if demand exceeds 500, you will sell all 500. On the other hand, if demand is under 500, you will have leftover units. These leftover items can disposed off for $7 each ($3 loss, the dealer will no longer buy these leftover units from you). What’s your decision? DecisionAnalysis BU.520.601 14 Example 1 .. Suppose you have 500 units of X in Demand: X 300 400 500 600 stock, purchased for $10 each. Dealer Pr(X) 0.30 0.45 0.20 0.05 sales price:$14, self sale price:$16 with salvage value:$7. Start with the tree having 2 branches (DB) at the decision point. There are no chance events in the dealer sale branch, For the self sale, there are 4 mutually exclusive possibilities. Dealer Sale Self sale DecisionAnalysis 500, 20% BU.520.601 15 Example 1 ... Suppose you have 500 units of X in Demand: X 300 400 500 600 stock, purchased for $10 each. Dealer Pr(X) 0.30 0.45 0.20 0.05 sales price:$14, self sale price:$16 with salvage value:$7. EMV = 2000 Payoff = 500*4 = 2000 Dealer Sale Payoff = 300*6 – 200*3 = 1200 Payoff = 400*6 – 100*3 = 2100 Self sale 500, 20% Payoff = 500*6 = 3000 Payoff = 500*6 = 3000 EMV = 0.3*1200 + 0.45*2100 + 0.2* 3000 + 0.05*3000 = 2055 DecisionAnalysis Your decision? BU.520.601 16 Risk profile is the probability distribution for the payoff associated with a particular action. Risk Profile 30% 300 Self Sale Payoff = 1200 400 45% Payoff = 2100 20% 500 5% 600 Payoff = 3000 Payoff = 3000 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. DecisionAnalysis BU.520.601 17 Example 3 We have received RFP (Request For Proposal). • We may not want to bid at all (our cost: 0) • If we bid, we will have to spend $5k for proposal preparation. Based on the information provided in the RFP, a quick decision is to bid either $115k or $120k or $125k. We must select among 4 alternatives (including no bid). • A quick estimate of the cost of the project (in addition to the preparation cost) is $95k. • Looks like we may have a competitor. • If we bid the same amount as the competitor, we will get the project because of our reputation with the client. • We have gathered some probabilities based on past experience. DecisionAnalysis BU.520.601 18 Example 3.. All numbers in thousand dollars Our bid (OB) must be 0 (no bid), 115, 120 or 125. Competitor’s bid (CB): 0, under 115, 115 to under 120, 120 to under 125, 125 and over. Assumption: If bids are equal, we get the contract. Information : Preparation cost: $5 + Cost of work : $95 = $100 total Profit for our bid Competitor’s bid 1. No bid 2a. Under $115 2b. $115 to under $120 2c. $120 to under $125 2d. Over $125 DecisionAnalysis 0 115 120 125 0 15 20 25 0 -5 -5 -5 0 15 -5 -5 0 15 20 -5 0 15 20 25 BU.520.601 Use mini-max, maxi-max, etc? There are probabilities involved. 19 Example 3… 1. There is a 30% probability that the competitor will not bid. 2. If the competitor does bid, there is (a) 20% probability of bid under $115. (b) 40% probability of bid $115 to under $120. (c) 30% probability of bid under $120 to under $125. (d) 10% probability of bid over $125. Actual Prob. Prob. Prob. Competitor’s bid 30% 30% - 1. No bid 14% 20% 2a. Under $115 28% 40% 2b. $115 to under $120 70% 21% 30% 2c. $120 to under $125 7% 10% 2d. Over $125 DecisionAnalysis BU.520.601 Profit for our bid 0 115 120 125 0 15 20 25 0 -5 -5 -5 0 15 -5 -5 0 15 20 -5 0 15 20 25 20 Example 3: Actual Prob. Competitor No bid $115 0 115 120 125 30% 1. No bid 14% 2a. < $115 0 15 20 25 0 -5 -5 -5 28% 2b. $115 to < $120 21% 2c. $120 to < $125 0 15 -5 -5 0 15 20 -5 0 15 20 25 7% 2d. > $125 $0 Profit for our bid Lose Payoff = (-5), Probability 14% Win Payoff = 15, Probability 86% bid (-5)*(0.14) + 15 * (0.86) = $12.2 DecisionAnalysis BU.520.601 21 Example 3: No bid $12.2 Bid $115 0 15 20 25 0 -5 -5 -5 28% 2b. $115 to < $120 21% 2c. $120 to < $125 0 15 -5 -5 0 15 20 -5 0 15 20 25 L -5, 14% W 15, 86% Our decision? $9.5 Bid $120 Bid= $125 $6.1 DecisionAnalysis 0 115 120 125 30% 1. No bid 14% 2a. < $115 7% 2d. > $125 $0 Profit for our bid Actual Prob. Competitor L -5, 42% W 20, 58% L W -5, 63% We will now use Excel to solve the problem. 25, 37% BU.520.601 22 Ex. 3: Excel =SUMPRODUCT(Profit_bid_115,Probabilities) =MAX(D9:G9) INDEX+MATCH HLOOKUP ? Value we are looking (12.2) is not in the ascending order in the table. DecisionAnalysis BU.520.601 23 Example 3: Sensitivity analysis What if 30% probability of no bid from competitor is incorrect? We can build a one variable data table. Variable: Competitor’s no bid probability. We select two outputs: bid and (corresponding maximum) profit. DecisionAnalysis BU.520.601 24 Ex. 3: DA and value of information Our decision was to bid $115 and EMV was $12.2. Suppose we get competitor’s bid information. Can we improve our profit? Profit for our bid Competitor’s bid 0 115 120 125 1. No bid 0 15 20 25 2a. Under $115 0 -5 -5 -5 2b. $115 to under $120 0 15 -5 -5 2c. $120 to under $125 0 15 20 -5 2d. Over $125 0 15 20 25 What is the probability? 0.30 0.7 * 0.2 = 0.14 0.7 * 0.4 = 0.28 0.7 * 0.3 = 0.21 0.7 * 0.1 = 0.07 EMV = 0.3*25+0.14*0+0.28*15+0.21*20+0.07*25 = 17.65 Earned Value of Perfect Information (EVPI) = $17.65 – $12.2 = $5.45 Sometimes we may have partial information. DecisionAnalysis BU.520.601 25 Example 3: Alternate method CB=0 15(.3)+11(.7) = $12.2 $0 No bid CB OB= $115 OB= $120 70% $9.5 OB= $125 $20 $5 -5(.2)+15(.4+.3+.1) = $11 30% $6.1 DecisionAnalysis 30% 20% <115 -$5 115 to <120 $15 40% 30% 120 to < 125 $15 10% >125 $15 70% bid Our $12.2 decision 30% $15 70% $25 -$2 EMV BU.520.601 Payoff 26 Example 3….. $12.2 $0 No bid OB= $115 This line indicates the decision made. $9.5 OB= $120 bid Values 12.2, 9.5 and 6.1 represent Expected Monetary Values (EMV). This is called folding back the decision tree. $12.2 OB= $125 $6.1 DecisionAnalysis BU.520.601 27 Utility theory Consider the gambling problems again. – Let us flip a fair coin once. – If you win I give you $102 – If I win, you give me $100 – How much will you pay me to play this game: $5, $2, $1, $0 ? Consider another gamble – Let us flip the same coin (500 times) with the same payoffs – How much will you pay me to play this game? DecisionAnalysis • • • • • • Different people will pay different amounts to play the first game Expected payoff in the first game is $1 but most people do not want to play the game at all. Why? Losing $100 is a bigger event than winning $102 Most people will play the second game. Still differ in how much they will pay. For most people a gain that is twice as big is not twice as good. A loss of twice as much is more than twice as bad. People’s attitude towards risk can be categorized as: risk averse, risk seeker and risk neutral. A common way to express it is through the decision-maker’s utility function. BU.520.601 28 Utility is a measure of relative satisfaction. We can plot a graph of amount of money spent vs. “utility” on a 0 to 100 scale. Typical shapes for different types of risk takers generally follow the patterns shown below. U(100) U(100) U(0) U(0) 0 100 Risk seeker U(100) 0 100 Risk averse U(0) 0 100 Risk neutral U(100) U(100) U(100) U(0) U(0) U(0) 0 100 0 100 0 100 Graphs above show that to achieve 50% utility, risk seekers will pay maximum, risk averse will pay minimum and risk neutral will pay an average amount. DecisionAnalysis BU.520.601 29