Decision Theory Chapter 1 PDF

INTRODUCTION - DECISION THEORY Komal Nadeem INITIAL THOUGHTS….  Examples of decision analysis at your workplace? DECISION THEORY  Decision theory is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factor...

INTRODUCTION - DECISION THEORY Komal Nadeem INITIAL THOUGHTS….  Examples of decision analysis at your workplace? DECISION THEORY  Decision theory is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical consequences to the outcome INTRODUCTION  Decision theory problems characterized by: 1. A list of alternatives 2. A list of possible future states of nature 3. Payoffs associated with each alternative/state of nature combination 4. An assessment of the degree of certainty of possible future events 5. A decision criterion LIST OF ALTERNATIVES  Set of mutually exclusive and collectively exhaustive decisions available  Sometimes ‘do nothing’ alternative too  E.g. real estate developer needs to decide on a future plan: 1. Residential proposal 2. Commercial proposal # 1 3. Commercial proposal # 2 STATES OF NATURE  Set of possible future conditions (or events) beyond the control of the decision maker that will be the primary determinants of the eventual consequence of the decision.  Mutually exclusive and collectively exhaustive  Main factor: whether or not shopping center is built and its size 1. No shopping center 2. Medium-sized shopping center 3. Large shopping center PAYOFFS  Estimated profits, revenues, costs, or other measures of value  Mostly financial measures  Weekly, monthly, annual amounts or present values of future cash flows  Number of payoffs depends on the number of alternative/states of nature combinations  In our example, three alternatives and three states of nature, so there are 3 x 3 possible payoffs  DEGREE OF CERTAINTY  Certainty ----------- Risk ----------- Uncertainty  Probability estimates for the various states of matter serve an important function if they can be obtained DECISION CRITERION  Embodies the decision maker’s attitudes towards the decision as well as the degree of certainty  E.g. Optimistic/Pessimistic/Maximize gains/Avoiding losses THE PAYOFF TABLE / DECISION MATRIX  Summarizes and organizes information – list of alternatives, states of nature, payoffs (also probabilities if available) States of nature S1 S2 S3 Alternative A1 V11 V12 V13 A2 V21 V22 V23 A3 V31 V32 V33 THE PAYOFF TABLE FOR REAL ESTATE EXAMPLE States of nature No center Medium Large Center Center Alternative Residential 400 1600 1200 Profit Commercial 600 500 1400 ($000) 1 Commercial -100 400 1500 2 DECISION MAKING UNDER CERTAINTY  Simplest scenario  Decision maker selects the alternative with best payoff in that state of nature  E.g. there is an announcement that no shopping center will be built  Developer can focus on the first column of the payoff table  Because the Commercial # 1 proposal has highest payoff in that column ($600,000), it would be selected DECISION MAKING UNDER COMPLETE UNCERTAINTY  Unable to estimate probabilities  Lacks confidence in available estimates of probabilities  Five approaches: 1. Maximin (Wald) 2. Maximax (Plunger) 3. Minimax Regret 4. Realism (Hurwicz) 5. Equal Likelihood (Laplace) 1. MAXIMIN  Conservative Approach  Setting a floor for the potential payoff – cannot be less than this  Identify the worst payoff for each alternative and then select the one with best of the worst payoffs  Considered pessimistic – protecting against the worst States of nature Worst Payoff No center Medium Large Center Center Alternative Residential 400 1600 1200 400 Profit Commercial 1 600 500 1400 500 MAX ($000) Commercial 2 -100 400 1500 -100 2. MAXIMAX  Optimistic Approach  Identify the best payoff for each alternative and then select the one with best of the best payoffs States of nature Best Payoff No center Medium Large Center Center Alternative Residential 400 1600 1200 1600 MAX Profit Commercial 1 600 500 1400 1400 ($000) Commercial 2 -100 400 1500 1500 3. MINIMAX REGRET  Maximin and Maximax criticized for being on the extremes  Develop an opportunity loss table – difference between each payoff and the best payoff in a column Opportunity Loss Table States of nature No center Medium Large Center Center Alternative Residential 200 0 300 Profit Commercial 1 0 1100 100 ($000) Commercial 2 700 1200 0 3. MINIMAX REGRET  Values in an opportunity loss table are potential regrets  Selecting an alternative to minimize the maximum regret  Identify the maximum OL in each row and then choose minimum Opportunity Loss Table States of nature Maximum OL No Medium Large center Center Center Alternative Residential 200 0 300 300 MIN Profit Commercial 1 0 1100 100 1100 ($000) Commercial 2 700 1200 0 1200  4. REALISM  Weighted average - compromise between maximax and maximin  Specify a degree of optimism – coefficient of optimism α (0 to 1)  Best payoff multiplied by α and worst multiplied by 1-α, results are added  Totals for all alternatives are compared and best is selected  For α = 0.3 Alternative Best Payoff Worst Payoff Residential 0.3*1600 0.7*400 =760 MAX Commercial 1 0.3*1400 0.7*500 =770 Commercial 2 0.3*1500 0.7*-100 =380 5. EQUAL LIKELIHOOD  Treats the states of nature as if each were equally likely and focuses on average payoff for each row – selecting the alternative with highest row average States of nature Average Payoff No center Medium Large Center Center Alternative Residential 400/3 1600/3 1200/3 =1067 MAX Profit Commercia 600/3 500/3 1400/3 =833 ($000) l1 Commercia -100/3 400/3 1500/3 =600 l2 USING EXCEL  Class 1 PRACTICE PROBLEM PRACTICE PROBLEM PRACTICE PROBLEM DECISION MAKING UNDER RISK  Partial uncertainty  When probabilities are given/can be estimated  Estimates by experienced managers or historical frequencies  The sum of probabilities for all states of nature must be 1.00  E.g. probability of no shopping center: 0.2, medium-sized shopping center: 0.5, large shopping center: 0.3 EXPECTED MONETARY VALUE (EMV)  Average payoff for each alternative  Best alternative with highest EMV EXPECTED MONETARY VALUE (EMV) States of nature No center Medium Large Center Center Alternative Residential 400 1600 1200 Profit Commercial 600 500 1400 ($000) 1 Commercial -100 400 1500 2  Residential EMV = 0.2(400)+0.5(1600)+0.3(1200) = $1240  Commercial 1 EMV = 0.2(600)+0.5(500)+0.3(1400) = $790  Commercial 2 EMV = 0.2(-100)+0.5(400)+0.3(1500) = $630 EXPECTED OPPORTUNITY LOSS (EOL)  Similar to EMV except that…  Table of Opportunity Loss is used  Alternative with smallest expected loss is chosen Opportunity Loss States of nature EOL Table No Medium Large center Center Center Alternativ Residential 200 0 300 =.2*200+.5*0+.3*300 = 130 e Commercial 0 1100 100 =.2*0+.5*1100+.3*100 = 580 Profit 1 ($000) Commercial 700 1200 0 =.2*700+.5*1200+.3*0 = 740 2 EXPECTED VALUE OF PERFECT INFORMATION  Waiting to move the decision into realm of certainty (but higher prices, cost of consultant, storage cost etc.)  The expected value of perfect information (EVPI) is the expected value obtained with perfect information minus the expected value obtained without perfect information (i.e. maximum EMV) – cost of going from risk to certainty  Removing uncertainties - If the developer knew that no center will be built, commercial 1 will be chosen $600; if medium sized center will be built, residential will be chosen $1600; if large center will be built, commercial 2 will be chosen $ 1500 EXPECTED VALUE OF PERFECT INFORMATION  We use original state of nature probabilities to weigh best payoffs under certainty – Expected Payoff under Certainty (EPC)  EPC = 0.2*600+.5*1600+0.3*1500 = $1370  EVPI = EPC – EMV = 1370 – 1240 = $130  EVPI is the upper bound on the amount of money the real estate developer would be justified in spending to obtain perfect information USING EXCEL  Class 1  Class 2  F  F  f. EVPI? DECISION TREES  Sequential decisions  Event node (uncertain): Circle  Decision node: Square  Rolling back DECISION TREES  Additional alternatives if no center: 1. Do nothing 2. Small shopping center 3. Park DECISION MAKING WITH ADDITIONAL (SAMPLE) INFORMATION (AKA EVSI)  Improve decision making with additional information  E.g. market survey, forecasting, delay a decision for clearer picture  Result: probability estimates get more accurate  Additional (sample) info incurs extra cost  Is it worth it???  E.g. Advertising manager – two proposals Market 70 30 Strong Weak Alternatives Print Media 40 20 ($000) Video Media 50 10  DECISION MAKING WITH ADDITIONAL INFORMATION  Option of testing the market (additional info) – revised probabilities for market strength  Testing the market would require $1000 Strong Market Weak Market 0.95 0.05 0.34 0.66 Strong Weak Strong Weak Alternatives Print 40 20 Alternatives Print 40 20 ($000) Media ($000) Media Video 50 10 Video 50 10 Media Media  Suppose the manager is able to determine the probability that market test will show a strong market is 0.59 and a weak market is 0.41 respectively COMBINED PROBABILITY AT NODE 2 = 39.308 DECISION MAKING WITH ADDITIONAL INFORMATION  Difference between the two options = 39308 - 38000 = $1308  Incurring cost = 1308 – 1000 = $308  Thus,  Expected value of sample information = Expected value with sample information – Expected value without sample information EVPI AND EVSI  The expected value of perfect information, or EVPI, is a theoretical number that says how much a business should pay to know with certainty the outcome of a decision. On the other hand, the expected value of sample information, or EVSI, is a real number that says how much actual information is worth. PRACTICE PROBLEM Assume that after learning about the reliability of surveys, manager estimates probability of low demand to be 0.35 PRACTICE PROBLEMS  Mike Dirr, vice-president of marketing for Super-Cola, is considering which of two advertising plans to use for a new caffeine-free cola. Mike estimates that the probability of complete advertising is 0.65. Payoff ($000) Limited Complete Advertising Advertising Plan I 800 1800 Plan II 1100 1200  Perform decision making under uncertainty and risk.  If he spends $50,000 in a market survey, to predict complete acceptance in 60% of the cases where there was acceptance and predicted limited acceptance 70% of the time when there was limited acceptance. Mike now feels that the probability of complete acceptance is 55%. Use this information to determine whether this survey should be used to help decide on an advertising plan. EFFICIENCY OF SAMPLE INFORMATION  Ratio of EVSI to EVPI  Efficiency of Sample Information = EVSI / EVPI  In the previous example, EPC = 0.70(50)+0.30(20) = $41,000 and EMV = $38,000, so EVPI = 41,000 – 38,000 = $3000  EVSI was $1308, so Efficiency = 1308 / 3000 = 0.436  This number ranges from 0 to 1, where being closer to 1 means sample information is close to perfect. COMPUTING THE PROBABILITIES  Suppose the manager takes out historical data from past records to get reliability information. It shows: Actual State of Nature Results of market test Strong Market Weak Market Shows strong market 0.80 0.10 Shows weak market 0.20 0.90  Which means that when market was actually strong, the market test correctly indicated this information 80 percent of the time and incorrectly indicated weak market 20 percent of the time.  These are conditional probabilities – showing reliability of sampling device PROBABILITY CALCULATIONS  Given that market test shows Strong market Actual Conditional Prior Joint Revised Market Probabilities Probabilities Probabilities Probabilities Strong 0.8 0.7 =0.8*0.7 = 0.56 =0.56/0.59 = 0.95 Weak 0.1 0.3 =0.1*0.3 = 0.03 =0.03/0.59 = 0.05 Marginal Probability of a strong market =0.56+0.03 = 0.59  Given that market test shows Weak market Actual Conditional Prior Joint Revised Market Probabilities Probabilities Probabilities Probabilities Strong 0.2 0.7 =0.2*0.7 = 0.14 =0.14/0.41 = 0.34 Weak 0.9 0.3 =0.9*0.3 = 0.27 =0.27/0.41 = 0.66 Marginal Probability of a strong market =0.14+0.27 = 0.41 SENSITIVITY ANALYSIS  Decision making under risk requires working with estimated values.  So decision maker must know the sensitivity of a decision to possible errors in estimation A GOOD READ  https://vpostrel.com/articles/operation-everything A GOOD READ  https://news.mit.edu/2023/automating-math-decision-making- under-uncertainty-0206 FURTHER DECISION TREE PRACTICE  http://people.brunel.ac.uk/~mastjjb/jeb/or/decmore.html THANK YOU