Decision under uncertainty with limited information

Transcription

1 Decision under uncertainty with limited information Efstratios Nikolaidis The University of Toledo December

2 Outline 1. Why decision-making is important 2. Decision analysis: a powerful method to make good decisions 3. Solving decision problems 4. Quantifying the value of information 5. How to define and estimate probabilities of one-time events 6. Lessons learnt 2

3 1. Why decision-making is important We all make decisions that affect our personal and professional life. The welfare of companies and countries depends on the ability of their leaders to make good decisions. 3

4 Example Drill for oil You own land in oil rich region Can sell it for $18,500 or drill Drilling costs $70,000 If no oil is found land becomes worthless Sell or drill? 4

5 Decision tree: Graphical representation of sequence of choices and uncertain events Shows profit for each combination of choices and outcomes of uncertain events. Uncertain event Decision Dril for Oil Decision Drill ($70,000) Dry Wet Soak $120, $270, Sell $18,

6 Both your decision and the outcomes of uncertain event affect profit Amount of oil Outcomes No oil Wet Soaking Drill or sell? Consequence Decision: Drill or sell? Drill outcome Consequence (profit) Drill No oil -$70,000 Drill Wet $50,000 Drill Soaking $200,000 Sell -- $18,500 6

7 In order to make a good decision you must consider: Alternatives and their consequences Probabilities of outcomes of uncertain events Your risk attitude (value of a sure amount of $18,500 vs. opportunity of a windfall and risk of a big loss) 7

8 Decisions in Product Development Decision Consequence Cost Performance Enterprise and engineering design variables Demand Profit Objective: maximize profit (price, cost, performance) Decision variables: Enterprise: price, volume Engineering: configuration, material, dimensions Our decisions affect both performance and cost 8 8

9 2. Decision analysis: a powerful method to make good decisions An approach that is based on common sense rules to make decisions in challenging situations. A good decision is based on careful consideration of alternatives, uncertainties and preferences. A good decision does not guarantee a good outcome, but it increases its likelihood, in the long run. 9

10 Elements of a decision Values and objectives Alternative courses of action Model for predicting consequences Uncertainty Probabilities of outcomes of uncertain events Decision criterion 10

11 Decision process Define and frame decision: understand the decision situation, examine values and preferences Identify options Structure decision Model preferences Model uncertainty Develop predictive model for the consequences of each alternative course of action Choose best course of action Assess sensitivity of decision to the assumptions Appraise decision quality, assess value of additional information Implement decision 11

12 Reducing risk by considering more alternatives No structure Drill ($70,000) Dry W et Soak $120, $270, Test ($10,000) Open structure Sell Drill $18, Dry ($70,000) W et $120, Soak $270, Closed structure Sell Drill $18, Dry ($70,000) W et $120, Soak $270, Test, drill or sell Sell $18, Drill Sell Dry ($70,000) W et $120, Soak $270, $18,

13 3. Solving decision problems Decision rule: The best alternative is the one with the highest expected monetary value Example, product development: Major household product manufacturer plans to produce innovative squeegee design Commissioned vendor to analyze strength under heavy use. Vendor reported that design is safe. Company s engineers reviewed report and are concerned that vendor made an erroneous assumption. This has probability Development and marketing cost: $50 million If vendor s assumption is correct the company will make $100 million revenue. Otherwise, revenue will be only $10 million. Can make and test 10 prototypes. Develop, test or cancel development? 13

14 Quantifying accuracy of test P(pass test/valid assumptions)=0.98 P(pass test/wrong assumptions)=0.1 Cost of test: $100,000 Total probability theorem: P(pass test)=0.76 Bayes rule: P(valid assumptions/pass test)= P(valid assumptions/fail test)=

15 Decision tree 15

16 Backward induction, first step: replace gambles with their expected profit Expected profit 16

17 Backward induction, second step: make decisions given what you know from the test 17

18 Backward induction, third step: make final decision Optimum strategy: First test product. Develop and market it if it passes test, otherwise cancel development. 18

19 4. Quantifying the value of information Principle: value of information from test = increase in expected profit from using the information from the test Value of test=$35.75-$27.5=$8.25 million Net value of test=value of test-cost=$8.15 million Before conducting a test estimate net value. Do not conduct a test with net negative value. 19

20 Value of test with perfect information Perfect information: test would always predict correctly if assumptions were valid Value of test=$37.5-$27.5=$10 million 20

21 5. How to define and estimate probabilities of one-time events Objective probability=long term relative frequency Example: P(heads in flip of fair coin)=0.5 Long-term relative frequency Objective measure Most people understand concept of objective probability 21

22 Objective probability cannot quantify uncertainty in most practical problems There is little data Too expensive to collect data We cannot conduct a repeatable experiment for one-ofa-kind events A particular student will pass the Ph.D. qualifying exams in spring 2011 Assumptions in a finite element analysis of a structure are erroneous Demand for Chevy Volt in US will not exceed 100,000 Price of crude oil will exceed $150 per barrel by December

23 Subjective probability can help model uncertainty Principle 1: Probability is a decision maker s (DM s) belief that an outcome will materialize Principle 2: DM avoids a risky venture that will result in sure loss Belief leads to inclination to act. Elicit it by observing how DM makes choices in the face of uncertainty. Observe inclination to accept gambles in controlled experiments 23

24 Objective vs. subjective probability Objective probability: how frequently does an earthquake stronger than 6 on the Richter scale occur in Southern California? Subjective probability: how likely is it that a proposition (demand for Chevy Volt<100,000) is true? 24

25 Estimating subjective probability of a candidate winning 2008 U.S. presidential election by using trading data This ticket is worth $1 only if Mr. Obama wins 2008 presidential election Maximum buying price reflected a gambler s belief that Mr. Obama would win election 25

26 26 Date P(win)=0.5 Ticket Price (cents) 90 P(win)= Trading data from 2008 U.S. presidential election (

27 Eliciting expert s probability Ticket Sure amount Ticket Sure amount Expert prefers ticket to sure amount. Expert prefers sure amount to ticket. $0 Ticket price=$p $1 Subjective probability=p Sure amount DM is decisive to avoid sure loss 27

28 Imprecision in probability elicitation 28

29 Eliciting a decision maker s probability distribution Estimating 5% percentile of water pump life if we cannot perform tests Real life experiment Reference experiment: wheel of fortune CDF 0.05 φ 2000 Life (hrs) Ticket 1: Worth $1 only if life 5% percentile 2000 hrs Ticket 1: Worth $1 only if needle settles in sector φ P(5% percentile 2000 hrs) = φ/

30 Combining judgments and data by using Bayes rule Example Judgment: 1 per 10 pumps fail on average Posterior = likelihood prior scaling constant Data: 1 out of 20 pumps failed Data: 10 out of 200 pumps failed PDF Prior Posterior Likelihood Likelihood PDF Prior Posterior Likelihood Likelihood Failure Probability Failure Probability Subjective probability converges to relative frequency and epistemic uncertainty decreases with amount of data 30

31 6. Lessons learnt Decision analysis is a power tool for making choices under uncertainty. It is based few on common sense rules. How to make good decisions Know what you want Make sure you are working on the right problem; ask your self if you are addressing the right question Consider many fundamentally different and creative alternatives. Do not discount luck: carefully assess the likelihood of uncertain events that affect the consequences of a decision. Compare alternative courses of action on the basis of their consequences not the actions themselves. Understand all important issues in a decision and include them in your model 31

32 In most practical decisions we do not have enough data to estimate relative frequencies. Objective probability is inadequate for modeling uncertainty. Subjective probability enables decision-maker to model uncertainty on the basis of both judgment and data. Subjective probability has a solid theoretical justification derived from first principles. Can combine judgment with data by using Bayes rule. Subjective probability converges to relative frequency with amount of data increasing. Ambiguity aversion leads to indecision. Some people s behavior is at odds with the precepts of subjective probability. 32