Decision Analysis under Uncertainty. Christopher Grigoriou Executive MBA/HEC Lausanne

Transcription

1 Decision Analysis under Uncertainty Christopher Grigoriou Executive MBA/HEC Lausanne

2 Introduction Examples of decision making under uncertainty in the business world; => Trade-off between bidding low to win the bid and bidding high to make a larger profit => Introducing a new product into the market (customers reaction), test market? => Insurance market All the problems have three common elements 1- The set of strategies available to the decision maker 2- The set of possible outcomes and the probabilities of these outcomes 3- A value model that prescribes monetary values for the various decision/outcome combinations 2

3 Case of an Insurance Market Uncertain prospect: during the next year probability p to have an accident Your wealth is w1 at the beginning of the year At the end of the year => wealth still w1 if no accident => wealth now w0<w1 if accident (w1-w0 = cost of repairing your car etc.) 3

4 The expected value of your wealth, w*, is the weighted average of the two possible outcomes: w* = p.w0 + (1-p).w1 Figure 1: Expected Wealth w0 w* w1 wealth The figure suggests a relatively high likelihood of accident (a low probability would place w* very close to w1) 4

5 If indifferent between - w* with certainty - the prospect of w0 with probability p and w1 with probability (1-p) => risk neutral Willing to pay an insurance premium: w1-w* for a policy that paid off w1-w0 in the event of an accident so that you get w* (w1-your premium) whatever happens 5

6 Example: A $ car / probability of a wreck is 1% => w1= ; w0= 0 ; w*= If the insurance company insures a large number of drivers, it can believe that its $ payouts will be made close to one in one hundred of its policy holders and it receives 200$ from each policy holder. Real insurance companies Costs other than claims payouts the premiums they charge exceed the amount they pay out The expected wealth of the consumer who does not insure exceeds the certain wealth of an ensured person Given a choice between an uncertain prospect with expected value of w* and alternatively w* with certainty, consumers prefer the certain alternative Consumers get more utility from the certain prospect 6

7 Expected utility with certain levels of utility 1- The following figure graphs consumer utility for the two points w0 and w1: => u(w0) is the utility you receive if you have suffered a clamity => u(w1) is the utility you receive if you do not have to worry about that clamity. => These u(w0) and u(w1) are the utility of having a certain level of wealth with certainty. 2- We can represent the different levels of utility depending on p and (1-p) Utility u(w1) p.u(w0)+(1-p).u(w1) u(w0) w0 w w* w1 Wealth 7

8 Expected utility and Insurance Premiums utility u(w1) u(w*) p.u(w0)+(1-p).u(w1) u(w0) w0 w w* w1 Wealth You exactly know, i.e. with certainty, the utility you would get with w0 or w1. Between w0 and w1, the expected utility not to be insured is less than the utility associated with w* (the case where we are insured) => the utility function for certain prospects that passes through (w0, u(w0)) and (w1, u(w1)) must lie above the line connecting the two points If the red line represents your utility, how much are you willing to pay for an insurance policy? => the consumer is indifferent between the uncertain prospect with expected value w* and w with certainty => the consumer starting with w1 would be willing to pay up to w1-w to avoid the uncertain prospect. 8

9 What do we know? What have we learnt?? (1) Risk aversion = the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain but possibly lower expected payoff. Choice between a bet of => either receiving 100$ (50%) or nothing (50%) => or receiving some amount with certainty. Risk neutral = indifferent between the bet and a certain 50$ payment Risk averse = accept a payoff of less than 50$ (e.g 40$) with probability 100% Risk loving = Required that the payment be more than 50$ (e.g 60$) to induce him to take the certain option over the bet. 9

10 What do we know? What have we learnt?? (2) The average payoff of the bet is called the «Expected (Monetary) Value» (50$) The certain amount accepted instead of the bet is called the «certainty equivalent», the difference between it and the expected value is called the risk premium 10

11 Definition and example: Imagine a decision maker must choose amongst three decisions (d1, d2 and d3) with three possible outcomes (O1, O2 and O3) Payoff tables = listing of payoffs for all decision-outcome pairs, positive values = gains/ negative = losses Outcome Decision O1 O2 O3 D D D Safe decision = chosing D1 D3 the riskier; greater possible gains and losses Decision makers must make rational decisions based on the information they have when the decisions must be made 11

12 Possible decision criteria The maximin criterion: choose the decision that maximizes the worst payoff (for a pessimistic decision maker) => D1 with payoff 10 Avoid large losses but also fails to consider large rewards => not commonly used The maximax: choose the decision that maximizes the best payoff (optimistic decision maker or risk taker) Focuses on large gains but ignores possibles losses => seldom used Maximin and maximax criteria make no reference to how likely each outcome is (decision makers typically have at least some idea of these likelihoods and ought to use this information in the decision making process). => if outcome O1 is very unlikely, then the maximin users are overly conservative => the same if O3 is quite unlikely, the maximax users take an unnecessary risk Expected Monetary value (EMV) => The EMV approach assesses probabilities for each outcome of each decision and then calculates the expected payoff from each decision For any decision, the EMV is the weighted average of the possible payoffs for this decision, weighted by the probabilities of the outcomes. We choose the decision with the largest EMV. 12

13 Expected Monetary Value The decision maker assesses the probabilities of the three outcomes (O1, O2, O3) as 0.4, 0.4 and 0.2 For each decision: EMV for D1: 10x x x0.2 = 10 EMV for D2: -10x x x0.2 = 12 EMV for D3: -30x0.4+30x0.4+70x0.2 = 14 The optimal decision is then to choose D3 since it has the largest EMV. => We ll never get 14$ (either -30 or +30 or +70) but on average, if running that decision many times we will make a gain of about 14$ 13

14 Sensitivity Analysis Changing slightly the inputs, how are the outputs (EMVs and the best decision) modified? Modify either the outcomes or the probabilities and see how the final decision change 14

15 Decision trees =>Used to analyse complex problems with a sequence of events (decisions and outcomes) 15

16 Methodology Decompose problem into chronological sequence of decisions and events Decisions: - You decide Events: - Others decide Determine all possible scenarios (sequences of decisions and events) For each scenario: Outcome? Likelihood? 16

17 Decision nodes and Chance nodes Decision node: You choose which way to go Chance node: Chance decides which way you go Decision 1 Event 1 Probability 1 Decision 2 Event 2 Probability 2 Decision 3 Event 3 Probability 3 Probabilities sum to 1 17

18 Example 0.5 Failure Marketing effort 0.5 Success 0.75 Failure... No Marketing effort 0.25 Success Posterior probabilities = Conditional probabilities: Depend on decisions and events preceding this event 18

19 Using Precision Tree or Treeplan Summarize the data (costs, revenues, probabilities) Structure the tree: Chronological sequence of decisions and events Insert the costs, revenues and probabilities for each branch Indicate whether you are minimizing or maximizing EMV Interpret the solution: - Expected Monetary Value - Risk Profile 19

20 The value of information How much is the information worth? Should we purchase it? => the answers to these questions are embedded in the decision tree itself Expected Value of the Perfect Information => the most we would be willing to pay for the sample information Price of Information = EMV with perfect information - EMV without information 20

21 Example 1 Assume you have to ship a gift, but there is a probability 0.4 that the shipment fails. In this case you have a loss of 80. Your total wealth is 100$. You have the opportunity to insure the gift and the insurance premium costs 30. How much are you willing to pay in order to know what is going on before choosing? 21

22 Example 2 2 risky investment plans. Plan A: prob. high market 0.8 payoff associated 100; prob. low market 0.2 payoff associated 20. Plan B: prob. bad event 0.2 payoff 10; prob. nothing change event 0.5 payoff 60; prob. good event 0.3 payoff Which one would you chose? 2- How much is your willingness to pay (two cases)? 22