Quantitative Decision Techniques 13/04/2009 Decision Trees and Utility Theory

Chapter Outline 4.1 Introduction 4.2 Decision Trees 4.3 How Probability Values Are Estimated by Bayesian Analysis 4.4 Utility Theory 4.5 Sensitivity Analysis

Introduction Decision trees enable one to look at decisions: with many alternatives and states of nature which must be made in sequence

Decision Trees A graphical representation where: a decision node from which one of several alternatives may be chosen l a state-of-nature node out of which one state of nature will occur

Thompson’s Decision Tree Fig A Decision Node A State of Nature Node Favorable Market Unfavorable Market Favorable Market Unfavorable Market Construct Large Plant Construct Small Plant Do Nothing

Five Steps to Decision Tree Analysis 1.Define the problem 2.Structure or draw the decision tree 3.Assign probabilities to the states of nature 4.Estimate payoffs for each possible combination of alternatives and states of nature 5.Solve the problem by computing expected monetary values (EMVs) for each state of nature node.

Decision Table for Thompson Lumber Alternative State of Nature Favorable Market ($) Unfavorable Market ($) Construct a large plant 200, ,000 Construct a small plant 100,000-20,000 Do nothing00 Probabilities0.50

Thompson’s Decision Tree Fig. 4.2 A Decision Node A State of Nature Node Favorable Market (0.5) Unfavorable Market (0.5) Favorable Market (0.5) Unfavorable Market (0.5) Constru ct Large Plant Construct Small Plant Do Nothing $200,000 -$180,000 $100,000 -$20,000 0 EMV =$40,000 EMV =$10,

2nd Decision Table for Thompson Lumber Alternative State of Nature Favorable Results Unfavorable Results Favorable Market Unfavorable Market Probabilities

Thompson’s Decision Tree -Fig. 3

Thompson’s Decision Tree -Fig. 4

Expected Value of Sample Information Expected value of best decision with sample information, assuming no cost to gather it Expected value of best decision without sample information EVSI =

Expected Value of Sample Information EVSI = EV of best decision with sample information, assuming no cost to gather it – EV of best decision without sample information = EV with sample info. + cost – EV without sample info. DM could pay up to EVSI for a survey. If the cost of the survey is less than EVSI, it is indeed worthwhile. In the example: EVSI = $49,200 + $10,000 – $40,000 = $19,200

Estimating Probability Values by Bayesian Analysis Management experience or intuition History Existing data Need to be able to revise probabilities based upon new data Posterior probabilities Prior probabilities New data Bayes Theorem

Example: Market research specialists have told DM that, statistically, of all new products with a favorable market, market surveys were positive and predicted success correctly 70% of the time. 30% of the time the surveys falsely predicted negative result On the other hand, when there was actually an unfavorable market for a new product, 80% of the surveys correctly predicted the negative results. The surveys incorrectly predicted positive results the remaining 20% of the time. Bayesian Analysis

Market Survey Reliability

Calculating Posterior Probabilities P(B  A) P(A) P(A  B) = P(B  A) P(A) + P(B  A’) P(A’) where A and B are any two events, A’ is the complement of A P(FM  survey positive) = [P(survey positive  FM)  P(FM)] / [P(survey positive  FM)  P(FM) + P(survey positive  UM)  P(UM)] P(UM  survey positive) = [P(survey positive  UM)  P(UM)] / [P(survey positive  FM)  P(FM) + P(survey positive  UM)  P(UM)]

Probability Revisions Given a Positive Survey Conditional Probability Posterior Probability State of Nature P(Survey positive|State of Nature Prior Probability Joint Probability FM 0.70* = 0.78 UM 0.20 * =

Probability Revisions Given a Negative Survey Conditional Probability Posterior Probability State of Nature P(Survey negative|State of Nature) Prior Probability Joint Probability FM 0.30* = 0.27 UM0.80* =

Utility Theory Utility assessment assigns the worst outcome a utility of 0, and the best outcome, a utility of 1. A standard gamble is used to determine utility values: When you are indifferent, the utility values are equal. Choose the alternative with the maximum expected utility EU(a i ) = u(a i ) = u(v ij ) P(  j )

Utility Theory $5,000,000 $0 $2,000,000 Accept Offer Reject Offer Red (0.5) Blue (0.5)

Utility Assessment Utility assessment assigns the worst outcome a utility of 0, and the best outcome, a utility of 1. A standard gamble is used to determine utility values. When you are indifferent, the utility values are equal.

Standard Gamble for Utility Assessment Best outcome Utility = 1 Worst outcome Utility = 0 Other outcome Utility = ?? (p)(p) (1-p) Alternative 1 Alternative 2

Figure 4.7 $10,000 U($10,000) = U(0)=0 $5,000 U($5,000)=p =0.80 p= 0.80 (1-p)= 0.20 Invest in Real Estate Invest in Bank

Utility Assessment (1 st approach) v* u(v*) = 1 x1 u(x1) = 0.5 x2 u(x2) = 0.75 (0.5) Lottery ticket Certain money Best outcome (v*) u(v*) = 1 Worst outcome (v – ) u(v – ) = 0 Certain outcome (x1) u(x1) = 0.5 (0.5) Lottery ticket Certain money x1 u(v*) = 0.5 Worst outcome (v – ) u(v – ) = 0 x3 u(x3) = 0.25 (0.5) Lottery ticket Certain money In the example: u(-180) = 0 and u(200) = 1 X 1 = 100  u(100) = 0.5 X 2 = 175  u(175) = 0.75 X 3 = 5  u(5) = 0.25 I II III

Utility Assessment (2 nd approach) Best outcome (v*) u(v*) = 1 Worst outcome (v – ) u(v – ) = 0 Certain outcome (v ij ) u(v ij ) = p (p)(p) (1–p) Lottery ticket Certain money In the example: u(-180) = 0 and u(200) = 1 For v ij =–20, p=%70  u(–20) = 0.7 For v ij =0, p=%75  u(0) = 0.75 For v ij =100, p=%90  u(100) = 0.9

Sample Utility Curve

Preferences for Risk Monetary Outcome Risk Avoider Risk Seeker Risk Indifference Utility

Example Point up (0.45) Point down (0.55) $10,000 -$10,000 0 Alternative 1 Play the game Alternative 2 Do not play the game

Utility Curve for Example

Using Expected Utilities in Decision Making Tack lands point up (0.45) Tack lands point down (0.55) Alternative 1 Play the game Alternative 2 Don’t play Utility