Decision Analysis-1 the Value of Experimentation

Value of Experimentation

Decision Analysis-1 The Value of Experimentation

• Should we perform the experiment? – Imperfect information - outcome is not always “correct” – What is the potential value of the experiment? • Two ways to evaluate the value of information – Expected value of perfect information (EVPI) • The value of having a “crystal ball” • This is a quick preliminary calculation • Provides an upper bound on the potential value of experimentation. If EVPI < Cost : don’t perform the experiment – Expected value of experimentation (EVE) • EVE is the difference between the expected payoff resulting from performing the experiment and the expected payoff without the experiment

Decision Analysis-2 Expected Payoff with Perfect Information

• Suppose the “crystal ball” could definitely tell us the true state of nature. Then we will pick the action with the maximum payoff for this true state of nature. • However, we don’t know in advance which state will be identified. So weigh the max payoff with prior probabilities. State of Nature

Action Oil Dry

Drill for oil 700 -100

Sell the land 90 90

Maximum payoff Prior probability 0.25 0.75

• E[PI] = expected payoff with perfect information = Decision Analysis-3 Expected Payoff with Perfect Information

Action Oil Dry

Drill for oil 700 -100

Sell the land 90 90

Maximum payoff 700 90 Prior probability 0.25 0.75

• E[PI] = expected payoff with perfect information = (700× 0.25) + (90× 0.75) = 242.5 Decision Analysis-4 Expected Value of Perfect Information

• Expected Value of Perfect Information: EVPI = E[PI] – E[OI] where E[OI] is expected payoff with original information (i.e., without experimentation) • EVPI for the Goferbroke problem:

Decision Analysis-5 Expected Value of Perfect Information

• Expected Value of Perfect Information: EVPI = E[PI] – E[OI] where E[OI] is expected payoff with original information (i.e., without experimentation) • EVPI for the Goferbroke problem = E[PI] – E[OI] = 242.5 – 100 = 142.5 • Since EVPI is greater than the cost of the experiment, 142.5 > 30, we should compute the expected value of the experiment

Decision Analysis-6 Expected Value of Experimentation

• We are interested in the value of the experiment. If the value is greater than the cost, then it is worthwhile to do the experiment. • Expected Value of Experimentation: EVE = E[EI] – E[OI] where E[EI] is expected payoff with experimental information

Decision Analysis-7 Goferbroke Example (cont’d)

• Expected Value of Experimentation: EVE = E[EI] – E[OI]

• For the Goferbroke problem E[EI] = E[payoff|USS]×P(USS) + E[payoff|FSS]×P(FSS) = (90×0.7) + (300×0.3) = 153 EVE = 153 - 100 = 53

Decision Analysis-8 Painting Problem

• Painting at an art gallery, you think is worth $12,000 • Dealer asks $10,000 if you buy today (Wed.) • You can buy today (Wed.) or wait until tomorrow (Thurs.): if not sold by then, it can be yours for $8,000 • Tomorrow (Thurs.) you can buy or wait until the next day (Fri.): if not sold by then, it can be yours for $7,000 • In any day, the probability that the painting will be sold to someone else is 50% • What is the optimal policy?

Decision Analysis-9 Drawer Problem

• Two drawers – One drawer contains three gold coins, – The other contains one gold and two silver. • Choose one drawer • You will be paid $500 for each gold coin and $100 for each silver coin in that drawer • Before choosing, you may pay me $200 and I will draw a randomly selected coin, and tell you whether it’s gold or silver and which drawer it comes from (e.g. “gold coin from drawer 1”) • What is the optimal decision policy? EVPI? EVE? Should you pay me $200?

Decision Analysis-10 Utility Theory

Decision Analysis-11 Validity of Monetary Value Assumption

• Thus far, when applying Bayes’ decision rule, we assumed that expected monetary value is the appropriate measure • In many situations and many applications, this assumption is inappropriate • For example, a decision maker’s optimal choice may depend on his/her “utility” for money • A decision maker’s utility is affected by his/her willingness to take risks

Decision Analysis-12 An Example

• Imagine you just graduated from college and owe $40,000 in educational loans to a bank. You have a rich aunt who offers you the following choice: – A 50-50 chance of winning $100,000 or nothing (expected value=0.5*100,000+0.5*0=50,000). – A gift of $40,000 with no uncertainty attached. • Which one would you accept? • So ... is Bayes’ expected monetary value rule invalid? No - because we can use it with the utility for money when choosing between decisions

Decision Analysis-13 Utility Examples

• Think of a capital investment firm deciding whether or not to invest in a firm developing a technology that is unproven but has high potential impact • How many people buy insurance? Is this monetarily sound according to Bayes’ rule? • Treatment for a disease – quality of life We’ll focus on utility for money, but in general it could be utility for anything (e.g., consequences of a doctor’s actions)

Decision Analysis-14 Outline

• Types of utility functions (risk averse, risk neutral, risk seeking) • Decision analysis with utility functions, fundamental property • How to construct utility functions for decision makers – Use fundamental property and answer “lottery” questions – Use an exponential function for risk averse decision makers

Decision Analysis-15 A Typical Utility Function for Money u(M)

4 • What does this mean? The decision maker values $500 only 3 times as much as $100 3 • The utility function has a decreasing slope at the amount of money 2 increases • decision maker has a decreasing marginal 1 utility for money (risk averse).

0 M $100 $250 $500 $1,000

Decision Analysis-16 Types of Utility Functions

• Risk-averse u(M) – Avoid risk

– Decreasing marginal utility for money M • Risk-neutral u(M) – Monetary value = Utility – Prizes money at its face value – Linear utility for money M • Risk-seeking (or risk-prone) u(M) – Seek risk – Increasing marginal utility for money M • Combination of these u(M) …

M Decision Analysis-17 Utility Theory and Decision Analysis

• Inclusion of utility theory in decision analysis is founded in some key ideas • Fundamental property:

The decision maker is indifferent between two alternative courses of action that have the same expected utility

• An optimal action is one that maximizes expected utility

Decision Analysis-18 Illustration of Fundamental Property

• Imagine an individual with the following utility function.

M 0 10,000 30,000 60,000 100,000 u(M) 0 1 2 3 4

• Suppose this individual has the opportunity to win $100,000 with probability p or nothing with probability 1-p. This person has the option of receiving a gift amount with certainty. Then the individual is indifferent between the following pairs of choices. p Guaranteed gift Expected utility amount 0.25 10,000 0.50 30,000 0.75 60,000 • As we shall see, this fundamental property can also be used to construct utility functions.

Decision Analysis-19 Illustration of Fundamental Property

• Imagine an individual with the following utility function.

M 0 10,000 30,000 60,000 100,000 u(M) 0 1 2 3 4

• Suppose this individual has the opportunity to win $100,000 with probability p or nothing with probability 1-p. This person has the option of receiving a gift amount with certainty. Then the individual is indifferent between the following pairs of choices. p Guaranteed gift Expected utility amount 0.25 10,000 4*0.25+0=1 0.50 30,000 4*0.50+0=2 0.75 60,000 4*0.75+0=3 • As we shall see, this fundamental property can also be used to construct utility functions.

Decision Analysis-20 Two Approaches to Constructing Utility Functions

• Ask the decision makers a series of “lottery” questions – Depends on the decision maker answering a series of difficult questions – Constructs utility function from the fundamental property • Assume a mathematical form (typically exponential) of the utility function – The exponential utility function is for risk averse decision makers – The decision maker only has to answer one question – Constructs utility function by estimating one parameter

Decision Analysis-21 Choosing between ‘Lotteries’

• Assume you were given the option to choose from two ‘lotteries’ – Lottery 1 0.5 $100,000 0.5 50:50 chance of winning $100,000 or $0 $0 – Lottery 2 1 Receive $50,000 for certain $50,000 • Which one would you pick? • How about between these two? 0.5 $100,000 0.5 – Lottery 1 $0 50:50 chance of winning $100,000 or $0 1 – Lottery 2 $40,000 Receive $40,000 for certain

Decision Analysis-22 What is Your Expected Utility?

• What is x so you are indifferent? 0.5 – Lottery 1 $100,000 0.5 50:50 chance of winning $100,000 or $0 $0 – Lottery 2 Receive x for certain 1 x

• What is p so you are indifferent? p $100,000 – Lottery 1 1-p p:1-p chance of winning $100,000 or $0 $0 – Lottery 2 1 $50,000 Receive $50,000 for certain

Decision Analysis-23 Goferbroke Example (with Utility)

• We need the utility values for the following possible monetary payoffs: 45° Monetary u(M) Payoff Utility

-130

-100 60

90 M 670

700

Decision Analysis-24 Constructing Utility Functions Goferbroke Example

• u(0) is usually set to 0. So u(0)=0 • Arbitrarily, set u(-130)=-150 • We ask the decision maker what value of p makes him/her indifferent between the following lotteries:

p u(700) 1 u(0) 1-p u(-130)

• The decision maker’s response is p=0.2 • Solve for u(700)

Decision Analysis-25 Constructing Utility Functions Goferbroke Example

• u(0) is usually set to 0. So u(0)=0 • Arbitrarily, set u(-130)=-150 • We ask the decision maker what value of p makes him/her indifferent between the following lotteries:

p u(700) 1 u(0) 1-p u(-130)

• The decision maker’s response is p=0.2 • Solve for u(700): 0.2*u(700) + 0.8*u(-130) = u(0) u(700) = (0 – 0.8*(-150)) / 0.2 = 600 Decision Analysis-26 Constructing Utility Functions Goferbroke Example

• We now ask the decision maker what value of p makes him/her indifferent between the following lotteries:

p u(700) 1 u(90) 1-p u(0)

• The decision maker’s response is p=0.15 • Solve for u(90)

Decision Analysis-27 Constructing Utility Functions Goferbroke Example

• We now ask the decision maker what value of p makes him/her indifferent between the following lotteries:

p u(700) 1 u(90) 1-p u(0)

• The decision maker’s response is p=0.15 • Solve for u(90): 0.15*u(700) + 0.85*u(0)=u(90) 0.15*600 + 0.85*0 = 90 = u(90)

Decision Analysis-28 Constructing Utility Functions Goferbroke Example

• We now ask the decision maker what value of p makes him/her indifferent between the following lotteries:

p u(700) 1 u(60) 1-p u(0)

• The decision maker’s response is p=0.1 • Solve for u(60)

Decision Analysis-29 Constructing Utility Functions Goferbroke Example

• We now ask the decision maker what value of p makes him/her indifferent between the following lotteries:

p u(700) 1 u(60) 1-p u(0)

• The decision maker’s response is p=0.1 • Solve for u(60): 0.1*u(700) + 0.9*u(0)=u(60) 0.1*600 + 0 = 60 = u(60)

Decision Analysis-30 Constructing Utility Functions Goferbroke Example

u(M) 800 Monetary 700 Payoff Utility 45° 600 -130 -150 500

-100 -105 400

60 60 300 90 90 200 100 670 580 0 -200 -100 0 100 200 300 400 500 600 700 800

700 600 -100 M -200

Decision Analysis-31 Exponential Utility Functions

• One of the many mathematically prescribed forms of a “closed- form” utility function

• It is used for risk-averse decision makers only • Can be used in cases where it is not feasible or desirable for the decision maker to answer lottery questions for all possible outcomes • The single parameter R is approximately the one such that the decision maker is indifferent between 0.5 R 1 0 0.5 and -R/2

Decision Analysis-32 Exponential Utility Functions

• Small R implies significant risk aversion • Large R implies small risk aversion (close to risk neutral)

Decision Analysis-33 Goferbroke Example (with Utility) Decision Tree

Decision Analysis-34