15.053 April 28, 2005 Decision Analysis 2 Utility Theory The Value of Information 1
Lotteries and Utility L1 $50,000 $ 0 Lottery 1: a 50% chance at $50,000 and a 50% chance of nothing. L2 $20,000 Lottery 2: a sure bet of $20,000 How many prefer Lottery 1 to Lottery 2? How many prefer Lottery 2 to Lottery 1? 2
Attitudes towards risk L1 $50,000 $ 0 Lottery 1: a 50% chance at $50,000 and a 50% chance of nothing. L $ K Lottery L : a sure bet of $ K Suppose that Lottery 1 is worth a sure bet of $K to you. If K = $25,000, then you are risk neutral. If K < $25,000, then you are risk averse or risk avoiding. If K > $25,000, then you are risk preferring. 3
Overview of Utility Theory If a person is risk averse or risk preferring, it is still possible to use decision tree analysis. Step 1: develop a mathematical representation of a person s attitude towards risk: a utility curve. Step 2. Solve decision trees assuming that the person wants to maximize expected utility. Next: constructing a utility curve 4
Incorporating Utility into Decision Trees Kaya is indifferent between Lottery 1 and Lottery 2 on the previous page. We next start drawing Kaya s Utility Curve u(k) =.p, where Kaya is indifferent between $K and p $50,000 L1(p) Note: u($50,000) = 1, and u($0) = 0. $ 0 1-p 5
Suppose that utilities can be treated with expected values Suppose that Kaya values this lottery at $20,000 $50,000 $ 0 Next: convert to utils 1 util util Then 0 util u($20,000) = 6
Utility 1 0.9 0.8 0.7 0.6 0 0.4 0.3 0.2 0.1 0 $0 $10,000 $20,000 $30,000 $40,000 $50,000 Dollars u(50,000) = 1 u(k) =, where Kaya is indifferent between $K and L1() $50,000 $ 0 u(0) = 0 7
Then one can compute other utilities Suppose that Kaya values this lottery at $8,000 $20,000 $ 0 Next: convert to utils (A util is our word for the basic unit of utility.).25 util util Then u($8000) =.25 0 util 8
Simple and Compound Lotteries Kaya is also indifferent between Lottery 3 and Lottery 4. What is Lottery 5 worth to Kaya? $20,000 L4 $8,000 L3 $ 0 Two simple lotteries $50,000 L5 $ 0 $ 0 A compound lottery: a lottery of lotteries. 9
Computing the value of a compound lottery $50,000 $20,000 L5 $ 0 $ 0 The certainty equivalent for Kaya of a simple lottery is the amount of money the lottery is worth to Kaya L5 $20,000 So, L5 is worth $8,000. $ 0 10
But L5 is equivalent to L6 $50,000 $8,000 L5 $ 0 $ 0.25 $50,000 $8,000 L6.75 $ 0 So, u($8,000) =.25 11
Utility 1 0.9 0.8 0.7 0.6 0 0.4 0.3 u(20,000) = u(0) = 0 u(k) =.25, where Kaya is indifferent between $K and $20,000 0.2 0.1 0 $0 $10,000 $20,000 $30,000 $40,000 $50,000 Dollars $ 0 12
Figuring out other utilities Suppose that Kaya values this lottery at $32,000 $50,000 $ 20,000 Next: convert to utils util Then u($32,000) = 13
1 0.9 0.8 0.7 Computing u -1 (.75) u -1 (1) = 50,000 u -1 () = 20,000 u(50,000) = 1 u(20,000) = Utility 0.6 0 0.4 0.3 u(k) =.75, where Kaya is indifferent between $K and $50,000 0.2 0.1 0 $0 $10,000 $20,000 $30,000 $40,000 $50,000 $ 20,000 Dollars 14
Computing the rest of the utility curve Suppose that u -1 (p) and u -1 (q) are known. Then u -1 ((p+q)/2) is the value K where Kaya is indifferent between $K and $u -1 (p) $ u -1 (q) Effective in practice. 15
On very fast computation of utility curves Although this approach can work well, it can be very time consuming to solicit the curve. And people are often inconsistent in their valuations of utilities. Next: an approach that requires only one data point. 16
Exponential Utility Curves Assumption: a person is risk averse Let u(x) = 1 e -x/r. R is called the risk tolerance. The higher R is, the more the person is tolerant of risk. The lower R, the less tolerant of risk. Next chart: We choose R so that it matches Kaya s certainty equivalent of the following lottery: $50,000 $ 0 17
Utility 1 0.9 0.8 0.7 0.6 0 0.4 0.3 0.2 0.1 0 An exponential utility function An exponential utility function $0 $10,000 $20,000 $30,000 $40,000 $50,000 Dollars 18
How to use utility functions in decision trees Step 1. Develop a utility function u( ). Step 2. Convert dollars to utils by replacing each outcome x by u(x). Step 3. Solve the decision tree using utils and using expected values Step 4. Convert utils back to dollars, by replacing each utility p by u -1 (p). 19
A Decision Problem Jabba has a choice of two lotteries. Lottery A Lottery B $160,000 $ 90,000 $ 0 $ 40,000 What is Jabba s best decision if u( $K ) = K? What is Jabba s best decision if u( $K ) = K? In this case, Jabba is risk neutral. 20
Maximizing the Expected Value $80,000 $160,000 $80,000 $ 0 $ 90,000 $65,000 $ 40,000 21
Maximizing the Expected Utility 200 400 $160,000 250 0 $ 0 250 300 $ 90,000 200 $ 40,000 u -1 (250) = 250 2 = 62,500 22
Does utility theory really work? Von Neuman and Morgenstern proved that utility theory is valid for a person if a certain set of very reasonable axioms are satisfied. In other words, it is probably never valid for a person, since people are inconsistent in their evaluations, which often depend on context. But, it s a reasonable approximation. Note: I am presenting the axioms for background information. You do not have to learn them. 23
The Axioms for utility theory Let X denote the decision maker. We write 1. r > r if X prefers r to r ; 2. r < r if X prefers r to r; 3. r r, if X is indifferent between r and r. Complete Ordering Axiom. Given two rewards r and r, exactly one of the following are true: r > r or r < r or r r. In addition, if r 1 > r 2, and r 2 > r 3, then r 1 > r 3. 24
Continuity Axiom If r 1 > r 2 and if r 2 > r 3, then there is a probability p such that X is indifferent between Lottery 1 and Lottery 2 below. Lottery 2 Lottery 1 r 2 p r 1 A sure bet of r 2 1-p r 3 25
Independence Axiom If r 1 r 2, then X is indifferent between Lottery 1 Lottery 2 p r 1 p r 2 1-p r 3 1-p r 3 Lottery 1 Lottery 2.7 $131.7 100.3 $60.3 $60 26
Unequal Probability Axiom If r 1 > r 2, and if p > q, then X prefer Lottery 1 to Lottery 2 Lottery 1 Lottery 2 p r 1 q r 1 1-p r 2 1-q r 2 Lottery 1 Lottery 2.8 $100.7 $100.2 $70.3 $70 27
Compound Lottery Axiom If two compound lotteries L1 and L2 have equal probabilities of the outcomes, then X is indifferent between L1 and L2..4 10 Value Prob. L1 0.6 5 5/8 0 0 5.3 10.2.8 L2 3/8.2 10 5 28
Summary of Utility Theory Attitudes towards risk can be incorporated into decision trees. Assessing a person s utility curve can be straightforward, and is sometimes very simple. Maximizing expected utility is a reasonable approximation to reality. It is correct when certain reasonable axioms are satisfied. 29
Next: The value of Information. Using decision analysis to assess the value of collecting information. 30
The value of information Tanyo Electronics has to decide between bringing one of two projects to market. The first product is a telephone that is also a small television. The second product is a telephone that also acts as an MP3- player. Based on market surveys, they anticipate the following: MP3-player:.4 chance of losing $40 million.6 chance of gaining $100 million Television: chance of gaining $50 million chance of gaining $60 million 31
Decision Tree (assume risk neutral) 55 1 MP3 44 2.4.6-40 100 TV 55 3 50 60 32
More detail on the tree Tanyo has the opportunity to conduct an extensive test market on whether the MP3 will be successful. Suppose that the test market will give perfect information on whether or not there will be a loss. How much is this information worth to Tanyo? EVWOI: Expected value with original information. This is the value of the original tree, which is $35 million. EVWPI: Expected value with perfect information. This is the value of the tree, assuming we can get perfect information (where the type of information is specified.) 33
Decision Tree (with perfect information) 82 1 Info says success.6 100 2 100 MP3 55 TV 4 5 0 1-40 100 50 60.4 Info says failure 55 3 MP3 TV -40 6 55 7 1 0-40 100 50 60 34
Expected Value of Perfect Information The Expected Value of Perfect Information: EVPI EVPI = EVWPI EVWOI = $82 million - $55 million = $27 million 35
Decision Tree: Suppose the initial probability of success for the MP3 is.7 Info says success 2 MP3 4-40 100 1 TV 5 50 60 MP3 6-40 100 Info says failure 3 TV 7 50 60 36
On conditional probabilities Suppose that I take two exams at random. What is the probability that the first has at least as high a grade as the second? Suppose that you learn that the second exam had a grade of 95. What is the probability that the first exam has at least as high a grade? Learning information causes probabilities to be updated. 37
The Expected Value of Imperfect Information Tanyo has the opportunity to conduct an extensive test market on whether the MP3 will be successful. But the Information is imperfect. Prob($100 MM if the test is successful) = 5/6 Prob(-$40 MM if the test is unsuccessful) = 3/4 Prob(test is successful) = 60% The probabilistic information is usually given in a more indirect manner than here. But the information above is needed. 38
Where the probabilities come from What is sometimes available is a joint probability table. $100 MM - $40 MM Total Pos Test Neg Test.1.6.1.3.4.6.4 Total Prob (test is positive) =.6 Prob ($100 MM Pos Test) = Prob( $100 MM and Pos Test)/ Prob( Pos Test) = /.6 Prob ( - $40 MM Neg Test) = Prob( - $40 MM and Neg Test)/ Prob( Neg Test) =.3/.4 39
Decision Tree (with perfect information) Info says success 76 2/3 76 2/3 MP3 1/6 5/6-40 100 68.6 55 TV 50 60-5 3/4-40.4 Info says failure 55 MP3 TV 55 1/4 100 50 60 40
The Expected Value of Imperfect Information EVWII: the expected value with imperfect information EVII: the expected value of imperfect information EVII = EVWII EVWOI = $68 MM - $55 MM = $13 MM. Note: a relatively small error rate in the test costs more than $14 million. 41
Summary on the Value of Information The value of information is the increase in the value of the decision problem if new information is provided. EVPI is the value of perfect information. EVPI = EVWPI EVWOI Decision trees often incorporate decisions about whether to gather information. 42