MATH 446/546 Homework 2: Due October 8th, 2014 Answer the following questions. Some of which come from Winston s text book. 1. We are going to invest $1,000 for a period of 6 months. Two potential investments are available: T-bills and gold. If the $1,000 is invested in T-bills, we are certain to end the 6-month period with $1,296. If we invest in gold, there is a 75% chance that we will end the 6-month period with $400 and a 25% chance we will end the 6-month period with $10,000. If we end up with x dollars, our utility function is given by u(x) = x 1/2. Should we invest in gold or T-bills. This problem may be considered as the comparison of the expected utility of two lotteries: T for the T-bills, and G for the investment in gold. Thus, T 1 $1, 296 and 0.75 $10, 000 G 0.75 $400 Here the expected utility of each is: E(U for T ) = (1296) 1/2 = 36. E(U for G) = 0.25(10, 000) 1/2 + 0.75(400) 1/2 = 40 As the expected utility of the investment in gold is higher we should invest in gold. 2. (The Allais Paradox) Suppose we are offered a choice between the following two lotteries: L 1 : With certainty we receive $1,000,000 L 2 : With probability 0.1, we receive $5,000,000 With probability 0.89 we receive $1,000,000 With probability 0.01 we receive $0.
Which lottery do you prefer? Here we can examine the expected value of each of the lotteries (assuming that we are risk neutral decision makers). Thus, EV (L 1 ) = 1, 000, 000. EV (L 2 ) = 0.1(5, 000, 000) + 0.89(1, 000, 000) + 0.01(0) = 1, 390, 000 Note that there is a chance here that you could miss out on any money at all, and a guaranteed million is hard to pass up. So with a chance of a loss most people will choose lottery L 1. Now consider the following two lotteries: L 3 : With probability 0.11 we receive $1,000,000 With probability 0.89 we receive $0 Which lottery do you prefer? L 4 : With probability 0.1, we receive $5,000,000 With probability 0.9 we receive $0. Here we can look again at the expected values as a reference point. EV (L 3 ) = 0.11(1, 000, 000) + 0.89(0) = 110, 000 EV (L 4 ) = 0.1(5, 000, 000) = 500, 000 It becomes clear here that most people would prefer lottery L 4. In this instance. Suppose that most people prefer L 1 to L 2, show that L 3 must have a larger expected utility than L 4. Page 2
If we consider the best and worst outcome for the given lotteries we see that utility theory assigns the values of u(0) = 0 and u(5, 000, 000) = 1 Before considering the assumption that L 1 pl 2 we should look at the expected utility of L 3 and L 4. Here E(U for L 3 ) = 0.11u(1, 000, 000) + 0.89u(0) = 0.11u(1, 000, 000) and E(U for L 4 ) = 0.1u(5, 000, 000) + 0.9u(0) = 0.1 Supposing that most people prefer L 1 to L 2 yields E(U for L 1 ) > E(U for L 2 ) = u(1, 000, 000) > 0.1u(5, 000, 000) +0.89u(1, 000, 000) + 0.01u(0) = u(1, 000, 000) > 0.1 + 0.89u(1, 000, 000) = 0.11u(1, 000, 000) > 0.1 = E(U for L 3 ) > E(U for L 4 ). This gives a paradox as most people would prefer Lottery L 4 over L 3. 3. Use prospect theory to explain the Allais Paradox in problem 2. Lets assume that users in general have a distorted view of probability. If we use the probability function discussed in class Lets see if there still exists a paradox. Here again the expected utility of Π(p) = 1.89799p 3.55995p 2 + 2.662549p 3 u(0) = 0, and u(5, 000, 000) = 1. and assuming that most people prefer Lottery 1 to Lottery 2 gives: Prospect for L 1 > Prospect for L 2 = u(1, 000, 000) > Π(0.1)u(5, 000, 000) + Π(0.89)u(1, 000, 000) + Π(0.01)u(0) = u(1, 000, 000) > 0.156862049(1) + 0.746389210981u(1, 000, 000) + 0 = u(1, 000, 000) > 0.61851488892 Page 3
If we now consider the prospect of Lottery 3 and Lottery 4 we have: Prospect for L 3 = Π(0.11)u(1, 000, 000) = 0.1692473577u(1, 000, 000) Prospect for L 4 = Π(0.1)u(5, 000, 000) + Π(0.9)u(0) = 0.156862049 Thus, if we consider relationship of preferring Lottery 4 to Lottery 3 we see that Prospect L 4 > Prospect L 3 = 0.156862049 > 0.1692473577u(1, 000, 000) = u(1, 000, 000) < 0.926821258 showing that with prospect theory there is room to eliminate the paradox present in expected utility theory as 0.61851488892 < u(1, 000, 000) < 0.926821258 when considering the prospects of the different lotteries under the assumption that: L 1 pl 2, and L 4 pl 3. 4. The IUP television network earns an average of $400,000 from a hit show and loses an average of $100,000 on a flop. Of all shows reviewed by the network, 25% turn out to be hits and 75% turn out to be flops. For $40,000, a market research firm will have an audience view a pilot of a prospective show and give its view about whether the show will be a hit or a flop. If a show is actually going to be a hit, there is a 90% chance that the market research firm will predict the show to be a hit. If the show is a actually going to be a flop, there is an 80% chance that the market research firm will predict the show to be a flop. Determine how the network can maximize its expected profits. Find the expected value of the research study, and determine the expected value of perfect information. Lets start by drawing a decision tree for the given situation. Page 4
$35,000 Air Show $25,000 Don't Air Show Hire Research Firm $35,000 Hit Flop $0 Predict Hit 0.375 Predict Flop 0.625 0.25 0.75 $400,000 -$100,000 $160,000 -$40,000 Air Show $160,000 Don't Air Show Air Show -$120,000 Don't Air Show Hit 0.6 Flop 0.4 -$40,000 Hit 0.04 Flop 0.96 -$40,000 $360,000 -$140,000 $360,000 -$140,000 Page 5
We need to find the following: P (Predict A Hit) P (Predict A Flop) We know that P (Hit) = 0.25 and P (Flop) Start by finding the Joint Probabilities fro the given likelihoods: P (Predict Hit Hit) = 0.9 P (Predict Flop Hit) = 0.1 P (Predict Hit Flop) = 0.2 P (Predict Flop Flop) = 0.8 The joint probabilities are: P (Predict Hit Hit) = P (Predict Hit Hit)P (Hit) = 0.9(0.25) = 0.225 P (Predict Hit Flop) = P (Predict Hit Flop)P (Flop) = 0.2(0.75) = 0.15 P (Predict Flop Hit) = P (Predict Flop Hit)P (Hit) = 0.1(0.25) = 0.025 P (Predict Flop Flop) = P (Predict Flop Flop)P (Flop) = 0.8(0.75) = 0.6 From the joint probabilities finding the probability of prediction we have: P (Predict Hit) = P (Predict Hit Hit)+P (Predict Hit Flop) = 0.225+0.15 = 0.375 and P (Predict Flop) = 1 0.375 = 0.625 Now we can find the Posterori Probabilities: P (Flop Predict Flop) = P (Flop Predict Flop)/P (Predict Flop) = 0.6 0.625 = 0.96 P (Flop Predict Hit) = P (Flop Predict Hit)/P (Predict Hit) = 0.15 0.375 = 0.4 P (Hit Predict Hit) = P (Hit Predict Hit)/P (Predict Hit) = 0.225 0.375 = 0.6 P (Hit Predict Flop) = P (Hit Predict Flop)/P (Predict Flop) = 0.025 0.625 = 0.04 Note the expected value with out information is $25,000. The expected value with sample information is $35, 000 + $40, 000 = $75, 000 Page 6
For the expected value of the research teams information we have: $75, 000 $25, 000 = $50, 000. The expected value with perfect information can be seen in the following figure: $100,000 Hit 0.25 Flop 0.75 $400,000 $0 Air Show Don't Air Show Ait Show Don't Air Show $400,000 $0 -$100,000 $0 Thus, the expected value of perfect information is $100, 000 $25, 000 = $75, 000. Page 7
546 Additional Homework 1. Although the Von Neumann-Morgenstern axioms seem plausible, there are many reasonable situations in which people appear to violate these axioms. For example, suppose a recent college graduate must choose between three job offers on the basis of starting salary, location of the job, and opportunity for advancement. Given two job offers that are satisfactory with regard to all three attributes, the graduate will decide between two job offers by choosing the one that is superior on at least two of the three attributes. Suppose he or she has three job offers and has rated each one as shown in the following table: Starting Salary Location Opp. for Advancement Job 1 E S G Job 2 G E S Job 3 S G E where E = excellent, G = good, and S = satisfactory. Show that the candidates preferences among these jobs violates the Complete Ordering Axiom. The complete ordering axiom states that for any two rewards r 1 and r 2 one of the following must be the case: r 1 p r 2 (preference of r 1 over r 2 ). r 2 p r 1 (preference of r 2 over r 1 ). r 1 i r 2 (indifferent between r 1 and r 2 ). Additionally we need to have a transitivity of preference. That is for r 1 pr 2 and r 2 pr 3 then r 1 pr 3. So for the above table of jobs we can that the decision maker will have the following preference (picking the one that is superior to the other in at least two categories). Note then Job 1 is prefered to Job 2 (based on Salary and Advancement) Job 2 is prefered to Job 3 (based on Salary and Location) Job 3 is prefered to Job 1 (based on Location and Advancement) Thus, the transitivity of preference assumption is violated. Page 8
2. You are given a choice between lotteries L 1 and L 2. In addition you are given the choice between L 3 and L 4 where: L 1 : Certain gain of $240 L 2 : 25% chance of $1000 75% chance of $0 L 3 : A sure loss of $750 L 4 : 75% chance of loosing $1000 25% chance of losing $0 Provided that 84% of people prefer L 1 over L 2, and 87% of people prefer L 4 over L 3 : (a) Explain why the choice of L 1 over L 2 and L 4 over L 3 contradicts expected utility maximization. One simple way to consider this is to add the preferred lotteries together and compare with the sum of the non-preferred lotteries: L 1 plus L 4 : 75% chance of -$760 and 25% chance of $240. L 2 plus L 3 : 75% chance of -$750 and 25% chance of $250. Note here that L 2 plus L 3 always yields a more favorable result. We can use expected utility theory to show potential miss ordering. Here we define u( 1000) = 0 and u(1000) = 1. Using E(U for L 1 ) = u(240) E(U for L 2 ) = 0.25u(1000) + 0.75u(0) = 0.25 + 0.75u(0) L 1 pl 2 = u(240) > 0.25 + 0.75u(0) = 4 3 u(240) 1 > u(0) (0.1) 3 E(U for L 3 ) = u( 750) E(U for L 4 ) = 0.25u(0) + 0.75u( 1000) = 0.25u(0) L 4 pl 3 = 0.25u(0) > u( 750) = u(0) > 4u( 750) and using (0.1) = 4 3 u(240) 1 3 > 4u( 750) = u(240) > 3u( 750) + 1 4 (0.2) Page 9
We can make note that using the following utility functions we can obtain: utility function reward x = 750 reward x = 0 reward x = 240 u(x) = 1 x + 1 2000 0.125 0.5 0.62 2 u(x) = 1 2000 x + 1000 0.3535 0.7071 0.7874 1/2 These values do not work in (0.2). (b) Can this anomalous behavior be explained? Prospect Theory allows us to have a distorted probability that will put the different lotteries back in the desired order of preference: Note Lottery 1 is preferred to Lottery 2 if and only if: u(240) > Π(0.25) + Π(0.75)u(0) and Lottery 4 is preferred to Lottery 3 if and only if: Π(0.25)u(0) > u( 750) Choosing values for Π(0.25) = 0.3, Π(0.75) = 0.6, u( 750) = 0.09, u(0) = 0.5, and u(240) = 0.7 will make the two lottery preferences consistent. There are other possibilities that will work. Page 10