Deck 4: Expected Utility Theory and Prospect Theory

Question 1

Suppose your current wealth (W) is $8000 and you obey the principles of expected utility theory. Suppose you are offered a gamble where you can win $4000 with one-half chance but you can lose $4000 with one half-chance. Suppose your utility function is defined as U(W)=2W. To you, the expected utility of this gamble is:

A) 16,000.
B) 8,000.
C) 12,000
D) 4,000.

Answer 1

16,000.

Question 2

Suppose your current wealth (W) is $8000 and you obey the principles of expected utility theory. Suppose you are offered a gamble where you can win $5000 with one-half chance but you can lose $5000 with one half-chance. Suppose your utility function is defined as U(W) = W^0.5. (The square root of your wealth). To you, the expected payoff from this gamble is:

A) 8,000.
B) 16,000.
C) 12,000.
D) 4,000

Answer 2

8,000.

Question 3

Suppose your current wealth (W) is $8000 and you obey the principles of expected utility theory. Suppose you are offered a gamble where you can win $5000 with one-half chance but you can lose $5000 with one half-chance. Suppose your utility function is defined as U(W) = W^0.5. (The square root of your wealth). To you, the expected utility of accepting this gamble is (approx.):

A) 84.39.
B) 89.44.
C) 86.39.
D) 8,000.

Answer 3

84.39.

Question 4

Suppose your current wealth (W) is $8000 and you obey the principles of expected utility theory. Suppose you are offered a gamble where you can win $5000 with one-half chance but you can lose $5000 with one half-chance. Suppose your utility function is defined as U(W) = W^0.5. (The square root of your wealth). To you, the Certainty Equivalent (CE) of this gamble is (approx.):

A) 7121.67.
B) 89.44.
C) 86.39.
D) 8,000.

Question 5

Suppose your current wealth (W) is $8000 and you obey the principles of expected utility theory. Suppose you are offered a gamble where you can win $5000 with one-half chance but you can lose $5000 with one half-chance. Suppose your utility function is defined as U(W) = W^0.5. (The square root of your wealth). For you the risk premium of this is (approx.):

A) 878.
B) 7119.98
C) 8,000.
D) 89.44.

Question 6

Assume, Elizabeth's utility function is: U(W) = W^0.5 and she operates under the tenets of expected utility theory. She is considering two job proposals:. Alternative 1: take a job at a bank with a certain salary of $54,000 per annum. Alternative 2: take a job with a start-up company, get a base salary of $4,000 per annum a plus a bonus of $100,000 per annum a with probability 0.5 (otherwise bonus = $0).

A) Elizabeth should choose Alternative 1 over Alternative 2 since the former yields expected utility of 232.4 while the latter yields expected utility of 192.9.
B) Elizabeth should choose Alternative 2 over Alternative 1 since the former yields expected utility of 192.9 while the latter yields expected utility of 232.4.
C) Elizabeth should choose Alternative 1 over Alternative 2 since the former yields expected utility of 89.4 while the latter yields expected utility of 86.44.
D) Elizabeth should choose Alternative 2 over Alternative 1 since the former yields expected utility of 232.4 while the latter yields expected utility of 192.9.

Question 7

Suppose your current wealth (W) is $10,000. Suppose you are offered a gamble where you can win $4000 with one-half chance but you can lose $4000 with one half-chance. Suppose your utility function is defined as U(W) = W^0.5. (The square root of your wealth). To you, the expected utility of accepting this gamble is:

A) 97.89.
B) 100.00.
C) 89.44.
D) 86.39.

Question 8

Suppose your current wealth (W) is $10,000. Suppose you are offered a gamble where you can win $4000 with one-half chance but you can lose $4000 with one half-chance. Suppose your utility function is defined as U(W) = W^0.5. (The square root of your wealth). To you, the Certainty Equivalent (CE) of this gamble is approximately:

A) 9582.5.
B) 100.00.
C) 10,000.
D) 417.6.

Question 9

Ken Jennings has just been offered a job with a start-up company. The job pays $30,000 guaranteed per year. On top of that that there is a ½ probability that he can get a $50000 performance bonus making a total of $80,000. Ken, operates under the assumptions of expected utility theory and in general, has a strong preference for a job that pays a fixed amount per year. Ken's utility of wealth function is given by U(W) = $\surd$ W. (U(W) is equal to the Square Root of W). Based on this information, you would conclude that:

A) Ken will be willing to accept any job that pays approximately $52,000 per year or higher.
B) Ken will be willing to accept any job that pays approximately $30,000 per year or higher.
C) Ken will be willing to accept a job that pays $50,000 per year.
D) Ken will be willing to accept a job that provides him with 200 utils of utility or less.

Question 10

Jim Holtzhauer is playing the tables at Vegas. He currently has $4000, which I will denote as 4K in order to keep things simple. He is looking at a bet where with ½ chance he can win 2K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)². (U(W) is equal to the W Squared). Based on this information, you would conclude that:

A) Jim would take the bet since his expected utility from taking the bet is 20K² utils while the expected utility of staying with 4K is 16K² utils.
B) Jim would not take the bet since his expected utility from taking the bet is 16K² utils while the expected utility of staying with 4K is 20K² utils.
C) Jim would not take this bet since he is risk averse and his expected utility from taking the bet is negative.
D) Jim would take this bet because the expected value of the gamble: ½ chance of winning 2K and ½ chance of losing 2K is positive.

Question 11

Jim Holtzhauer is playing the tables at Vegas. He currently has $4000, which I will denote as 4K in order to keep things simple. He is looking at a bet where with ½ chance he can win 2K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)². (U(W) is equal to the W Squared). Based on this information, you would conclude that:

A) Jim has no Certainty Equivalent since he always prefers to gamble rather than settle for a certain amount.
B) Jim's certainty equivalent is 4K.^.
C) Jim's certainty equivalent is 4.47K.
D) Jim is risk-averse.

Question 12

When respondents are given the following two choices A and B, a majority choose B over A.
Choice A: $5,000,000 with probability .10, $1,000,000 with probability .89; 0 with probability .01.
Choice B: $1,000,000 with probability 1.
When respondents are given the following two choices C and D, a majority choose C over D.
Choice C: $5,000,000 with probability .10 and 0 with probability .90.
Choice D: $1,000,000 with probability .11 and 0 with probability .89.
These choices are a violation of the principles of Expected Utility theory, which suggests that if one chooses B over A, then that person should choose D over C.
A potential explanation for this pattern of behaviour is the following.

A) In A, the 0.89 probability on $1,000,000 is underweighted, making the gamble relatively less appealing than the certain million. However, in C and D, the low probabilities of both gambles are now being over-weighted, making C the relatively more attractive choice.
B) This is an example of the well-known conjunction fallacy, where people evaluate the probability of a subset of an event as being higher than the event itself.
C) In A, the 0.89 probability on $1,000,000 is over-weighted making this gamble less appealing than the certain million. However, in C and D, the low probabilities of both gambles both gambles are now being under-weighted, making C the relatively more attractive choice.
D) People judge choices from some reference point and here the reference point is $1 million.

Question 13

Which of the following is NOT a good description of the phenomenon of Loss Aversion?

A) A risk neutral person will not accept any gamble that involves the possibility of a loss even if the expected value of the gamble (or even the expected utility) is positive.
B) The magnitude of value increase from a gain of a particular size is smaller than the magnitude of the value decrease from an equivalent loss.
C) The increase in value from a gain of a particular size is smaller than the increase in value from avoiding an equivalent loss.
D) People are risk seeking in losses and risk averse in gains.

Question 14

The fact that people often ask for a much higher price for a good they possess than they are willing to pay to buy the same good is an example of:

A) The endowment effect.
B) Priming.
C) Framing.
D) The availability bias.

Question 15

Which of the following is more likely to result in negotiation deadlocks?

A) Over-confident expectations and a loss frame.
B) Over-confident expectations and a gain frame.
C) Realistic expectations and a loss frame.
D) Realistic expectations and a gain frame.

Question 16

Respondents are given the following choices:
1) Choose between Gamble A: Win $4000 with probability 0.8 or Gamble B: Win $3000 for sure.
2) Choose between Gamble C: Lose $4000 with probability 0.8 or Gamble D: Lose $3000 for sure.
80% of respondents choose Gamble B over Gamble A; i.e. they prefer to win $3000 for sure over $3200 in expectation. But 92% of those same respondents chose Gamble C over Gamble D; i.e., they prefer to lose $3200 in expectation than lose $3000 for sure.
A potential explanation of this pattern of choices is that:

A) People are risk seeking in losses but risk averse in gains.
B) People are risk seeking in gains but risk averse in losses.
C) People are risk neutral in losses but risk averse in gains.
D) People evaluate gambles from a reference point and here the reference point is not clear, leading to inconsistent choices.

Question 17

Suppose you have won $1,000 on a game show. In addition to these winnings, you are now asked to choose between: 1. Gamble A: Win $1000 with 0.50 probability or Gamble B: Win $500 for sure.
On the other hand, suppose you have won $2,000 on a game show and are then asked to choose between:
Gamble C: Lose $1,000 with 0.50 probability or Gamble D: Lose $500 for sure.
These two gambles are obviously identical in terms of final wealth states and probabilities. However, subjects are much more likely to choose the risk averse B and the risk seeking C. This suggests that participants:

A) Are making their decisions over changes in wealth and are anchoring their choices on the basis of an initial reference point, rather than the final asset positions and wealth levels.
B) Underweight the 0.5 probability after they win $1000 but overweight that same probability after they win $2000.
C) Behave in accordance with expected utility theory since Gambles A and C yield higher expected value compared to Gambles B and D respectively.
D) Overweight the 0.5 probability after they win $1000 but underweight that same probability after they win $2000.

Question 18

Gamble A: Win $1000 with 0.50 probability or Gamble B: Win $500 for sure.
On the other hand, suppose you have won $2,000 on a game show and are then asked to choose between:
Gamble C: Lose $1,000 with 0.50 probability or Gamble D: Lose $500 for sure.
These two gambles are obviously identical in terms of final wealth states and probabilities. However, subjects are much more likely to choose the risk averse B and the risk seeking C. This suggests that participants:

A) Are making their decisions over changes in wealth and are anchoring their choices on the basis of an initial reference point, rather than the final asset positions and wealth levels.
B) Underweight the 0.5 probability after they win $1000 but overweight that same probability after they win $2000.
C) Behave in accordance with expected utility theory since Gambles A and C yield higher expected value compared to Gambles B and D respectively.
D) Overweight the 0.5 probability after they win $1000 but underweight that same probability after they win $2000.

Question 19

Todd has $1000. He is given a choice of flipping a coin, heads wins $1000 while tails loses $500. Todd refuses to accept the gamble. A possible explanation is that:

A) Todd is loss averse; so he prefers to avoid the gamble since his expected utility will be negative even though the expected value of the gamble is positive.
B) Todd is risk neutral; so he prefers to avoid the gamble even though both the expected utility and the expected value of the gamble are positive.
C) Todd is risk neutral; so he prefers to avoid the gamble since both the expected utility and the expected value of the gamble are negative.
D) Todd is loss averse; so he prefers to avoid the gamble even though both the expected utility and the expected value of the gamble are positive.

Question 20

The risk premium of a gamble is defined as:

A) The difference between the expected value of the gamble and the Certainty Equivalent; the amount you are willing to forego to avoid the gamble.
B) The amount of money that makes an individual indifferent between receiving that amount for certain and taking on the gamble.
C) The difference between the expected value of the gamble and the expected utility of the game.
D) The initial endowment you started with prior to being faced with the gamble.

Question 21

Suppose I bought some Harvard mugs valued at $10.98. I gave half of the class a mug. These are the sellers. The other half do not have any mugs. They are the buyers. Neither side knows that true value of the mugs. Now supposed I asked the sellers to name a price at which they are willing to sell the mugs. At the same I asked the buyers to name a price at which they are willing to buy the mugs. It is likely that on average:

A) The sellers will ask for a price higher than $10.98; the buyers will state a price less than $10.98. This is due to the endowment effect.
B) The buyers will state a price higher than $10.98; the sellers will ask for a price less than $10.98. This is due to the endowment effect.
C) The buying price and the selling price will be equal, since MLD students are all perfectly rational.
D) The sellers will ask for a price higher than $10.98; the buyers will state a price less than $10.98. This is due to the sellers' over-confidence.

Question 22

Consider two sides negotiating over how to split a surplus. It is more likely that they will arrive at a mutually agreeable outcome if the two sides:

A) Are realistic and adopt a gain frame.
B) Are over-confident and adopt a gain frame.
C) Are realistic and adopt a loss frame.
D) Are over-confident and adopt a loss frame.

Question 23

It is generally well-known that buying lottery tickets do not make sense since the expected value of the lottery ticket is typically less than the price we pay for the ticket. One way to rationalize the act of buying such tickets is to appeal to:

A) The fact that people tend to over-estimate small probabilities.
B) The endowment effect.
C) The conjunction fallacy.
D) Preference reversals.

Question 24

Jim Holtzhauer is playing the tables at Vegas. He currently has $10,000, which I will denote as 10K in order to keep things simple. He is looking at a bet where with ½ chance he can win 6K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)^2. (U(W) is equal to the W Squared). Based on this information, you would conclude that:

A) Jim will not accept this gamble since the expected utility of the gamble is less than the utility he gets from holding on to the 10K he has now.
B) Jim will accept this gamble since the expected utility of the gamble is more than the utility he gets from holding on to the 10K he has now.
C) Jim's Certainty Equivalent is 24K^2.
D) Jim's Certainty Equivalent is [(1/2)(8K)^2+(1/2)(16K)^2].

Question 25

Jim Holtzhauer is playing the tables at Vegas. He currently has $10,000, which I will denote as 10K in order to keep things simple. He is looking at a bet where with ½ chance he can win 6K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)^2. (U(W) is equal to the W Squared). Based on this information, you would conclude that:

A) Jim will accept the gamble since his expected utility from accepting the gamble is 160K^2 while the expected utility from keeping the 10K is 100K^2.
B) Jim will not accept the gamble since his expected utility from accepting the gamble is 100K^2 while the expected utility from keeping the 10K is 160K^2.
C) Jim will not accept since he is risk averse.
D) Jim's Certainty Equivalent is [(1/2)*4K2+(1/2)36K2].

Question 26

Jim Holtzhauer is playing the tables at Vegas. He currently has $10,000, which I will denote as 10K in order to keep things simple. He is looking at a bet where with ½ chance he can win 6K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)^2. (U(W) is equal to the W Squared). Based on this information, you would conclude that:

A) Jim will not accept the gamble since he is risk averse.
B) Jim will not accept the gamble since his expected utility from accepting the gamble is 100K^2 while the expected utility from keeping the 10K is 160K^2.
C) Jim will not accept the gamble since his expected utility from accepting the gamble is 160K^2 while the expected utility from keeping the 10K is 100K^2.
D) Jim's Certainty Equivalent is approximately 12.65K.

Question 27

Jim Holtzhauer is playing the tables at Vegas. He currently has $10,000, which I will denote as 10K in order to keep things simple. He is looking at a bet where with ½ chance he can win 6K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)^2. (U(W) is equal to the W Squared). Based on this information, you would conclude that:

A) Jim will be willing to give up 2K to avoid this gamble.
B) Jim will need to be paid at least 2.65K in order not to accept the gamble.
C) Jim will need to be paid at least 6K in order not accept the gamble,
D) Jim's Certainty Equivalent is approximately 10K.

Question 28

Denote by "W" is the amount of wealth you are holding. Currently you have $5000. Consider a gamble where there is a half-chance of winning $1000 and a half-chance of losing $500. Assuming standard preferences, a risk neutral person will:

A) Accept this bet since the expected final wealth is $5250.
B) Reject this bet since (s)he wants to avoid the possibility of loss.
C) Accept this bet since the expected final wealth is $5000 and there is no possibility of loss.
D) Reject this bet since the expected final wealth is less than $5000.

Question 29

The rule at Shadows is: If you are drinking, you must be 18. In order to check the validity of this rule, you must check:

A) Everyone who is drinking and everyone under the age of 18.
B) Everyone who is drinking and everyone above the age of 18.
C) Everyone not drinking.
D) Everyone who is above 18.

Question 30

Suppose you took a group of people and you gave half of the group a University of Auckland mug. The mugs are valued at $15.99 but the group is not (necessarily) aware of the price. All those who have a mug are called "sellers" while all those without a "mug" are buyers. Now you ask the buyers to name a maximum price at which they are willing to buy the cup. You ask the sellers to name a minimum price at which they are willing to sell a cup. It is highly likely that:

A) The average buying price is less than the average selling price; this is due to the endowment effect.
B) The average buying price is less than the average selling price; this is due to the sunk cost fallacy.
C) The average selling price is less than the average buying price; this is due to the endowment effect.
D) The average selling price is less than the average buying price; this is due to the sunk cost fallacy effect.

Question 31

Ken Jennings has just been offered a job with a start-up company. The job pays $30,000 guaranteed per year. On top of that that there is a ½ probability that he can get a $50000 performance bonus making a total of $80,000. Ken operates under the assumptions of expected utility theory and in general, has a strong preference for a job that pays a fixed amount per year. Ken's utility of wealth function is given by U(W) = W^0.5. (U(W) is equal to the Square Root of W). Show that if Ken has choice between this job and another job then he will choose the job with the start-up company as long as the other job pays less than approx. $52,000 per year.

Question 32

Ken Jennings has just been offered a job with a start-up company. The job pays $30,000 guaranteed per year. On top of that that there is a ½ probability that he can get a $50000 performance bonus making a total of $80,000. Ken operates under the assumptions of expected utility theory and in general, has a strong preference for a job that pays a fixed amount per year. Ken's utility of wealth function is given by U(W) = W^0.5. (U(W) is equal to the Square Root of W). Show that if Ken has a choice between this job and another job that pays $60,000 per year then he will choose the other job over the job with the start-up company.

Question 33

Assume, Elizabeth's utility function is: U(W) = W^0.5 and she operates under the tenets of expected utility theory. She is considering two job proposals:. Alternative 1: take a job at a bank with a certain salary of $54,000 per annum. Alternative 2: take a job with a start-up company, get a base salary of $4,000 per annum a plus a bonus of $100,000 per annum a with probability 0.5 (otherwise bonus = $0). Show that Elizabeth would prefer Alternative 1 over Alternative 2 based on expected utility calculations.

Question 34

Jim Holtzhauer is playing the tables at Vegas. He currently has $10,000, which I will denote as 10K in order to keep things simple. He is looking at a bet where with ½ chance he can win 6K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)^2. (U(W) is equal to the W Squared). Should Jim accept this gamble? If yes, why? If not, then why not? Does Jim have a certainty equivalent in this case? If yes, then what is this amount? Briefly explain whether this will imply Jim actually giving up money or Jim having to be given extra money in order to forego the gamble.

Question 35

Emilio has just bought a fancy new 65-inch OLED Smart television for $5,000.00. Suppose there is a 10% chance that the television will topple over from his entertainment unit and get smashed to pieces and a 90% chance that nothing will happen. Emilio is risk averse and his utility of wealth is U(W) = W^0.5. What is Emilio's (i) certainty equivalent and (ii) risk premium? How much is the maximum that Emilio should be willing to pay to take out insurance on his TV? How would you answers change if the probability of breakage was 1% rather than 10%?

Question 36

For the Value Function in Prospect Theory, the magnitude of value increase from a gain of a particular size is smaller than the magnitude of the value decrease from an equivalent loss. Draw a neat diagram with the Value Function and explain what this means.

Question 37

Suppose you have won $1,000 on a game show. In addition to these winnings, you are now asked to choose between:
Alternative 1. Gamble A: Win $1000 with 0.50 probability or Gamble B: Win $500 for sure.
On the other hand, suppose you have won $2,000 on a game show and are then asked to choose between:
Alternative 2. Gamble C: Lose $1,000 with 0.50 probability or Gamble D: Lose $500 for sure.
It is frequently observed that a majority choose Gamble B for Choice 1 while they choose Gamble C in Alternative 2.
Show that the final wealth levels are not different for the two alternatives. If the final wealth levels are not different then why do people choose Gamble B over Gamble A in Alternative 1, while they choose Gamble C over D for Alternative 2?

Question 38

Suppose Ana's current wealth (W) is $8000 and Ana obeys the principles of expected utility theory. Suppose she is offered a gamble where she can win $5000 with one-half chance but she can lose $5000 with one half-chance. Suppose her utility function is defined as U(W) = W^0.5. (The square root of her wealth). What is the expected monetary payoff of this gamble? What is the expected utility of the gamble? What is Ana's certainty equivalent? What is her risk premium? If Ana were offered this gamble, then on the basis of the utility function defined above, would she accept the gamble or would she prefer on to hang on to her $8000 endowment?

Question 39

Liam currently has $10,000. He is offered a gamble where with one-half chance he can win $2000 but with one-half chance he can lose $2000. (a) Show that Liam is indifferent between accepting this gamble and sticking with this initial endowment of $10,000 if Liam's utility function is given by U(W) = W. (b) However, if Liam's utility function is given by U(W) = Square Root (W), then Liam strictly prefers to stick with his initial endowment of $10,000 rather than accepting this gamble. (Hint: In answering this question, you should denote utilities using 2 decimal points rather than rounding them up to the nearest whole number. Make sure you show your work.) (c) Draw TWO NEAT graphs with wealth (W) on the x-axis and Utility (U) on the y-axis to depict the utility functions in Part (a) and Part (b).

Question 40

A group of participants were given the following gambles to choose from: (a) winning $4000 with 80% chance and $0 with 20% chance OR (b) winning $3000 for sure. 80% of respondents chose gamble (b) over gamble (a). The same group of participants were then given a choice between (c) losing $4000 with 80% change and losing $0 with 20% chance OR (d) losing $3000 for sure. Now 92% of the participants choose gamble (c) over (d).
(a) Why do these choices represent an anomaly? What kind of preferences over gains and losses can explain this behaviour? (b) What do such preferences imply for the shape of the underlying value (utility) function? Draw a neat diagram to illustrate this value function. A neatly drawn diagram should be sufficient to answer this question. (c) On the diagram you drew for Part (b), show that the increase in value from a gain of a particular size is smaller than the increase in value from avoiding a loss of similar size. Either use coloured pencils or clearly mark (using letters A, B, C etc.) the relevant gains and losses on your diagram. Briefly explain your answer so that it is clear what you are showing on your diagram.