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11 Apr 2019

You have no money in your pocket or in your bank account. You are proposed to participate in one of two lotteries A or B, which could alleviate your poverty. With lottery A, there is a 90 % chance that you receive a payoff of $0 and a 10% chance that you receive a payoff of $400. With lottery B, there is a 0.50 chance that you receive a payoff of $30 and a 0.50 chance that you receive a payoff of $50.

a. Verify that these two lotteries have the same expected value but that lottery A has a bigger variance than lottery B [Note: the variance is a measure of the risk of the lottery. If I have not defined it in class before you do your homework, look at the handouts which are on SmartSite for the definition of variance]

b. Suppose that your utility function for wealth is u(x) = â x + 500 and that you are an expected utility maximizer. Which lottery do you prefer? Are you risk averse, risk neutral or risk lover?

c. Suppose that your utility function for wealth is u(x) = x + 500 and that you are an expected utility maximizer. Which lottery do you prefer? Are you risk averse, risk neutral or risk lover?

d. Suppose that your utility function for wealth is u(I) = (x+500)2 and that you are an expected utility maximizer. Which lottery do you prefer? Are you risk averse, risk neutral or risk lover

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Nestor RutherfordLv2

12 Apr 2019

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Question 1. Suppose that a decision maker is confronted with a choice between (i) the certainty of a $200 gain and (ii) a gamble where there is a 0.40 probability that he will gain $600 and a 0.60 probability that he will lose $20. Moreover, suppose that the utilities he attaches to -$20, $200, and $600 are -5, 10 and 25 respectively. Which course of action will he choose?

(a) .The gamble, since the expected utility from the gamble is greater than O.

(b) The gamble, since the expected utility from the gamble exceeds that from the certainty of a $200 gain.

(d) The certainty 'of a $200 gain, since the expected utility from the certainty of a $200 gain exceeds that from the gamble.

(e) He will be indifferent between the two, since both actions yield the same expected utilities.

2. An increase in monetary gain of $1 is associated with smaller and smaller increases in utility as the monetary gain increases in size. People with utility functions of this sort are (a) risk neutral

(b) risk averters

(d) utilitarians

3. An increase in monetary gain of $1 is associated with larger and larger increases in utility as the monetary gain increases in size. People with utility functions of this sort are

(a) risk neutral

(b) risk averters

(d) utilitarians

Question 4. You are trying to decide whether or not to join your friends down at the local refreshment centre. Unfortunately, you have an economics exam tomorrow. You have tried to keep up with the assignments and believe that if you do not study any more you have a 0,8 chance of getting at least a 70 on the exam, If you are lucky, however, and the lecturer does not ask any of the material you are still confused about, you will get a 90 (20% chance). However, if you stay horne and study you have a 90% chance of getting an 85 and a l0%'chance of getting a 75, If you wish to maximize the expected points on this exam, what should you do?

Question 5. Bud Weiser's Neumann-Morganstern utility function can be represented by the following equation U = 9 + M - 0.01M^2 where U is utility and M is monetary gain (in thousands of dollars), He has the opportunity to invest $20,000 in The Bourbon Cowboy Bar and Grill. He believes that there is a 2/3 chance that he will lose his entire investment and a 1/3 chance that he will gain $30,000.

(i) If he makes the investment, what is his expected utility?

(ii) Should he make the investment?

pucerat135

Suppose you own a house in Mobile, AL. Your house is currently worth $200,000. Suppose

there is a 5% chance that a ood destroys your house, in which case it will be worth $0. You

have $50,000 in your bank account.

a. What is the expected value of your house?

b. You have a utility function over wealth (w ) given by U (w ) =

w . What is the least you

would be willing to accept to sell your house?

c. Why are the answers to the above two questions different?

1- Two friends on a trip, A and B, meet by chance and want to dine together, although both recognize that A prefers to eat at Chili's and B prefers to eat at Steak-n-Shake. Each wonders what to say and accept in negotiating a joint decision. Let O₁ = both eat at Stake-n-Shake, O₂ = A eats at Stake-n-Shake and B eats at Chili's, O₃ = B eats at Stake-n-Shake and A at Chili's, and O₄ = both eat at Chili.

Write an ordinal utility function for each of the friends in respect of the possible outcomes:

Friend A: u_A(O₁) = Friend B: u_B(O₁) =

u_A(O₂) = u_B(O₂) =

u_A(O₃) = u_B(O₃) =

u_A(O₄) = u_B(O₄) =

Construct a 2 x 2 matrix that models this situation, including players, strategies, and the ordinal utilities of each player over the possible outcomes. Make sure you label everything (i.e., strategies, players, etc.).

2- Using the situation described above:

Construct an interval utility function for each player, assuming the following: A is indifferent between (i) its second most preferred outcome and a lottery composed of a .6 chance of its most preferred outcome and a .4 chance of its least preferred outcome, and (ii) A is indifferent between its third-most preferred outcome and a lottery with a .9 chance of its second most preferred outcome and a .1 chance of its least preferred outcome. However, B is indifferent between (i) its second most preferred outcome and a lottery composed of a .7 chance of its most preferred outcome and a .3 chance of its least preferred outcome, and (ii) B is indifferent between its third-most preferred outcome and a lottery with a .8 chance of its second most preferred outcome and a .2 chance of its least preferred outcome.

Construct a 2 x 2 matrix that models the situation, as described above, including players, strategies, and the interval utilities of each player for the possible outcomes.

3- An angry faction, A, is considering whether or not to challenge a status quo power, B, by highjacking B's plane and threatening to kill the passengers if B does not release some terrorists previously imprisoned by B, and whether to carry out its threat or release the passengers if B does not concede to its demands, were the demands made. Let the possible outcomes be: O₁ = A does not highjack the plane, O₂ = The passengers are saved after B releases the imprisoned terrorists in response to the threat made by A, O₃ = the passengers are killed by A after B refuses to submit to A's demand for the release of prisoners, O₄ = the passengers are released by A after B refuses to submit to A's demands. Assume (a) that the best outcome for B, but the worst for A, is for the passengers on the plane to be released after a highjacking, given the reputation for toughness B would gain at A's expense, (b) that the best outcome for A would be for B to release the prisoners after A makes its threat, (c) that A prefers to kill the passengers after a failed attempt to obtain the release of the terrorists over doing nothing at all to obtain the release of the terrorists, and (d), of course, B prefers having to do nothing to have to concede to A's demands, if A decides to challenge B.

Write an ordinal utility function for the angry faction and the status quo power in respect of those outcomes:

Angry A: u_A(O₁) = Power B: u_B(O₁) =

u_A(O₂) = u_B(O₂) =

u_A(O₃) = u_B(O₃) =

u_A(O₄) = u_B(O₄) =

b. Construct a 2 x 2 matrix that models this situation, including players, strategies, and the ordinal utilities of each player over the possible outcomes. Make sure you label everything (i.e., strategies, players, etc.).

4- Why, as game theorists, do we want to obtain an interval measurement of preferences rather than an ordinal measurement? In particular, what does an interval measurement allow us to do that an ordinal measurement doesn't? Are there any limitations on interval measurements? Be specific.

5- What is the difference between decisions under risk and decisions under certainty? What significance does this have for the relevant decision principles for a given game? If a decision principle used for games under uncertainty is applied to a game under risk and it contradicts the decision principle normally used for games under risk, which decision principle should be used, and why? Give an example.

6- Name the two types of irrational actions and explain what they are. Give an example of each. What relevance do irrationalities have for game theory?

7- Josiah, a young man, is considering whether or not to ask Lydia to go to prom with him. Lydia is either willing to go with him or not. Josiah surmises that there is a .6 chance that Lydia would say yes. If Josiah does not ask Lydia out, he will find out whether or not she would have accepted his offer afterward. Let O₁ = Josiah asks and Lydia accepts, O₂ = Josiah asks and Lydia denies, O₃ = Josiah doesn't ask and Lydia would have accepted, and O₄ = Josiah doesn't ask and Lydia wouldn't have accepted. Josiah prefers O₁ to O₄, O₄ to O₃, and O₃ to O₂. That is, O₁ > O₄ > O₃ > O₂.

a) Assume that Josiah is indifferent between O₄ and a lottery with a .9 chance of O₁ and a .1 chance of O₂, and likewise assume that he is indifferent between O₃ and a lottery with a .4 chance of O₁ and a .6 chance of O₂. Construct an interval utility function for Josiah over the respective outcomes:

Josiah: u(O₁) =

u(O₂) =

u(O₃) =

u(O₄) =

b) Use these utilities to construct a 2 X 2 matrix representing the situation. Use the Bayesian decision principle to determine which strategy Josiah would take, were he rational.

c) We treated this game as a game against nature, or a parametric game, rather than a strategic game. Which of these two types of games would be a more realistic model of the described situation? Why or why not?

8- It's Saturday night and Josiah is trying to decide where to take his new girlfriend, Lydia, out to dinner. He has two options: Taco Bell and Macaroni Grill. Lydia prefers Macaroni Grill to Taco Bell, and Josiah prefers what Lydia prefers. There is a problem, however. Martin works at both Taco Bell and Macaroni Grill, and he is practically Josiah's stalker. If Josiah is anywhere near Martin, Martin will find Josiah and ruin dinner for Lydia, which is all that Josiah cares about. Luckily, Martin is either at Macaroni Grill or Taco Bell, not both. Unluckily, neither Josiah nor Lydia has any idea which restaurant Martin is working at or any way of finding out. Lydia prefers eating at Taco Bell without putting up with Martin to eating at Macaroni Grill while putting up with Martin. If Lydia has to put up with Martin, she would still rather eat at Macaroni Grill than Taco Bell.

a) What are the relevant outcomes?

O₁ =

O₂ =

O₃ =

O₄ =

b) Construct an ordinal utility function for these outcomes. Remember, Josiah only cares about what Lydia prefers.

u(O₁) =

u(O₂) =

u(O₃) =

u(O₄) =

c) Construct a 2 X 2 matrix using the utilities from above. Consider two decision principles: maximin and the Bayesian decision principle + the principle of insufficient reason. Determine if each principle can actually be applied to this game. For any principle which does apply, use it to determine Josiah's rational strategy, if the principle recommends one.

9- Consider the following game between play S and player T.

	T1	T2	T3	T4
S1	1, -1	4, -4	10, -10	1, -1
S2	9, -9	5, -5	7, -7	6, -6
S3	3, -3	1, -1	8, -8	5, -5
S4	10, -10	3, -3	1, -1	2, -2

a) Determine the saddlepoint strategies of both players. Show your work.

b) Remove T4 and S4 from the game and determine the outcome using dominance reasoning. Show your work.

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