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: uA(O1) = Friend B: uB(O1) =
uA(O2) = uB(O2) =
uA(O3) = uB(O3) =
uA(O4) = uB(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: uA(O1) = Power B: uB(O1) =
uA(O2) = uB(O2) =
uA(O3) = uB(O3) =
uA(O4) = uB(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.
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: uA(O1) = Friend B: uB(O1) =
uA(O2) = uB(O2) =
uA(O3) = uB(O3) =
uA(O4) = uB(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: uA(O1) = Power B: uB(O1) =
uA(O2) = uB(O2) =
uA(O3) = uB(O3) =
uA(O4) = uB(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.
