MGF 1107
SUPPLEMENTAL HOMEWORK/ CHAPTER 15
1. Consider the following two-person game between players ROW and COLUMN. ROW has two possible strategies R1 and R2. COLUMN has the strategies C1 and C2.
a) What is the minimax of this game?
b) What is the maximin of this game?
c) Is the minimax equal to the maximin?
d) Is there an element of this matrix that is both a row minimum and a column maximum?
e) Does this game have a saddlepoint?
f) What strategies should ROW and COLUMN use?
g) What is the value of this game?
h) Is this a fair game?
2. In a presidential campaign, there are two candidates, a Democrat and a Republican, and two types of issues, domestic and foreign issues. Pollsters tell the Democrat that if both candidates campaign on domestic issues, he will gain 4 points in the polls. If both campaign on foreign issues, the Democrat will gain 3 points in the polls. If the Democrat concentrates on domestic issues while the Republican campaigns on foreign policy, the Democrat will lose 2 points in the polls. Finally, if the Democrat campaigns on foreign policy and the Republican concentrates on domestic policy, the Democrat will drop 1 point.
a) Summarize this information in a payoff matrix with the Democrat as the row player and domestic policy strategy 1 for each player.
b) What is the best strategy for each player?
c) If both players use the strategy you found in part (b), how many points should the Democrat rise or fall in the polls?
3. The number of cases of African flu has reached epidemic levels. The disease is known to have two strains with similar symptoms. There are two medicines available: the first is 60% effective against the first strain and 40% effective against the second. The second medicine is completely effective against the second strain but totally ineffective against the first. You are the Surgeon General. Use the payoff matrix below to determine what mixture of medicines you should recommend to the public.
| |
Strain 1 |
Strain 2 |
|
Medicine 1 | .6 | .4 |
|
Medicine 2 | 0 | 1 |
4. The City of Miami is negotiating a new contract with the union representing sanitation workers. The union and the city can use either aggressive or conciliatory strategies. Suppose the following matrix gives the payoffs corresponding to the four possibilities. The payoffs represent the increase in hourly wage won for the union members.
| |
City aggressive |
City conciliatory |
|
Union aggressive | $2 | $3.75 |
|
Union conciliatory | $1 | $2.50 |
Find the optimal strategies for the union and the city and find the value of the game.
5. Consider the following two-person game between players ROW and COLUMN. ROW has the possible strategies R1 and R2 and R3. COLUMN has the strategies C1 and C2 and C3.
| |
C1 |
C2 |
C3 |
|
R1 | 1 | 3 | 0 |
|
R2 | 2 | -1 | -6 |
|
R3 | 2 | 3 | -1 |
a) What is the minimax of this game?
b) What is the maximin of this game?
c) Is the minimax equal to the maximin?
d) Does this game have a saddlepoint?
e) Is there an element of this matrix that is both a row minimum and a column maximum?
f) What strategies should ROW and COLUMN use?
g) What is the value of this game?
h) Is this a fair game?
i) Does this game have pure or mixed strategies?
6. Consider the following two-person game between players ROW and COLUMN. ROW has three possible strategies R1 and R2 and R3. COLUMN has the strategies C1 and C2 and C3.
| |
C1 |
C2 |
C3 |
|
R1 | 2 | 0 | -1 |
|
R2 | -2 | 1 | -1 |
|
R3 | 0 | 1 | 1 |
a) What is the minimax of this game?
b) What is the maximin of this game?
c) Does this game have a saddlepoint?
d) Is R1 dominated
by any other row?
e) Is R3 dominated by any other row?
f) Does C2 dominate any other column?
g) Does C3 dominate any other column?
h) Eliminate all possible rows and columns. What matrix remains?
i) What strategies should ROW and COLUMN play?
j) What is the expected payoff to ROW?
k) Which player has the advantage in this game?
7. Kuwait and Qatar are major oil-exporting countries. Suppose they decide to play by the following
rules:
i) There are only two prices they can charge: high (H) or low (L).
ii) At the start of each month they must decide which price they will
charge, and are not permitted to change prices during the month.
Their monthly gross revenue (in millions of dollars) is indicated in the
following payoff matrix:
| |
Qatar H |
Qatar L |
|
Kuwait H | (9, 9) | (5, 17) |
|
Kuwait L | (17, 5) | (6, 6) |
a) Does either player have a dominant strategy?
b) What is the minimax strategy for each player?
c) Find all Nash equilibria.
d) Can the players benefit by cooperating?
8. The state legislature is about to vote on two bills that authorize
the construction of new roads in Miami and Ft. Lauderdale. If the two cities join forces, they can muster
enough political power to pass the bills, but neither city can do it alone.
If a bill is passed, it will cost the taxpayers of both cities a million
dollars and the city in which the roads are constructed will gain $10 million.
The bills were introduced in the closing minutes of the annual legislative
session, so legislators have no time to discuss cooperating. Since a city always supports its own bond issue,
the legislators have only two possible strategies: supporting or not supporting,
their neighboring city. To explain how
the payoffs in the matrix below were computed, suppose Miami supports Ft. Lauderdale’s
bond issue, but Ft. Lauderdale doesn’t support Miami’s.
Then Ft. Lauderdale’s bond issue passes, each city pays a million dollars,
Ft. Lauderdale gets $10 million and Miami gets nothing.
The net effect is that Ft. Lauderdale gets $9 million and Miami loses $1
million.
| |
Ft. Lauderdale supports Miami’s bond |
Ft. Lauderdale doesn’t support Miami’s bond |
|
Miami supports Ft. Lauderdale’s bond | (8, 8) | (-1, 9) |
|
Miami doesn’t support Ft. Lauderdale’s bond | (9, -1) | (0, 0) |
b) What is the minimax strategy for each player?
c) Find all Nash equilibria.
d) Can the players benefit by cooperating?
9. During the period 1980-1 in Poland, the Solidarity labor union challenged
the ruling Communist party. The party
had two choices: reject (R) or accept (A) the limited autonomy of plural social
forces set loose by Solidarity. Rejection would, if successful, restore the monolithic structure
underlying Communist rule. Acceptance
would allow political institutions other than the Communist party to participate
in some meaningful way in the formulation of public policy. Solidarity also had two strategies: reject
(R) or accept (A) the monolithic structure of the country. Rejection would put pressure on the government
to limit severely the extent of the state’s authority in political matters.
Acceptance would significantly reduce the chances of Solidarity or other
independent institutions to alter certain state activities.
The two strategies available to each side give rise to four possible outcomes,
with the associated payoffs being the rankings of each possible outcome as theorized
by NYU political scientist Steven J. Brams:
| |
Communist party R |
Communist party A |
|
Solidarity A | (2, 4) | (3, 3) |
|
Solidarity R | (1, 2) | (4, 1) |
a) Does either player have a dominant strategy?
b) What is the minimax strategy for each player?
c) Find all Nash equilibria.
d) Can the players benefit by cooperating?
10. Consider the following matrix for a zero-sum game between player ROW
and player COLUMN.
| |
C1 |
C2 |
C3 |
C4 |
C5 |
C6 |
C7 |
|
R1 | 5 | 4 | 1 | 6 | 0 | 5 | 4 |
|
R2 | 6 | 4 | 3 | 7 | -1 | 9 | 6 |
What strategies should ROW and COLUMN play?
11. Two Republican Senators, McCain and Lott, plan to introduce similar, but slightly different, campaign finance reform bills. After discovering the plans of their colleague, each senator has two choices: proceed with plans to introduce their bill or drop their plans and allow their colleague’s bill to be considered alone. There are four possible outcomes, which we list from most favorable (4) to least favorable (1), from Senator McCain’s point of view.
4) McCain proceeds and Lott drops his bill
3) Lott proceeds and McCain drops his bill
2) Both senators drop their bills.
1) Both senators proceed with their bills (This is the least desirable option since it would lead to a nasty intra-party squabble.)
After considering the four possible outcomes from Senator Lott’s point of view,
a) write the payoff matrix for this game with Senator McCain as the row player.
b) Does either player have a dominant strategy?
c) What is the minimax strategy for each player?
d) Find all Nash equilibria.
e) Can the players benefit by cooperating?
12. Repeat the previous problem assuming that both senators care more about getting some bill passed than avoiding a divisive fight among Republicans, but still think that introducing only one bill is the preferred route.
13. Bill McCollum and Bill Nelson ran against each other for U.S. Senator in 2000. One issue was whether the candidates would accept soft money donations. Each candidate had two strategies: to accept (A) soft money or to not accept (A¢) soft money. There are four possible outcomes, which we list from most favorable (4) to least favorable (1), from Mr. McCollum’s point of view.
4. McCollum accepts soft money and Nelson doesn’t. (This would give McCollum a big fund-raising advantage.)
3. Both turn down soft money. (No one has a fund-raising advantage and the media will stop asking him about the ethics of accepting soft money.)
2. McCollum turns down the soft money, but Nelson accepts it. (While this puts McCollum at a fund-raising disadvantage, he now has a great issue to use against Nelson in the campaign.)
1. Both accept the soft money. (Not only does McCollum not have a fund-raising advantage, but he will continue to take heat in the media for accepting soft money.)
Assuming, Mr. Nelson has the same ranking of the outcomes,
a) write the payoff matrix for this game assuming Mr. Nelson is the row player.
b) Does either player have a dominant strategy?
c) What is the minimax strategy for each player?
d) Find all Nash equilibria.
e) Can the players benefit by cooperating?
14. Find the optimal strategy for each player and the value of the game.
| a)
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b)
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| c)
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d)
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| e)
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f)
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Answers:
1a) –1 1b) –1 1c) yes 1d) yes 1e) yes
1f) ROW should use strategy R1 and COLUMN should use strategy C1.
1g) –1 1h) no
2a)
2b) The optimal strategy for the Democrat is to spend 40% of his time on domestic issues and 60% on foreign policy. The Republican should divide his time evenly.
2c) The Democrat will rise one point in the polls.
3) Patients should take medicine 1 about 5/6 of the time and medicine 2 about 1/6 of the time.
4. Both the union and the city should use aggressive strategies; the value is $2.
5a) 0 5b) 0 5c) yes 5d) yes 5e) yes
5f) ROW should use strategy R1 and COLUMN should use strategy C3.
5g) 0 5h) yes 5i) pure
6a) 1 6b) 0 6c) no 6d) no 6e) no 6f) no
6g) yes (column 2)
6h) 
6i) ROW should play
R1 25% of the time and R3 75% of the time. COLUMN
should play C1 and C3 each 50% of the time.
6j) ½ 6k) ROW
7a) Both countries have a dominant strategy of charging the low price.
7b) The minimax strategy for both countries is to charge the low price.
7c) (6, 6) is the only Nash equilibrium.
7d) They can both benefit by cooperating since if they agree to both charge the higher price, both will get higher payoffs. (This explains why the countries benefit by belonging to a cartel like OPEC.)
8a) Both cities have a dominant strategy to not support the other’s bond.
8b) The minimax strategy for both cities is to not support the other’s bond.
8c) (0, 0) is the only Nash equilibrium.
8d) They can both benefit by cooperating since if they agree to both support the other’s bond, both will get higher payoffs.
9a) The Communist party has a dominant strategy of rejection.
9b) The minimax strategy for Solidarity is Acceptance. For the party, it is rejection.
9c) The only Nash equilibrium is the outcome (2, 4)
9d) No. The party already receives its best outcome without cooperation.
10. ROW should play R1 and COLUMN should play C5.
11a)
| |
Lott proceeds |
Lott drops bill |
|
McCain proceeds | (1, 1) | (4, 3) |
|
McCain drops bill | (3, 4) | (2, 2) |
11b) No
11c) The minimax strategy for both McCain and Lott is to drop their bills.
11d) (4, 3) and (3, 4) are both Nash equilibria.
11e) Both can improve their payoffs by cooperating.
12a)
| |
Lott proceeds |
Lott drops bill |
|
McCain proceeds | (2, 2) | (4, 3) |
|
McCain drops bill | (3, 4) | (1, 1) |
12b) No
12c) The minimax strategy for both McCain and Lott is to proceed with their bills.
12d) (4, 3) and (3, 4) are both Nash equilibria.
12e) Both can improve their payoffs by cooperating.
13a)
| |
McCollum A |
McCollum A¢ |
|
Nelson A | (1, 1) | (4, 2) |
|
Nelson A¢ | (2, 4) | (3, 3) |
13b) No
13c) The minimax strategy for both is to not accept soft money.
13d) (2, 4) and (4, 2) are Nash equilibria.
13e) It doesn’t help to cooperate.
14a) ROW should play R3 and COLUMN should play C1 or C2. The value of the game at the saddlepoint is 2.
14b) ROW should play R1 1/6 of the time and R2 5/6 of the time. COLUMN should play C2 2/3 of the time and C3 1/3 of the time. The value of the game is 14/3.
14c) ROW should play R1 3/7 of the time and R2 4/7 of the time. COLUMN should play C1 2/7 of the time, never play C2, and play C3 5/7 of the time. The value of the game is 8/7.
14d) ROW should play R1 7/12 of the time and R2 5/12 of the time. COLUMN should play C4 5/12 of the time and play C5 7/12 of the time. The value of the game is 49/12.
14e) ROW should play R1 3/7 of the time and R2 4/7 of the time. COLUMN should play C1 4/7 of the time and C2 3/7 of the time. The value of the game is 5/7.
14f) ROW should play R1 1/2 of the time and R2 1/2 of the time. COLUMN should play C1 3/5 of the time and C2 2/5 of the time. The value of the game is 1.