Sep 2012 --MATH1020 Math for Non-Science-- Assignment1—Prof. Tsang
Multiple Choice Questions
1. If a Nash Equilibrium exists which of the following is (are) true?





a.
b.
c.
d.
e.
the game is a zero-sum game
there is no better payoff given the opponent's action
the payoff for each player is maximized
the payoffs for all players are optimized
all of the above
Solutions: b
A Nash Equilibrium is the best a player can do given another player's strategy.
a, the game may be a nonzero-sum game. c, the corresponding payoffs of Nash equilibrium is
not the best among all payoffs. d, we can’t generally think Nash equilibrium
optimize the payoff of all palyers.
2. What type of strategy do rational players choose not to play?





a.
b.
c.
d.
e.
not optimized strategy
weakly dominant strategy
not a dominant strategy
dominated by at least one other strategy
absolutely dominant
Solutions: d
A strictly dominated strategy is one that rational players do not play.
A rational player will take absolutely dominant strategy if there is any.
We can choose others like a, b, or c if there is not absolutely dominant strategy.
3. What term in game theory describes a strategy that should be employed regardless of an
opponent's action?





a.
b.
c.
d.
e.
optimized
dominant
weakly dominated
sub-dominant
none of the above.
Solutions:
b
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Sep 2012 --MATH1020 Math for Non-Science-- Assignment1—Prof. Tsang
A dominant strategy is the best strategy for a player regardless the action taken by other
player.
4. Consider a two player game between “Player 1” and “Player 2”. “Player 1” has two available strategies:
“Strategy A” and “Strategy B.” “Player 2” has three available strategies: “Strategy c,” Strategy d,” and
“Strategy e.” If “Strategy A” of “Player 1” is a “Best Reply” to a choice of “Strategy c” by “Player 2,” then
A. “Strategy A” must be a dominant strategy for “Player 1.”
B. “Strategy A” cannot be a dominant strategy for “Player 1.”
C. “Strategy B” cannot be a dominant strategy for “Player 1.”
D. “Strategy B” is a dominated strategy for “Player 1.”
Solutions: C
We can’t determine whether “Stragety A” is a dominant strategy or not for Player 1, as we don’t know
whether “Strategy A” of “Player 1” is a “Best Reply” to a choice of “Strategy d” and “Strategy e” by
“Player 2”. But we can determine “Strategy B” cannot be a dominant strategy for “Player 1”, as we know
that “Strategy B” is not a “Best Reply” if Player 2 takes a choice of “Strategy c”. Likewise, we can’t
determine whether “Strategy B” is a dominated strategy or not for “Player 1.
5. Which of the following is NOT one of the “three basic elements of a game”?
A. Players.
B. Dominant strategy.
C. All possible actions.
D. Payoffs.
Solutions: B
Players, All possible actions(strategies) and Payoffs are three basic elements of a game.
There may not be dominant strategies in a game.
For questions 6 through 9, consider the 2 player simultaneous move game below:
6. For this game
A. neither Player 1 nor Player 2 has a dominant strategy.
B. both Player 1 and Player 2 has their own dominant strategy.
C. Player 1 has a dominant strategy but Player 2 does not.
D. Player 2 has a dominant strategy but Player 1 does not.
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Sep 2012 --MATH1020 Math for Non-Science-- Assignment1—Prof. Tsang
E. none of the above.
Solutions:
A
Player 2
C
Player 1
A 20,
B 8,
D
25 40, 15
35 60, 55
We find neither Player 1 nor Player 2 has a dominant strategy after we marked the best one for each player
given the choice of the opponent. As the numbers we marked is not in a row or column.
7. If Player 1 were to choose “Strategy A,” then the best response for Player 2 would be
to choose
A. “Strategy A.”
B. “Strategy B.”
C. “Strategy C.”
D. “Strategy D.”
Solutions: C
From the payoff matrix, we can get that If Player 1 were to choose “Strategy A,” then Player 2 taking
“Strategy C” would get better payoff than taking “Strategy D”.
8. Which of the following “pairs of strategies” is a Nash Equilibrium?
A. Player 1 chooses “Strategy A”; Player 2 chooses “Strategy C.”
B. Player 1 chooses “Strategy B”; Player 2 chooses “Strategy C.”
C. Player 1 chooses “Strategy B”; Player 2 chooses “Strategy D.”
D. More than one of the above answers is correct.
Solutions:
D
Player 2
C
Player 1
A 20,
B 8,
D
25 40, 15
35 60, 55
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Sep 2012 --MATH1020 Math for Non-Science-- Assignment1—Prof. Tsang
From the above payoff matrix, we can get that there are two Nash equilibriums (A, C) and (B, D) in this
game. When the two players are at a Nash equilibrium, for example (A, C), once Player 1 changes his
strategy from A to B while Player 2 keeps, Player 1’s payoff will get worse.
9. In this game
A. “A” is the dominated strategy for Player 1.
B. “B” is the dominated strategy for Player 1.
C. “C” is the dominated strategy for Player 2.
D. “D” is the dominated strategy for Player 2.
E. none of the above.
Solutions:
E
Player 2
C
Player 1
A 20,
B 8,
D
25 40, 15
35 60, 55
We find neither Player 1 nor Player 2 has a dominated strategy after we marked the worst one for each
player given the choice of the opponent. As the numbers we marked is not in a row or column.
Problems
10. Albert and Victoria are roommates. Each of them prefers a clean room to a dirty room, but neither likes
to clean the room. If both clean the room, they each get a payoff of 5. If one cleans and the other doesn’t
clean the room, the person who does the cleaning has a utility of 0, and the person who doesn’t clean the
room has a utility of 8. If neither cleans the room, the room stays a mess and each has a utility of 1. The
payoffs from the strategies “Clean” and “Don’t Clean” are shown in the box below.
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Sep 2012 --MATH1020 Math for Non-Science-- Assignment1—Prof. Tsang
a.
b.
c.
d.
Is there any dominant strategy for Albert?
Is there any dominant strategy for Victoria?
Find the equilibrium.
Is the equilibrium you found a Nash equilibrium? Explain your answer.
Solution:
In this game, notice that if Victoria chooses to clean, then Albert will be better off not cleaning than he would be if he
chose to clean. Likewise if Victoria chooses not to clean, Albert is better off not cleaning than cleaning. Therefore
“Don’t Clean” is a dominant strategy for Albert. Similar reasoning shows that no matter what Albert chooses to do,
Victoria is better off if she chooses “Don’t Clean”. Therefore the outcome where both roommates choose “Don’t
Clean” is the dominant strategy equilibrium. It is interesting to notice that this is true, even though both persons would
be better off if they both chose the strategy “Clean”.
11．The Battle of the Bismarck Sea The setting of this game is in the South Pacific during 1943. Admiral
Imamura must transport Japanese troops from the port of Rabaul in New Britain, across the Bismarck Sea to
New Guinea. The Japanese fleet could either travel north of New Britain, where it is likely to be foggy and
easier for the Japanese ships to hide, or south of New Britain, where the weather is likely to be clear. U.S.
Admiral Kenney hopes to bomb the Japanese troop ships. Kenney has to choose whether to concentrate his
reconnaissance aircrafts on the Northern or the Southern route. Once he finds the convoy, he can bomb it
until its arrival in New Guinea. Kenney’s staff has estimated the number of days of bombing time for each
of the outcomes. The payoffs to Kenney and Imamura from each outcome are shown in the box below. The
game is modeled as a“zero-sum game.” For each outcome, Imamura’s payoff is the negative of Kenney’s
payoff.
a. Is there any dominant strategy for Kenney?
b. Is there any dominant strategy for Imamura?
c. Is there any equilibrium, Nash equilibrium? Explain your result.
Solution:
This game does not have dominant strategy equilibrium, since there is no dominant strategy for Kenney. His best
choice depends on what Imamura does. However, going South is a weakly dominated strategy for Imamura. Since
there are only 2 choices, Imamura should go North.
The only Nash equilibrium for this game is where Imamura chooses the northern route and Kenney concentrates his
search on the northern route. To check this, notice that if Imamura goes North, then Kenney gets an expected two days
of bombing if he (Kenney) chooses North and only one day if he (Kenney) chooses South. Furthermore, if Kenney
concentrates on the north, Imamura is indifferent between going north or south, since he can be expected to be
bombed for two days either way. Therefore if both choose “North,” then neither has an incentive to act differently.
You can verify that for any other combination of choices, one admiral or the other would want to change.
As things actually worked out, Imamura chose the Northern route and Kenney concentrated his search on the North.
After about a day’s search the Americans found the Japanese fleet and inflicted heavy damage on it.
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