MATH1020 Math for Non-Science--Chap1 Assignment—Dr. Tsang
Multiple Choice Questions
1. If a Nash Equilibrium exists which of the following is (are) true?





a.
b.
c.
d.
e.
there are at least two players
there is no better payoff given others' actions
the payoff for each player is maximized
the payoffs for all players are optimized
all of the above
The correct answer is b. A Nash Equilibrium is the best a player can do given another player's
strategy.
2. What type of strategy do rational players not choose to play?





a.
b.
c.
d.
e.
below dominant
weakly dominant
unplayable dominant
strictly dominated
absolutely dominant
The correct answer is d. A strictly dominated strategy is one that rational players do not
play.
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.
The correct answer is b. A dominant strategy is one where the best choice for a player is
constant no matter the action another player takes.
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. More than one of the above answers is correct.
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MATH1020 Math for Non-Science--Chap1 Assignment—Dr. Tsang
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.
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 each have a “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.
e. none of the above.
7. If Player 1 were to choose “Strategy A,” then the best reply for Player 2 would be
to choose
A. “Strategy A.”
B. “Strategy B.”
C. “Strategy C.”
D. “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.
9. If “Player 2” were to “randomize” and choose “Strategy C” with probability 1/4
and choose “Strategy D” with probability 3/4 then “Player 1”
A. would maximize her own payoff by choosing “Strategy A.”
B. would maximize her own payoff by choosing “Strategy B.”
C. would maximize her own payoff by choosing “Strategy A” with
Probability 1/3 and choosing “Strategy B” with probability 2/3
D. has no control over her own payoff (since the value of her expected payoff
is the same for any randomization over her two available strategies).
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MATH1020 Math for Non-Science--Chap1 Assignment—Dr. Tsang
Sample problem
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.
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.
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 This game is set in the South Pacific in 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, or south of New
Britain, where the weather is likely to be clear. U.S. Admiral Kenney hopes to bomb the troop ships. Kenney
has to choose whether to concentrate his reconnaissance aircraft 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.
This game does not have a dominant strategy equilibrium, since there is no dominant strategy for Kenney. His best
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MATH1020 Math for Non-Science--Chap1 Assignment—Dr. Tsang
choice depends on what Imamura does. 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.
12. The Free Kick A soccer player has been awarded a free kick. The only player allowed to defend
against his kick is the opposing team’s goalie. The kicker has two possible strategies. He can try to kick the
ball into the right side of the goal or he can try to kick the ball into the left side of the goal. There is not time
for the goalie to determine where the ball is going before he must commit himself by jumping either to the
left or to the right side of the net. Let us suppose that if the goalie guesses correctly where the kicker is
going to kick, then the goalie always stops the ball. The kicker has a very accurate shot to the right side of
the net, but is not so good at shooting left. If he kicks to the right side of the net and the goalie jumps left,
the kicker will always score. But the kicker kicks to the left side of the net and the goalie jumps to the right,
then the kicker will score only half of the time. This story leads us to the following payoff matrix, where if
the kicker makes the goal, the kicker gets a payoff of 1 and the goalie a payoff of 0 and if the kicker does not
make the goal, the goalie gets a payoff of 1 and the kicker a payoff of 0.
This game has no Nash equilibrium in pure strategies. There is no combination of actions taken with certainty such
that each is making the best response to the other’s action. The goalie always wants to be where the kicker is kicking
and the kicker always wants to kick where the goalie isn’t. What we can find is a pair of equilibrium mixed strategies.
13. Battle of the Sexes This is a famous game, known to game theorists as “The Battle of the Sexes.” The
story goes like this. Two people, let us call them Michelle and Roger, although they greatly enjoy each
other’s company, have very different tastes in entertainment. Roger’s tastes run to ladies’ mud wrestling,
while Michelle prefers Italian opera. They are planning their entertainment activities for next Saturday night.
For each of them, there are two possible actions, go to the wrestling match or go to the opera. Roger would
be happiest if both of them went to see mud wrestling. His second choice would be for both of them to go to
the opera. Michelle would prefer if both went to the opera. Her second choice would be that they both went
to see the mud wrestling. They both think that the worst outcome would be that they didn’t agree on where
to go. If this happened, they would both stay home and feel bad.
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MATH1020 Math for Non-Science--Chap1 Assignment—Dr. Tsang
a. Is this a zero-sum game?
b. Find two Nash equilibria in pure strategies for this game.
c. Find a Nash equilibrium in mixed strategies.
14. The Hawk-Dove Game One fascinating and unexpected application of game theory occurs in biology.
This problem is based on an example developed by the biologist John Maynard Smith to illustrate the uses
of game theory in the theory of evolution. Males of a certain species frequently come into conflict with other
males over the opportunity to mate with females. If a male runs into a situation of conflict, he has two
alternative “strategies.” A male can play “Hawk” in which case he will fight the other male until he either
wins or is badly hurt. Or he can play “Dove,” in which case he makes a display of bravery but retreats if his
opponent starts to fight. If an animal plays Hawk and meets another male who is playing Hawk, they both
are seriously injured in battle. If he is playing Hawk and meets an animal who is playing Dove, the Hawk
gets to mate with the female and the Dove slinks off to celibate contemplation. If an animal is playing Dove
and meets another Dove, they both strut around for a while. Eventually the female either chooses one of
them or gets bored and wanders off. The expected payoffs to each of two males in a single encounter depend
on which strategy each adopts. These payoffs are depicted in the box below.
Neither type of behavior, it turns out, is ideal for survival: a species containing only hawks would have a
high casualty rate; a species containing only doves would be vulnerable to an invasion by hawks or a
mutation that produces hawks, because the population growth rate of the competitive hawks would be much
higher initially than that of the doves.
Now while wandering through the forest, a male will encounter many conflict situations of this type.
Suppose all of the other males in the forest act like Doves. Any male that acted like a Hawk would find that
his rival always retreated and would therefore enjoy a payoff of 10 on every encounter.
If a male acted like a Dove when all other males acted like Doves, he would receive an average payoff of 4.
If you know that you are meeting a Dove for sure, it pays to be a Hawk.
If all the other males acted like Hawks, then a male who adopted the Hawk strategy would be sure to
encounter another Hawk and would get a payoff of −5.
If instead, this male adopted the Dove strategy, he would again be sure to encounter a Hawk, but his payoff
would be 0.
If everyone plays Hawk, it would be profitable to play Dove.
Since there is not an equilibrium in which everybody chooses the same strategy, we might ask whether there
might be an equilibrium in which some fraction of the males chose the Hawk strategy and the rest chose the
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MATH1020 Math for Non-Science--Chap1 Assignment—Dr. Tsang
Dove strategy.
Suppose that the fraction of a large male population that chooses the Hawk strategy is p. Then if one acts
like a Hawk, the fraction of one’s encounters in which he meets another Hawk is about p and the fraction of
one’s encounters in which he meets a Dove is about 1−p. Therefore the average payoff to being a Hawk
when the fraction of Hawks in the population is p, must be p×(−5)+(1−p)×10 = 10−15p.
Similarly, if one acts like a Dove, the probability of meeting a Hawk is about p and the probability of
meeting another Dove is about (1 − p). Therefore the average payoff to being a Dove when the proportion of
Hawks in the population is p will be p × 0 + (1 − p) × 4.
Write an equation that states that when the proportion of the population that acts like Hawks is p, the payoff
to Hawks is the same as the payoffs to Doves. 4 − 4p = 10 − 15p.
Solve this equation for the value of p such that at this value Hawks do exactly as well as Doves. This
requires that p = 6/11.
In the graph below, the blue line is the average payoff to the Dove strategy when the proportion of the male
population who are Hawks is p. The red line represents the average payoff to the Hawk strategy.
If the proportion of Hawks is slightly greater than e, which strategy does better? Dove
If the proportion of Hawks is slightly less than e, which strategy does better? Hawk.
If the more profitable strategy tends to be adopted more frequently in future plays, then if the strategy
proportions are out of equilibrium, will changes tend to move the proportions closer toward equilibrium or
further away from equilibrium? Closer.
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