2011 MATH1020 Math for Non-Science—Assignment 2—Prof. Tsang
1. Pure strategy problems: State whether the following games (with their payoff matrixes given) have
unique pure strategy solutions, and if so what kind of solution they are and how they can be found.
(a)
Row Player | Column Player Left
Center
Right
Up
4. 3
2, 7
0, 4
Down
5, 5
5, -1
-4, -2
(b)
Row Player | Column Player
Up
Middle
Down
Left
3, -3
1, 6
5, 7
Center
4, 8
-5, -6
5, -1
Right
7, 2
8, 9
-4, -2
Mixed strategy problems
2. 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 assume 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 consideration 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.
(a) Are there actions taken by the players with certainty such that each will get the best result independent of the
other’s action?
(b) Does this game have Nash equilibrium in pure strategies?
(c) Can you find mixed strategies for the players so that each will get the best result regardless of the other’s
action?
3. Hide and Seek (Battle of the Sexes)
This is a famous game, known to game theorists as “The Battle
of the Sexes”, but actually more like “Hide and Seek”. The story goes like this. Once upon a time, there
were a girl and a boy, whose names were Michelle and Roger. They did not know each other very much but
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2011 MATH1020 Math for Non-Science—Assignment 2—Prof. Tsang
they loved each other at first sight. They greatly enjoyed each other’s company, but had very different tastes
in entertainment. Roger liked to watch wrestling, while Michelle preferred opera. They were shy to date the
other openly, and tried to meet the other at their favorite spots, the wrestling or opera places. 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 wrestling. His second choice would be to meet Michelle at the opera. Michelle
would prefer if both went to the opera. Her second choice would be to meet Roger at the wrestling place.
They both felt bad if they did not meet each other at their favorite places. But the worst outcome would be
that Roger went to the opera and Michelle went to the wrestling. Each went to the other’s favorite spot and
found out the other was not there. The payoff matrix below summaries the reward of this dating game.
Roger \ Michelle
Wrestling
Opera
Wrestling
2, 1
-1, -1
Opera
0, 0
1, 2
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.
4. 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.
Fill in the answers in the following questions:
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 ______ on every encounter.
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2011 MATH1020 Math for Non-Science—Assignment 2—Prof. Tsang
If a male acted like a Dove when all other males acted like Doves, he would receive an average payoff of
______.
If you know that you are meeting a Dove for sure, it pays to be a __________________.
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 _________.
If instead, this male adopted the Dove strategy, he would again be sure to encounter a Hawk, but his payoff
would be __________.
If everyone plays Hawk, it would be profitable to play ___________.
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
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 ___________________________.
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____________________.
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. _________________________.
Solve this equation for the value of p such that at this value Hawks do exactly as well as Doves. This
requires that p= e=________________.
If the proportion of Hawks in the population is slightly greater than e, which strategy does better?
_____________
If the proportion of Hawks is slightly less than e, which strategy does better?. _______________
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? _____________
5. In the Paradox of the Chair’s Position, three voters A, B, and C are electing the next chairperson among
them. Voter A, the current chair, has 3 votes. Voters B and C have 2 votes each. The winner will be the
person who gets most votes. Voter A’s preference is (A>B>C). Voter B’s preference is (B>C>A). Voter C’s
preference is (C>A>B).
(a)Under the assumption that A votes for himself, construct a payoff matrix (Table 1) using the actions
available to B and C, as shown below:
B\C
Vote for A
Vote for B
Vote for C
Vote for A
A 3,1,2
A 3,1,2
A 3,1,2
Vote for B
A 3,1,2
Vote for C
A 3,1,2
where the first character in the 9 cells representing the 9 possible outcomes is the winner of the voting game
and the 3 numbers that follow are the payoffs to the 3 players A, B and C respectively. The payoffs are
defined as: 3 for the first preference, 2 for the second preference, and 1 for the last preference getting elected.
Complete the payoff matrix above by filling in the blanks.
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2011 MATH1020 Math for Non-Science—Assignment 2—Prof. Tsang
(b)If A votes for B, construct another payoff matrix (Table 2) using the actions available to B and C, in a
similar way:
B\C
Vote for A
Vote for B
Vote for C
Vote for A
A 3,1,2
Vote for B
Vote for C
Complete the payoff matrix above by filling in the blanks.
(c)If A votes for C, construct another payoff matrix (Table 3) using the actions available to B and C, in a
similar way:
B\C
Vote for A
Vote for B
Vote for C
Vote for A
A 3,1,2
Vote for B
Vote for C
Complete the payoff matrix above by filling in the blanks.
(d)Comparing the three Tables (1, 2, and 3), which action will A take so that he can get the highest payoff
with most certainty?
(e)Knowing that A will take the action determined from (d), use one of the three Tables to decide what
action B & C will take, and hence who will be the final winner?
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