Oct 2012 --MATH1020 Math for Non-Science—Assignment2—Prof. Tsang
Multiple Choice Questions
1. Consider the 2 player simultaneous move game below:
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 she does not know what action Player 2 will take).
Problems
2. Assume the following payoff matrix (showing only the row player’s payoff) for a 2-person zero-sum
game. Follow the steps below to find the pure strategy solution for this game.
a. Is there any dominated strategy for the row player? Eliminate it if there is any.
b. Is there any dominated strategy for the column player after the previous step? Eliminate it if there is
any.
c. Now, you should be able to reduce the payoff matrix to a 3x3 matrix. Find the maximin & the
minimax of this game.
d. Is there a saddle point for this problem? Is there a pure strategy solution for this game?
3. Morra There are many forms of this game. In its simplest form, each player hides one or two
coins. Then each player guesses how many coins the opponent has hidden. There is a win if and only
if one player guesses correctly and the other one guesses incorrectly. In case of a win an amount
equal to the total number of hidden coins are paid to the winner.
Strategies available to both players are:
(1/1) Hide one coin, guess “one coin”,
(1/2) Hide one coin, guess “two coins”,
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Oct 2012 --MATH1020 Math for Non-Science—Assignment2—Prof. Tsang
(2/1) Hide two coins, guess “one coin”,
(2/2) Hide two coins, guess “two coins”.
The payoff matrix for this zero-sum game can be written as:
1/1
1/2
2/1
2/2
1/1
0
1
-1
0
1/2
-1
0
0
2
2/1
1
0
0
-2
2/2
0
-2
2
0
a. Does the saddle point exist in this game? Is there a pure strategy solution for this game?
b. If you assume your opponent mixed all her strategies with equal probabilities, find the expected
payoff for each of your strategy. Is there an optimal strategy that can offer you better payoff?
c. Since the game is fair to both players, your opponent will also notice there is an optimal strategy that
offers better payoff. If she mixes all her other strategies with equal probabilities and the optimal
strategy with 40% probability, recalculate the expected payoff for each of your strategy. Do you still
have an optimal strategy that can offer you better payoff?
4. 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.
a.
b.
c.
d.
Is this a zero-sum game?
Is there any dominated strategy? Any dominant strategy?
Is there any Nash equilibrium in pure strategies for this game? Find it/them.
Find a Nash equilibrium in mixed strategies.
5. 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
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Oct 2012 --MATH1020 Math for Non-Science—Assignment2—Prof. Tsang
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.
Fill in the blanks or select the correct answer in the following analysis of this game.
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
on every encounter.
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 ( Dove/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
.
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 ( Dove/Hawk ).
Since there is no 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 probability of one’s encounters in which he meets another Hawk is p and the probability of
one’s encounters in which he meets a Dove is
. 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 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
.
In the graph below, the blue line is the average payoff to the Dove strategy when the proportion of the male
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Oct 2012 --MATH1020 Math for Non-Science—Assignment2—Prof. Tsang
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/Hawk
If the proportion of Hawks is slightly less than e, which strategy does better? Dove/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 toward/further away.
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