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Chapter 30 NAME
Game Applications
Introduction. As we have seen, some games do not have a “Nash equi-
librium in pure strategies.” But if we allow for the possibility of “Nash
equilibrium in mixed strategies,” virtually every game of the sort we are
interested in will have a Nash equilibrium.
The key to solving for such equilibria is to observe that if a player is
indifferent between two strategies, then he is also willing to choose ran-
domly between them. This observation will generally give us an equation
that determines the equilibrium.
Example: In the game of baseball, a pitcher throws a ball towards a batter
who tries to hit it. In our simplified version of the game, the pitcher can
pitch high or pitch low, and the batter can swing high or swing low. The
ball moves so fast that the batter has to commit to swinging high or
swinging low before the ball is released.
Let us suppose that if the pitcher throws high and batter swings low,
or the pitcher throws low and the batter swings high, the batter misses
the ball, so the pitcher wins.
If the pitcher throws high and the batter swings high, the batter
always connects. If the pitcher throws low and the batter swings low, the
batter will connect only half the time.
This story leads us to the following payoff matrix, where if the batter
hits the ball he gets a payoff of 1 and the pitcher gets 0, and if the batter
misses, the pitcher gets a payoff of 1 and the batter gets 0.
Simplified Baseball
Pitcher
Batter
Swing Low Swing High
Pitch High 1,0 0,1
Pitch Low .5,.5 1,0
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 batter always wants to swing
the same place the pitcher throws, and the pitcher always wants to throw
to the opposite place. What we can find is a pair of equilibrium mixed
strategies.
366 GAME APPLICATIONS (Ch. 30)
In a mixed strategy equilibrium each player’s strategy is chosen at
random. The batter will be willing to choose a random strategy only if
the expected payoff to swinging high is the same as the expected payoff
to swinging low.
The payoffs from swinging high or swinging low depend on what the
pitcher does. Let πPbe the probability that the pitcher throws high and
1−πPbe the probability that he throws low. The batter realizes that if
he swings high, he will get a payoff of 0 if the pitcher throws low and 1
if the pitcher throws high. The expected payoff to the batter is therefore
πP.
If the pitcher throws low, then the only way the batter can score is
if pitcher pitches low, which happens with probability 1 −πP.Eventhen
the batter only connects half the time. So the expected payoff to the
batter from swinging low is .5(1 −πP).
These two expected payoffs are equalized when πP=.5(1 −πP). If
we solve this equation, we find πP=1/3. This has to be the probability
that the pitcher throws high in a mixed strategy equilibrium.
Now let us find the probability that the batter swings low in a mixed
strategy equilibrium. In equilibrium, the batter’s probability πBfrom
swinging low must be such that the pitcher gets the same expected payoff
from throwing high as from throwing low The expected payoff to the
pitcher is the probability that the batter does not score.
If the pitcher throws high, then the batter will not connect if he
swings low, but will connect if he swings high, so the expected payoff to
the pitcher from pitching high is πB.
If the pitcher throws low, then with probability (1 −πB), the batter
will swing swing high, in which case the pitcher gets a payoff of 1. But
when the pitcher throws low, the batter will swing low with probability πB
and connect half the time, giving a payoff to the pitcher of .5πB. Therefore
the expected payoff to the pitcher from throwing low is (1 −πB)+.5πB=
1−.5πB. Equalizing the payoff to the pitcher from throwing high and
throwing low requires πB=1−.5πB. Solving this equation we find that
in the equilibrium mixed strategy, πB=2/3.
Summing up, the pitcher should throw low two-thirds of the time,
and the batter should swing low two-thirds of the time.
Calculus 30.1 (2) Two software companies sell competing products. These prod-
ucts are substitutes, so that the number of units that either company sells
is a decreasing function of its own price and an increasing function of the
other product’s price. Let p1be the price and x1the quantity sold of
product 1 and let p2and x2be the price and quantity sold of product
2. Then x1= 1000 90 −1
2p1+1
4p2and x2= 1000 90 −1
2p2+1
4p1.
Each company has incurred a fixed cost for designing their software and
writing the programs, but the cost of selling to an extra user is zero.
Therefore each company will maximize its profits by choosing the price
that maximizes its total revenue.
NAME 367
(a) Write an expression for the total revenue of company 1, as a function
2
(b) Company 1’s best response function BR1(·) is defined so that BR1(p2)
is the price for product 1 that maximizes company 1’s revenue given
that the price of product 2 is p2. With the revenue functions we have
specified, the best response function of company 1 is described by the
with respect to p1and solve for the revenue-maximizing p1given p2.)
(c) Use a similar method to solve for company 2’s best response function
(e) Suppose that company 1 sets its price first. Company 2 knows the
price p1that company 1 has chosen and it knows that company 1 will
not change this price. If company 2 sets its price so as to maximize its
revenue given that company 1’s price is p1, then what price will company
2choose? p2=$90 + p1/4If company 1 is aware of how company
2 will react to its own choice of price, what price will company 1 choose?
30.2 (1) Here is an example of the “Battle of the Sexes” game discussed
in the text. 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 sulk.
368 GAME APPLICATIONS (Ch. 30)
Battle of the Sexes
Michelle
Wrestling Opera
(a) Is the sum of the payoffs to Michelle and Roger constant over all
outcomes? No. Does this game have a dominant strategy equilibrium?
No.
(b) Find two Nash equilibria in pure strategies for this game. Both
(c) Find a Nash equilibrium in mixed strategies. Michele
chooses opera with probability 2/3and
30.3 (1) This is an example of the game of “Chicken.” Two teenagers
in souped-up cars drive toward each other at great speed. The first one
to swerve out of the road is “chicken.” The best thing that can happen
to you is that the other guy swerves and you don’t. Then you are the
hero and the other guy is the chicken. If you both swerve, you are both
chickens. If neither swerves, you both end up in the hospital. A payoff
matrix for a chicken-type game is the following.
Chicken
Leroy
Swerve Don’t Swerve
NAME 369
(a) Does this game have a dominant strategy? No. What are the two
(b) Find a Nash equilibrium in mixed strategies for this game. Play
30.4 (0) I propose the following game: I flip a coin, and while it is in the
air, you call either heads or tails. If you call the coin correctly, you get
to keep the coin. Suppose that you know that the coin always comes up
(a) Suppose that the coin is unbalanced and comes up heads 80% of
the time and tails 20% of the time. Now what is your best strategy?
(b) What if the coin comes up heads 50% of the time and tails 50% of the
time? What is your best strategy? It doesn’t matter.
(c) Now, suppose that I am able to choose the type of coin that I will toss
(where a coin’s type is the probability that it comes up heads), and that
you will know my choice. What type of coin should I choose to minimize
(d) What is the Nash mixed strategy equilibrium for this game? (It may
help to recognize that a lot of symmetry exists in the game.) I
30.5 (0) Ned and Ruth love to play “Hide and Seek.” It is a simple
game, but it continues to amuse. It goes like this. Ruth hides upstairs or
downstairs. Ned can look upstairs or downstairs but not in both places.
If he finds Ruth, Ned gets one scoop of ice cream and Ruth gets none. If
370 GAME APPLICATIONS (Ch. 30)
he does not find Ruth, Ruth gets one scoop of ice cream and Ned gets
none. Fill in the payoffs in the matrix below.
Hide and Seek
Ned
Ruth
Upstairs Downstairs
Upstairs 1,0 0,1
Downstairs 0,1 1,0
(a) Is this a zero-sum game? Yes. What are the Nash equilibria in
(b) Find a Nash equilibrium in mixed strategies for this game.
Ruth hides upstairs and Ned searches
(c) After years of playing this game, Ned and Ruth think of a way to
liven it up a little. Now if Ned finds Ruth upstairs, he gets two scoops of
ice cream, but if he finds her downstairs, he gets one scoop. If Ned finds
Ruth, she gets no ice cream, but if he doesn’t find her she gets one scoop.
Fill in the payoffs in the graph below.
Advanced Hide and Seek
Ned
Ruth
Upstairs Downstairs
Upstairs 2,0 0,1
Downstairs 0,1 1,0
NAME 371
(d) Are there any Nash equilibria in pure strategies? No. What mixed
time will Ned find Ruth? 1/2.
30.6 (0) Perhaps you have wondered what it could mean that “the meek
shall inherit the earth.” Here is an example, based on the discussion in
the book. In a famous experiment, two psychologists∗put two pigs—a
little one and a big one—into a pen that had a lever at one end and a
trough at the other end. When the lever was pressed, a serving of pigfeed
would appear in a trough at the other end of the pen. If the little pig
would press the lever, then the big pig would eat all of the pigfeed and
keep the little pig from getting any. If the big pig pressed the lever, there
would be time for the little pig to get some of the pigfeed before the big
pig was able to run to the trough and push him away.
Let us represent this situation by a game, in which each pig has two
possible strategies. One strategy is Press the Lever. The other strategy
is Wait at the Trough. If both pigs wait at the trough, neither gets any
feed. If both pigs press the lever, the big pig gets all of the feed and the
little pig gets a poke in the ribs. If the little pig presses the lever and
the big pig waits at the trough, the big pig gets all of the feed and the
little pig has to watch in frustration. If the big pig presses the lever and
the little pig waits at the trough, then the little pig is able to eat 2/3
of the feed before the big pig is able to push him away. The payoffs are
as follows. (These numbers are just made up, but their relative sizes are
consistent with the payoffs in the Baldwin-Meese experiment.)
Big Pig–Little Pig
Little Pig
Big Pig
Press Wait
Press −1,9−1,10
Wait 6,4 0,0
∗Baldwin and Meese (1979), “Social Behavior in Pigs Studied by
Means of Operant Conditioning,” Animal Behavior
372 GAME APPLICATIONS (Ch. 30)
(a) Is there a dominant strategy for the little pig? Yes, Wait. Is
there a dominant strategy for the big pig? No.
(b) Find a Nash equilibrium for this game. Does the game have more than
(Incidentally, while Baldwin and Meese did not interpret this experiment
as a game, the result they observed was the result that would be predicted
by Nash equilibrium.)
30.7 (1) Let’s have another look at the soccer example that was discussed
in the text. But this time, we will generalize the payoff matrix just a little
bit. Suppose the payoff matrix is as follows.
The Free Kick
Goalie
Kicker
Kick Left Kick Right
Jump Left 1,0 0,1
Jump Right 1-p,p 1,0
Now the probability that the kicker will score if he kicks to the left
and the goalie jumps to the right is p. Wewillwanttoseehowthe
equilibrium probabilities change as pchanges.
(a) If the goalie jumps left with probability πG, then if the kicker kicks
right, his probability of scoring is πG.
(b) If the goalie jumps left with probability πG, then if the kicker kicks
left, his probability of scoring is p(1 −πG).
(c) Find the probability πGthat makes kicking left and kicking right lead
to the same probability of scoring for the kicker. (Your answer will be a
function of p.) πG=p
1+p.
