Strategic Behavior – Spring 2012
Homework Assignment 3 - Answers
Due: Wednesday, March 14, 2012
(Please write legibly; if in doubt, type your assignment.)
This homework assignment is based on the material in chapters 4 and 7 of the course text.
1.
(Chapter 4, Question 2) Consider a modification of driving conventions, shown in the Figure
below, in which each player has a third strategy: to zigzag on the road. Suppose if a player
chooses zigzag, the chances of an accident are the same whether the other player drives on
the left, drives on the right, or zigzags as well. Let that payoff be 0, so that it lies between
(-1), the payoff when a collision occurs for sure, and 1, the payoff when a collision does
not occur. Find all pure strategy Nash equilibria.
Drive Left
Drive Left
1, 1
Drive Right -1, -1
Drive Zigzag 0, 0
Drive Right
-1, -1
1, 1
0, 0
Drive Zigzag
0, 0
0, 0
0, 0
Answer: In addition to (drive left, drive left) and (drive right, drive right), (zigzag, zigzag)
is also a Nash equilibrium.
2.
(Chapter 4, Question 4) Consider the two-player game illustrated in the figure below.
a. For each player, derive those strategies that survive the iterative deletion of strictly
dominated strategies.
b. Derive all strategy pairs that are Nash equilibria.
X
Y
A 4, 0 2, 1
B 2, 2 3, 4
C 2, 3 1, 2
Z
3, 2
0, 1
0, 3
Answer:
a) A strictly dominates C as it gives a strictly higher payoff for each strategy of player 2.
Neither A nor B is strictly dominated as A gives the highest payoff when player 2 uses X,
and B gives the highest payoff when player 2 uses Y. None of player 2’s strategies is
strictly dominated as X is best when player 1 uses C, Y is best when player 1 uses B, and Z
is best when player 1 uses A (or C). After round 1, the surviving strategies are then A and B
for player 1 and X, Y, and Z for player 2. After deletion of strategy C, player 2 can delete
strategy X, which is dominated by Y. This leaves the following strategies that survive
IDSDS: A, B, Y, and Z.
b) There are two Nash equilibria: (B,Y) and (A, Z).
3.
(Based on question 1, chapter 7) Reproduced below is the telephone game from Section 4.2.
Find all Nash equilibria in pure and mixed strategies.
Winnie
Call Wait
Colleen
Call
0,0
2,3
Wait
3,2
1,1
Answer: Let p be the probability that Colleen chooses call, and q be the probability that
Winnie chooses call. For Colleen to be indifferent, Winnie must randomize in such a way
that Colleen’s expected payoff from call is the same as her expected payoff from wait:
q x 0 + (1-q) x 2 = q x 3 + (1-q) x 1
q=¼
If Winnie calls with probability 1/4, then Colleen is indifferent between calling and
waiting.
For Winnie to find it optimal to randomize, she has to receive the same expected payoff
from her two pure strategies, which requires that Colleen choose p so that:
p x 0 + (1-p) x 2 = p x 3 + (1-p) x 1
p=¼
The game is symmetric, so it makes sense that the probabilities are symmetric (i.e., the
equilibrium is symmetric). This game has three Nash equilibra:
(Call, Wait) [Colleen chooses call and Winnie chooses wait.]
(Wait, Call) [Colleen chooses wait and Winnie chooses call.]
(¼, ¼) [Both choose call with probability ¼ and wait with probability ¾.]
4.
(Question 2, chapter 7) The count is three balls and two strikes, and the bases are empty.
The batter wants to maximize the probability of getting a hit or a walk, while the pitcher
wants to minimize this probability. The pitcher has to decide whether to throw a fast ball or
a curve ball, while the batter has to decide whether to prepare for a fast ball or a curve ball.
The strategic form of this game is shown below. Find all the Nash equilibria in pure and
mixed strategies.
Pitcher
Fastball Curveball
Batter
Fastball
.35, .65
.3,.7
Curveball
.2,.8
.5,.5
Answer: There is no Nash equilibrium in pure strategies. To see if there is a mixed strategy
NE, let p be the frequency with which a batter prepares for a fastball, and q be the frequency
with which a pitcher throws a fastball.
p x .65 + (1-p) x .8 = p x .7 + (1-p) x .5
p = 6/7 86%
q x .35 + (1-q) x .3 = q x .2 + (1-q) x .5
q = 4/7 57%
NE: (6/7, 4/7). [This means that in equilibrium, the pitcher throws a fastball 57% of the
time, and the batter prepares for a fastball 86% of the time.]
5.
Briefly explain your subject for your term paper in this class. Include the scenario, players,
and players’ objectives. Recall that your paper is to be 7 to 10 pages (double-spaced with
one-inch margins), and should discuss a scenario from a book, a movie, or a historical
event.