Download Answers

Strategic Behavior – Spring 2012
Homework Assignment 4 - Answers
Due: Wednesday, April 4, 2012
(Please write legibly; if in doubt, type your assignment.)
This homework assignment is based on the material in chapters 7 and 8 of the course text.
1.
(Question 8, chapter 7) It is the closing seconds of a football game, and the losing team has
just scored a touchdown. Now down by only one point, the team decides to go for a twopoint conversion that, if successful, will win the game. The offense chooses between three
possible running plays: run wide left, run wide right, and run up the middle. The defense
decides between defending against a wide run and a run up the middle. The payoff to the
defense is the probability that the offense does not score, and the payoff to the offense is
the probability that it does score. Find all pure and mixed-strategy Nash equilibria. This
question is not easy; think carefully.
Offense
Run wide left Run up middle Run wide right
Defense
Defend against wide run
.6,.4
.4,.6
.6,.4
Defend against run up the middle
.3,.7
.5,.5
.3,.7
Let l be the probability that Offense chooses run wide left, r be the probability that it chooses
run wide right, and 1 – l – r be the probability of middle. The defense is indifferent between its
two strategies when:
l x .6 + (1 – l – r) x .4 + r x .6 = l x .3 + (1 – l - r) x .5 + r x .3
l + r = .25
Let d be the probability that Defense picks defend against wide run. Offense’s expected payoff
from its three pure strategies is:
Wide left: d x .4 + (1 – d) x .7 = .7 - .3d
Middle: d x .6 + (1 – d) x .5 = .5 - .1d
Wide right: d x .4 + (1 – d) x .7 = .7 - .3d
For the offense to be indifferent among them, it must be true that:
.7 - .3d = .5 + .1d
d = .5
This means a Nash equilibrium is any strategy pair in which the defense defends against the
outside run with probability .5 and the offense runs up the middle with probability .75. The
offense can divide up .25 between left and right outside in any way; it is not correct to force the
offense to divide equally between left and right. This percentage must be specified to allow
flexibility. [While (.5; .125) is a NE, it is not the ONLY: (.5; .1, .75) is also.]
Answer: There is one Nash equilibrium; it is in mixed strategies: (1/2, q2 = 3/4), or (.5, q2 = .75),
or (1/2, 1- q2 = .25).
2.
(Question 4, chapter 8) Benjamin Franklin once said, “Laws too gentle are seldom obeyed;
too severe, seldom executed.” To flush out what he had in mind, the game [below] has
three players: a lawmaker, a (typical) citizen, and a judge. The lawmaker chooses among a
law with a gentle penalty, one with a moderate penalty, and a law with a severe penalty.
In response to the law, the citizen decides whether or not to obey it. If she does not obey
it, then the judge decides whether to convict and punish the citizen. Using subgame
perfect Nash equilibrium, find values for the unspecified payoffs (those with letters, not
numbers) that substantiate Franklin’s claim by resulting in lawmaker’s choosing a law with
a moderate penalty.
Lawmaker
Gentle
Moderate
Severe
Citizen
Obey
Judge
Do not obey
Obey
Do not obey
Convict
8
a
c
Do not convict
6
b
d
Do not obey
10
4
8
10
4
8
10
4
8
Obey
Convict
8
e
g
Do not convict
6
f
h
Convict
Don’t
8
i
k
The objective is to find values so that a subgame perfect Nash equilibrium has the following properties:
1) When the law is gentle, the citizen disobeys the law even though he will be convicted
6
j
l
2) When the law is moderate, the citizen obeys the law because he will be convicted
3) When the law is severe, the citizen disobeys the law because he will not be convicted.
Answer: For 1) to occur, we need c > d (so the judge will convict the citizen) and a > 4 (so the citizen
prefers to disobey the law and be convicted than to obey the law). For 2) to occur, we need g > h (so the
judge will convict the citizen) and 4 > e (so the citizen prefers to obey the law rather than disobey and be
convicted). For 3) to occur, we need l > k (so the judge will not convict the citizen and j > 4 (so the
citizen prefers to disobey the law and not be convicted than obey it). By backward induction, the
lawmaker then faces these three alternatives:
Choose a gentle law and receive a payoff of 8 because the citizen disobeys the law and is convicted
Choose a moderate law and receive a payoff of 10 because the citizen obeys the law
Choose a severe law and receive a payoff of 6 because the citizen disobeys the law and is not
convicted.
The lawmaker optimally chooses a moderate law.
3.
(Question 6, chapter 8) In 1842, the Sangamo Journal of Springfield, Illinois, published
letters that criticized James Shields, the auditor of the State of Illinois. Although the letters
were signed “Rebecca,” Shields suspected that it was state legislator Abraham Lincoln who
penned the letters. As shown in [the figure below], Shields considered challenging Lincoln
to a dual, and, as history records, Shields did challenge Lincoln. In response to a challenge,
Lincoln could avoid the duel, or; if he chose to meet Shield’s challenge, he had the right to
choose the weapons. We will also allow Lincoln to decide whether to offer an apology of
sorts. If he decides to go forward with a duel, then Lincoln has four choices: propose guns,
propose guns and offer an apology, propose swords, or propose swords and offer an
apology. In response to any of the four choices, Shields must decide to either go forward
with the duel or stop the duel (and accept the apology if offered). Find all subgame perfect
Nash equilibria.
Shields
5
6
Do not challenge
Challenge
Lincoln
Avoid
Guns
Guns & apology Swords
Swords & apology
Shields
10
1
Stop
0
5
Go
8
2
Stop
6
4
Go
Stop
8
2
0
5
Go
2
3
Stop
6
4
Go
2
3
Begin with Shield’s final four decision nodes. If Lincoln proposes guns, then Shields will go forward with
the dual, as his payoff is 8, while it is 0 if he were to back down. If Lincoln proposes guns with an
apology, then Shields still goes forward with the duel, as his payoff is 8 again, while it is 6 if he were to
back down. If Lincoln proposes swords, then Shields will go forward with the duel as his payoff is 2,
while it is 0 if he were to back down. Finally, if Lincoln proposes swords but with an apology, then
Shields will stop the duel and accept the apology, as the payoff is 6, while it is 2 if he were to actually
battle Lincoln using swords. Substituting that part of the game with the associated payoffs derived using
backward induction gives the game shown below.
Shields
Do not challenge
5
6
Challenge
Lincoln
Avoid
Shields
10
1
Guns
8
2
Guns & apology Swords
8
2
2
3
Swords & apology
6
4
Lincoln chooses between his five choices in the event that Shields has challenged him to a duel.
Lincoln’s optimal choice is to accept the challenge, propose that they sue swords, and offer an apology.
In response to that choice, Shields will accept the apology and the duel will be avoided, which means a
payoff of 4 for Lincoln. If Lincoln makes any of the other three choices that involve going forward with
the duel, his payoff is no higher than 3, as the duel will commence. Avoiding the duel is even worse,
with a payoff of 1. Moving up to Shields’s initial decision node, his payoff is 5 from not challenging
Lincoln and is 6 from doing so. Thus, Shields challenges Lincoln.
Answer: The subgame perfect Nash equilibrium strategy profile is [challenge/go/go/go/stop; swords
and apology].
4.
Suppose you are analyzing a simple two-stage bargaining game, in which the first move is a
proposal of how to divide $4, which must be either $3 for the proposer and $1 for the
responder or $2 for each. The responder sees the proposal and either accepts (which
implements the split) or rejects the offer, which results in $0 for each. The responder’s
strategy must specify a reaction to each possible proposal. First, indicate the pure strategy
Nash equilibrium(ia). Then indicate how the concept of subgame perfect Nash equilibrium
can eliminate seeming irrationality within a Nash equilibrium. Your answer should explain
the value of the concept with respect to the usefulness of game theory and Nash
equilibrium in general.
Proposer
P gets $3; R gets $1
Each gets $2
Responder
Accept
Reject
3
1
Accept
2
2
0
0
Reject
0
0
Responder
Proposer
A/A A/R R/A R/R
P = $3; R = $1 3, 1 3,1 0,0 0,0
P = $2; R = $2 2,2 0,0 2,2 0,0
Answer: There are three Nash Equilibria:
(P = $3; R = $1, A/A), which is SPNE
(P = $3; R = $1, A/R)
(P = $2; R = $2, R/A)
It would not be optimal for the Proposer to offer $2, $2, knowing he could instead offer $3, $1 and get a
higher payoff of $3. While in strategic form offering $2, $2 appears as a NE, the extensive form clearly
indicates that $3, $1 is optimal since the Proposer gets $3 rather than $2. SPNE adds predictive power
by ruling out equilibria that are sustained by a player making an irrational choice.
5.
Carefully write the first section of your paper. This should include the introduction to your paper
generally, the scenario you are modeling, the players, and their objectives. It also should include
your extensive form diagram and your strategic form diagram. This part of your homework
assignment should be typed (your extensive and strategic form diagrams may be handwritten and
submitted with the problems) and sent to me via e-mail as a word document.