# University of California Irvine Game Theory Questions

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Problem Set 6, Econ 171
Question 1. Carl and Karen have four candy bars to divide between them One
is a Snickers bar, one is a Milky Way, one is a Kit-Kat and one is a Baby Ruth.
They have decided to divide them in the following way. Karen gets to choose
a bar first. Carl can then choose two of the remaining bars. Karen will then
get the bar that has not been chosen as well as the bar she chose Both know
cach others preferences. Karen likes Snickers best (payo? 4), Milky Way next
(payo? 3), then Kit-Kat (payo? 2) and last Baby Ruth (payo? 1) Carl likes
Milky Way best (payo? 4), Baby Ruth next (payo? 3), then Kit-Kat (payo?
2) and last Snickers (payo? 1). Their total payo?s are the sum of their payo?s
from the two candy bars they get.
A) Show this game in extensive form.
B) How many strategies are possible for Karen? How many strategies are
possible for Carl?
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C) Show the extensive form for each of the proper subgames of this game
and identify the Nash equilibrium in each of them
D) What are the strategies used by each player in a subgame perfect Nash
equilibrium. What are the total payo?s to each in this equilibrium?
E) Find a Nash equilibrium profile that is not subgame perfect and has a
higher payo? for Carl than the subgame perfect Nash. Can you describe this
strategy in simple language that a kid could understand?
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Question 2.
A) For the game shown above, find all of the subgame perfect Nash equilibrium strategy profiles.
B) For this game, are there any Nash equilibria that are not subgame perfect?
If so, find it or them.
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Question 3. Alice and Bob had a terrible fight last week. Alice doesnt want
to see Bob ever again. Bob would like to see Alice and try to smooth things
over. This Saturday night there are two parties. Each of them plans to go to
one of the parties. Alice would prefer Party A to Party B, if it werent for the
question of where Bob will be. Alices payo? from going to Party A would be 3
if Bob is not there and 3 Y if Bob is there. Her payo? from Party B would
be 1 if Bob is not there and 1 Y if Bob is there. Bob would otherwise prefer
Party B to Party A, but he would like to be at the same party that Alice goes
to. Bobs payo? form Party A would be 2 if Alice is there and 0 if Alice is not
there. His payo? from Party B would be 3 if Alice is there and 1 if Alice is not
there.
Bob chooses a party and goes there before Alice decides where to go. Alice
has a friend who will phone her and tell her which party Bob went to. After
finding out where Bob went, Alice decides which party to go to. Bob knows
that this is what Alice will do.
A) Show this game in extensive form.
B) Show this game in strategic form.
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C) Find all of the pure strategy Nash equilibria if Y = 1.
D) Find all of the pure strategy Nash equilibria if Y = 3.
E) Find all of the subgame perfect pure strategy Nash equilibria if Y = 1.
F) Find all of the subgame perfect pure strategy Nash equilibria if Y = 3.
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Problem 4. Val and Earl are at work on a very hot afternoon. There is a
job waiting for them. It takes only one person to do the job. The one who
does it must crawl through a mucky culvert to connect a pipe. Once the pipe is
connected, they can both go to a comfortable tavern for a beer. The game tree
is shown below, where at the terminal nodes, the top number is Vals payo? and
the bottom number is Earls payo?.
At the beginning of play, Earl is working on another job, but Val could do
the job and if he does it right away, they can both go out for beer. Val could
choose to stall, waiting for Earl to come back. If Val decides to stall, then when
Earl gets back, he could do the job, or he could stall. If Earl chooses to stall,
Val could either do the job, or stall again, and so on… If Val and Earl both
stall until Earls final decision node at the end of the workday, Earl can either
do the job, or they can both leave the job undone. The payo?s to both from
leaving the job undone are given by the variable X.
A) How many proper subgames does this game have?
B) List all of the possible strategies for Earl and all of the possible strategies
for Val.
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