University of California Irvine Game Theory Economics Worksheet
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Problem Set 8, Econ 171
Problem 1. Two people are quarreling over a resource. Person 2 does not know
whether Person 1 is weak or strong. She assigns a probability of p to Person 1
being strong. Person 1 knows exactly how strong both people are. Each person
must choose to fight or to yield without knowing whether the other player will
fight or yield.
A person who yields will get a payo? of 0, no matter what the other person
does. If either person fights and the other yields, the person who fights will get
a payo? of 1. If both players fight, then: (a) if Player 1 is weak, Player 2 will
get a payo? of 1 and Player 1 will get -2 (b) if Player 1 is strong, Player 2 will
get a payo? of -2 and Player 1 will get a payo? of 1.
A) Show this game in extensive form, where Nature moves first by making
Player 1 either weak or strong.
B) In this formulation as a Bayesian game, list the strategies that are possible
for Player 1. Also list the strategies that are possible for Player 2.
C) If Player 2 uses the strategy Yield, what strategy is the best response for
Player 1. (Remember that a strategy specifies what she will do if she is weak
and what she will do if she is strong.)
D) For what values of p is there a Bayes-Nash equilibrium in which Player
2 yields and Player 1 fights, no matter whether she is weak or strong? Explain
your answer.
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D) Suppose that player 2 chooses the strategy Fight. What is the best
response for Player 1?
E) If Player 1 fights if strong and yields if weak, what is the expected payo?
to Player 2 from playing the strategy Fight?
F) For what values of p is there a Bayes-Nash equilibrium in which Player 2
fights and Player 1 fights if strong and yields if weak? Explain your answer.
Problem 2. A professor plays a simplified poker game with one of his students.
The rules are as follows. The student and the professor each put $1 in a pot.
The professor randomly draws a card from a deck of cards that consists of 4
kings and 4 queens. The professor looks at the card that he drew but does
not show it to the student. The professor then must either fold or bet.
If the professor folds, the game ends and all of the money in the pot goes to
the student, so the professor loses $1 and the student wins $1. If the professor
bets, then he must put another dollar in the pot. In this case, the student must
decide whether to fold or bet. If the student folds, the game ends and all of
the money in the pot goes to the professor. If the student bets, she must add
another dollar to the pot and then the professor must show the card that he
has drawn. If the card is a king, the professor gets all of the money that is in
the pot. If the card is a queen, the student gets all of the money in the pot.
A) Is this a constant sum game?
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B) Show this game in extensive form.
C) How many strategies are possible for the professor? How many for the
student?
D) Show this game in strategic form.
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E) How many pure strategy Nash equilibria does this game have?
Problem 3. Here we look for a mixed strategy Bayes-Nash equilibrium in the
simplified poker game of Question 3.
A) If the student always folds, what is the professors best response strategy?
B) if the student always bets, what is the professors best response strategy?
C) Suppose that the student chooses to bet with probability b > 0 and to fold
with probability 1 b. For what probability b would the professor be indi?erent
between folding and betting when he has drawn a queen?
D) Suppose that the student knows that the professor always bets when he
draws a king and that he bets with probability q when he draws a queen. If the
professor bets, what is the probability that he has drawn a king? (your answer
will depend on q).Explain.
E) Suppose that the professor always bets when he draws a king and bets
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with probability q when he draws a queen. For what probability q would the
student be indi?erent between betting and folding?
F) What are the mixed strategies for the professor and the student in a
mixed strategy Nash equilibrium?
Problem 4. An antique table is for sale in a sealed bid auction. It will go
to the high bidder at the price the high bidder bids. You dont know if it is a
fake or not, but you do know that 20% of all antiques that look like this one are
fakes. You are not able to have an appraiser examine it. If it is a fake, you will
know this after you buy it and it will be worthless to you. If it is real, it will be
worth $1,000 to you. You know that the only other possible bidder is an expert
antique dealer has also looked at this table and you know that this dealer can
always tell a fake from a real antique. You know that if the antique dealer finds
that it is a fake, she will bid zero for it. If she finds that it is real, she will bid
$500 for it if she has a similar table in stock, and she will bid $800 for it if she
does not have another table like it in stock. Suppose that you believe that it is
equally likely that the dealer has another table like it in stock.
A) What is the probability that the antique dealer will bid $500 for the
table? What is the probability that she will bid $800 for the table?
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B) If you bid $300 for the table and you are the high bidder, what is the
probability that the table is a genuine antique? What is your expected profit
(or loss) if you bid $300? Explain your answers.
C) If you bid $501 for the table, what is the probability that you will be the
high bidder. What is the probability that the table is genuine if you bid $501
and are the high bidder? What is your expected profit (or loss) if you bid $501?
Explain your answers.
D) If you bid $801 for the table, what is the probability that you will be
the high bidder? What is your expected value for the table if you are the high
bidder? What is your expected profit (or loss) if you bid $801?
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D) Suppose you could choose to bid any amount between 0 and $1000 for the
table. What bid would maximize your expected payo?? Explain your answer.
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6 Pages
Tags:
values
economics
Game Theory
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