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Game Theory Problems

QUESTION 1. (Flat Tire and Coordination) Two college students, Al and Bob, very
confident about their mid-term exam performance, decided to attend a party the weekend
before the final exam. The party was so good that they overslept the whole Sunday. Instead
of taking the final exam unprepared on Monday, they pleaded to the professor to give them a
make-up exam. Their excuse was a flat tire without a spare and any help. The professor,
surprisingly, agreed. On Tuesday morning, the professor placed them in separate rooms and
handed them the test. The test had just one question: which tire?
.

(a) Suppose the final score is entirely based on the make-up test. If Al and Bob give the
same answer on the flat tire question, both pass the exam, yielding a payoff of 10 for each of
them. If Al and Bob, however, give different answers, both fail the exam, getting payoff
zero. Write down the game table of this simultaneous-move game between Al and Bob.

.

(b) Find all pure-strategy Nash equilibria of this game.

.

(c) Find two mixed-strategy Nash equilibria of this game with the following requirements:
in one of the mixed Nash equilibrium, each player randomizes using two pure strategies,
and in the other mixed Nash equilibrium, each player randomizes using three or four pure
strategies.

QUESTION 2. (The Employee Monitoring Game) Consider the employee monitoring
game discussed in class (Lecture Notes 3, page 18): Suppose an employee in a company can
either work or shirk. The manager of the company cannot directly observe whether the
employee is working or not. However, the manager can pay a cost of $10 to find out the
employees behavior. If the manager finds out that the employee shirks, the manager can
legally pay nothing to the employee. If, however, the manager has no hard evidence about
the employees shirking, the manager has to pay the employee $100. We assume that the
manager will get a payoff of $200 if the employee works and $0 if the employee shirks. In
addition, we assume that it costs the employee $50 if he works.
1. Write down the game table of the above simultaneous-move game. What are the

(complete) strategy sets of the players?
2. Find all (pure-strategy and mixed-strategy) Nash equilibria for the above game.

QUESTION 3. (First-Price Sealed-Bid Auction) Alice is selling her 2000 Chevrolet
cavalier to her friends, Bob and Charles. It is commonly known that Bob attaches a value of
$6000 to Alice’s old car, while Charles’s value of the car is $6500. Alice designs the
following auction to sell her car: First, she asks each of them to write his bid on a piece of
paper. Then Bob and Charles give their bids to Alice. Notice that when Bob and Charles
write down their bids, they don’t know each other’s bid (hence the name “sealed bid”). After
Alice receives the sealed bids, the bids are shown to everyone and the car will be sold to the
person who has a higher bid. When there is a tie (Bob and Charles bid the same amount),
then Alice would flip a fair coin to decide who will get the car.

.

(a) Write down the normal form representation of this game.

.

(b) Find weakly dominated strategies for both Bob and Charles. Is “bidding 0” a weakly
dominated strategy for either player? Explain.

.

(c) Find the set of all pure-strategy Nash equilibria for this game.

QUESTION 4. (Cournot Duopoly Revisited) Consider the Cournot duopoly model where
the (inverse) demand is P (Q) = a – Q. The two firms now have asymmetric marginal costs:
c1 for firm 1 and c2 for firm 2.
(a) What is the Nash equilibrium if 0 < ci < a/2, i ? {1,2}for each firm? (b) What is the Nash equilibrium if 0 < c1 < c2 < a, but 2c2 > a+c1?

QUESTION 5. (Dominance by Mixed Strategies and Mixed Strategy Nash Equilibrium)
Consider the following 3 x 3 simultaneous-move game (“3 x 3” means that both players
have three pure strategies).
L

C

R

T`

1, 1

0, 0

2, 0.3

M

0, 0

1, 1

1, 0.3

B

0.3, a 0.3, b

0.5, c

(a) Given that all you know about the game is the above game table and that a, b and c is
that they are three arbitrary constants, find all strictly dominated strategies for the players
(Hint: One strategic consequence of introducing mixed strategies to a game is that now a
pure strategy can be strictly dominated by a mixed strategy, which is the case for one of the
players in the above game.)
(b) Find all (pure-strategy and mixed-strategy) Nash equilibria for the above game.

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