Geir B. Asheim, 23 September 2009
ECON5200 ADVANCED MICROECONOMICS, fall 2009
Problems for the third seminar
Problem 1
What is meant by Nash equilibrium in a strategic game? Consider an
auction between two bidders, bidder 1 and bidder 2. Bidder 1 has a valuation
that equals 10 and bidder 2 has a valuation that equals 4. These valuations
are commonly known among the players. The bids can be integers from 0
to 12; i.e. there are thirteen different bids that can be made by each bidder:
0, 1, 2, . . . , 12. If the bids are different, the bidder that submits the highest
bid wins. If the bids are equal, each wins with equal probability. In both
cases, the winner pays his own bid. What is the set of Nash equilibria of
this game?
Problem 2
What is meant by a Bayesian game? Consider, as in Problem 1, an
auction between two bidders, bidder 1 and bidder 2. However, assume now
that each bidder knows his own valuation, but not the valuation of the
other bidder. The valuations are uniformly and independently distributed
on {0, 1, 2} × {0, 1, 2}. I.e., for each of integers 0, 1 and 2, the one bidder
assigns probability 1/3 to the event that the other’s valuation equals this
integer. The bids can be integers from 0 to 2; i.e., there are three different
bids that can be made by each bidder: 0, 1, and 2. If the bids are different,
the bidder that submits the highest bid wins. If the bids are equal, each
wins with equal probability. In both cases, the winner pays his own bid.
Find a symmetric Nash equilibrium of the resulting Bayesian game.
Problem 3
Consider the “guess the number” game: n players pick an integer in the
set {1, 2, . . . , 99, 100}. A given prize is shared by the players choosing the
integer(s) closest to 32 of the mean.
(a) For each strategy in {1, 2, . . . , 99}, show that this strategy can be
a best responses to some belief (i.e., no strategy in {1, 2, . . . , 99} is
strictly dominated).
1
2
(b) Find a pure or mixed strategy that strictly dominates 100 (implying
that 100 is a never-best response).
(c) Use O&R Definition 55.1 to show that 1 is the only rationalizable
strategy.
(d) How many rounds of iterated elimination of strictly dominated strategies are (at least) needed to eliminate all strategies above 67? How
many rounds of iterated elimination of strictly dominated strategies
are (at least) needed to eliminate all strategies but the only rationalizable strategy, 1?
(e) Show that there is a unique Nash equilibrium, in which all players
choose 1.
Problem 4
(a) In a δ-discounted infinitely repeated game, what is a simple strategy
profile?
(b) In a δ-discounted infinitely repeated game, how can it be checked
that a simple strategy profile is a subgame perfect equilibrium? In
particular, why does the one deviation property hold?
L R
T x, x -1, 5
B 5, -1 0, 0
(c) Let a δ-discounted infinitely repeated game have the stage given
above. Consider the following paths:
(a(1)t ) = ((T, R), (B, L), (T, R), (B, L), . . . )
(a(2)t ) = ((B, L), (T, R), (B, L), (T, R), . . . )
For what values of x and the discount factor δ will the simple strategy
profiles σ((a(1)t ), (a(1)t ), (a(2)t )) and σ((a(2)t ), (a(1)t ), (a(2)t )) be
subgame perfect equilibria?
Problem 5
What is meant by a sequential equilibrium?
Problem 6
Consider the following game, where there are three players: One worker
and two firms. The structure of the game is as follows: Nature chooses
first (with probability equal to 21 for both alternatives) whether the worker
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is productive or not. The worker is told about his own productivity and
chooses whether to go through an educational program or not. The firms
observes whether the worker has gone through the educational program or
not, but does not observe whether the worker is productive or not. Then
each of the firms choose simultaneously a wage from the set {0, 2, 4}. The
probability that a firm employs the worker is 1 if the competitor chooses
a lower wage, 21 is the competitor chooses the same wage, and 0 if the
competitor chooses a higher wage. The players’ payoffs are as follows:
• A worker (independently of productivity) without education: wage.
• A productive worker with education: wage − 3.
• An unproductive worker with education: wage − 5.
• Firm if worker is productive: employment probability · (8 − wage).
• Firm if worker is unproductive: employment probability · (0 − wage).
The structure of the game is commonly known.
Does there exist a sequential equilibrium where an unproductive worker
go through the educational program? Find a (separating) sequential equilibrium in which the worker go through the educational program if and only
if he is productive.
Problem 7
Consider the game of Problem 6. Find a (pooling) sequential equilibrium
in which the worker (independently of productivity) does not go through
the educational program. If you conclude that the wage level for a worker
without education is uniquely determined, report what this wage level is.
If you conclude that there are (pooling) separating equilibria with different
wage levels for a worker without education, try to argue that one of these
wage levels is more “reasonable”.
Problem 8
In the equilibrium/a of Problem 7 (in which the worker independently
of productivity does not go through the educational program), what limitations does the consistency requirement of the sequential equilibrium concept
impose on what the firms believe about the productivity about a worker who
does not follow his equilibrium strategy, but goes through the educational
program after all? Can the two firms in this/these equilibrium/a have different beliefs about a worker with education?