Review Sheet and Practice Final Exam: FE450 Fall 2011
Key Ideas:
1) Be able to define do a Harsanyi Transformation. That is, take a static game with uncertainty,
create a new game matrix and solve for the pure strategy NE.
2) Be able to explain the differences between Auctions: first-price, second-price, English, Dutch,
Vickery, open/closed, sealed/oral, etc.
3) Be able to solve for the NE for first-price and second-price sealed-bid auctions with IPV
(independent private values).
4) Be able to set up a game tree for a perfect Bayesian equilibrium. Identify payoffs and information
nodes.
5) Be able to identify separating equilibria in a signaling game. Be able to identify pooling equilibria.
The exam will have 5 of 6 problems (20 points each); 2 from the new material and the rest comprehensive
(so study all the questions)
Suppose that you are involved in a legal suit. If you hire a “good” lawyer you will win the lawsuit
75% of the time. If you hire a “bad” lawyer, you will win the lawsuit 25% of the time. Suppose
that you find a lawyer who is willing to represent you, and demands a wage, w. If you do not hire
the lawyer or the lawyer does not seek your case, you will lose the case with certainty. If you win
you earn $20,000, if you lose you earn $0.
The ordering of the game is as follows:
1) Nature moves first and determines if the lawyer is good (G) or bad (B).
2) The lawyer decides to go to a prestigious law school (P) or a non-prestigious school. For the
“good” lawyer, a prestigious law school costs $3,000 and for the “bad” lawyer a prestigious law
school costs $6,000. The cost for a non-prestigious law school (NP) is equal to $0 for both types
of lawyers.
3) After deciding on qualifications (where to go to law school), the lawyer can decide to seek your
case or not. If the lawyer does not seek your case, she can earn $4000 elsewhere (her opportunity
cost is $4,000).
4) If the lawyer seeks your case, she demands a wage, w. The lawyer’s utility function is the wage
minus their cost for education and the amount they can earn elsewhere.
5) You observe the wage demand, w and then decide to hire or not hire the lawyer.
Illustrate this with an appropriate game tree. Find the wages for a separating equilibrium.
1.
2.
The US Military has instituted a draft for the current war on terrorism. Once a person is qualified
for service, they would have to make a decision of whether to volunteer or be drafted. If a person
serves, the person will lose all civilian wages, and also take on the burden of possibly being
wounded or worse. This cost of service has been estimated to be $1200. However, if a person
volunteers, they will receive a bonus of $700 to offset some of the cost for serving. If a person
does not serve, they earn $0. Suppose there are two people, and that if neither person volunteers, a
person will be chosen randomly to serve (each player has a ½ chance of being chosen to serve if
neither volunteers):
a) Set up the appropriate game if each person makes his or her decision sequentially. Be sure to
calculate the appropriate payoffs (payoffs should be numbers).
b) What does the bonus have to be in order for all service members to be just indifferent to
volunteering and being drafted?
c) Given the $700 bonus, what does the probability of being drafted have to be in order for player 1
to be just indifferent to volunteering and being drafted? That is, when will person 1 volunteer if
the probability of being drafted is no longer equal to ½ for each person but the probability is now
based on personal characteristics such as speed, intelligence, and character?
3.
Consider the following modification to the battle of the sexes game. Husband (H) and Wife (W)
have to decide whether to go to the football game (F) or the opera (O). Suppose that the husband
is not sure about the wife’s preferences. In particular, he does not know if his wife likes to be with
him (Type = Loving) or if she prefers to go to either event by herself (Type = Leaving). Let the
payoffs associated with each type be given in the following table:
Type: LOVING
Type: LEAVING
Wife
Wife
F
O
F
O
Husband
F
4,2
0,0
Husband
F
4,0
0,3
O
0,0
1,3
O
0,2
1,0
Note that the payoffs listed above illustrate that a loving type wife prefers to be with her husband while a
leaving type wife prefers to be alone. Assume the wife knows her type, the husband attaches a probability
p to the type loving and (1-p) to the type leaving, and that the wife knows her husband’s estimate of p.
a) Find an equilibrium for this game when p=0 (ie, the wife is type Leaving).
A pure strategy Bayes-Nash equilibrium consists of a pure strategy for the husband and a pure strategy for
each type of wife. For this question, you can list a set of pure strategies in the following order {Husband
strategy, Loving Wife strategy, Leaving Wife strategy}. Given the setup, the following pure strategy
profiles are possible: {F,F,O}{F,F,O}{F,O,F}{F,O,O}{O,F,F}{O,F,O}{O,O,F}{O,O,O}
b) What is the lowest value of p for which the pure strategy profile {F,F,O} is a Bayes-Nash
equilibrium (hint: it might be useful for you to draw the extensive form game tree for this game,
with nature moving first to determine the type of wife).
c) For high values of p, there are multiple pure strategy Bayes-Nash Equilibria (to see this, think
about the possible Equilibria when p=1). What is the lowest value of p for which there are
multiple pure strategy Bayes-Nash Equilibria, and what are they?
OLDER MATERIAL:
4. Consider a market with only two firms producing outputs q 1 and q2 and the following attributes:
Demand Curve: P= 30 – ¼ Q;
Q=q1+q2
Marginal Costs: MC=0 for both firms
Fixed Costs:
FC = 0 for both firms
a) Assume the two firms are Cournot competitors (i.e. they compete in quantities). Find the
Cournot-Nash Equilibrium. BE SURE TO CLEARLY LABEL THE EQUILIBIRUM
QUANTITIES, THE MARKET CLEARING PRICE AND THE PROFITS OF BOTH
FIRMS
b) Assume the two firms decide to collude, each charging the monopoly price and producing ½
of the monopoly output. What quantities will each firm produce, what will be the market
clearing price, and what will be each firms profits?
c) If one firm decides to cheat on the agreement that both firms will produce ½ of the monopoly
output, what quantity will the cheating firm produce? What will be the corresponding
profits for each of the two firms (the ‘cheater’ and the firm that makes ½ monopoly output,
the ‘colluder’)?
d) Now imagine that the game is infinitely repeated. Describe a grim strategy that has each firm
produce ½ of the monopoly output. For what range of the discount factor (δ), if any, can this
grim strategy be supported as a subgame perfect equilibrium?
5.
Consider the game paper, scissors, rock. Each player chooses one of the three strategies (paper,
scissors, or rock). The choices are made simultaneously and without either player knowing what
the other has chosen. The outcomes are determined as follows: paper beats rock, rock beats
scissors, and scissors beats paper. A draw occurs whenever the players choose the same strategy.
The payoffs associated with these outcomes are as follows. A paper-rock outcome results in the
exchange of $1 from rock to paper. Similarly, a rock-scissor outcome results in the exchange of $1
from scissor to rock. However, a scissor-paper outcome results in an exchange of $5 from paper to
scissor. In the case of a draw, no money is exchanged.
a) Draw the normal form of this game, and verify that no pure strategy equilibria exist.
b) State the equations that must be solved in order to calculate the mixed strategy equilibrium of
this game.