Econ 106G
Spring 2013
Practice Problems 5
You do not need to submit answers to these problems. I still encourage you to
think them through thoroughly and write down your answers before looking at
the solutions.
1. Game Theory Concepts, True or False: Answer if the following statements are
true or false. If they are false, give an example that contradicts the statement.
(a) In every game every player has a strictly dominant strategy.
(b) Every normal-form game has a Nash equilibrium in pure strategies.
(c) Every normal-form game with 2 players and 2 actions has a Nash equilibrium in
pure or mixed strategies.
(d) Every …nite extensive-from game with perfect information has a backward induction solution.
(e) Every Subgame-Perfect Nash Equilibrium is a Nash equilibrium.
(f) Every Nash Equilibrium is a Subgame-Perfect Nash equilibrium.
2. The Final Exam: When a professor is giving an exam, his utility is decreasing in the
e¤ort it takes him to make up new questions but he also cares about how students learn
and pass the class (see below). He has two choices: Make up new questions (“new”)
or reuse questions from old exams (“old”). In preparing for the exam, students also
have two choices: spend their time trying to learn the course material (“learn”), or
trying to …nd out what questions were asked in the past and memorize the answers
(“memorize”).
As for the payo¤s, assume that the professor su¤ers an e¤ort cost of e = 1 when he
makes up new questions (this is his utility both in the case (“new”,“learn”) and in the
case (“new”,“memorize”)), a frustration cost of f = 4 when he used old questions and
students just memorized them, and a utility of 0 when he used an old exam but the
students still learned the course material (professors are unfortunate creatures who can
at most get a utility of 0. How sad...).
Students get a utility of 1 when they prepared for the exam correctly, i.e. study old
exams if the professor gives old questions, and learn the course material if he gives new
questions, and a utility of 0 otherwise.
(a) Model this situation as a 2
2 normal-form game including payo¤s!
(b) Show that this game admits no Nash equilibrium in pure strategies!
(c) Calculate the best-response functions in mixed strategies and draw them in (p; q)space (where p; q 2 [0; 1] are the mixing probabilities of the student and the
professor) for this game!
1
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(d) What is the mixed-strategy Nash equilibrium (p ; q ), both for the above numbers
and as a function of f and e?
(e) 30 years later the professor has grown old and mellow and his frustration cost has
decreased to f = 2. How does the Nash equilibrium (p ; q ) change as a reaction!
3. Limit Capacity: Two …rms, A and B, will engage in Cournot competition for widgets.
Before they do so, either …rm can invest into their production capacity by building
either a small plant, a large plant, or no plant at all. A small plant costs $50; 000 and
has a capacity of producing up to 100 widgets. A large plant costs $175; 000 and has
unlimited capacity. The timing is
Investment stage:
– First, …rm A invests either small or large
– Then, …rm B invests small, large, or not at all
Production stage: Firms simultaneously produce qA ; qB widgets (if they invested,
and up to their capacity)
Marginal costs of production are 0 and the inverse demand function is given by p (Q) =
900 Q, where Q = qA + qB is the total number of produced widgets.
(a) Assume for the moment that …rm B does not exist, or equivalently that it does
not invest. What is the optimal strategy for …rm A?
(b) Sketch the game tree and solve for subgame-perfect equilibrium with generalized
backward induction! (Hint: To do so, start by solving the six production subgames, i.e. solving for the equilibrium qA ; qB after both …rm invested large, both
…rms invested small, etc.) Why does …rm A choose to build a di¤erent size plant
than in part (a)?
(c) If …rm B could commit to a plant size (small, large, or none) in advance, and A
would have to respond optimally to this action, which plant size would B choose?
(I.e. First B invests small, large or not at all, then A invests small or large, and
then they play Cournot.)
(d) Now consider a variation of the game, where the …rms simultaneously decide to
invest into no plant, a small plant or a large plant (A can also invest into no plant
now), before engaging in Cournot competition: What are the subgame-perfect
equilibria of this game? (Hint: The investment stage is now a simultaneous 3 3
game, in which the …rms’payo¤s are given by the payo¤s calculated in part (b).)
4. Group Projects: Two students work on a joint project for presentation in class.
Each one of them can exert e¤ort x1 ; x2 2 [0; 1] on the project. The material is very
tedious, so e¤ort x implies a cost of 3x. On the other hand their grade in the class
depends crucially on the quality of the presentation. More speci…cally, if they exert
2
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e¤ort x1 ; x2 , their gross payo¤ from the resulting grade is 2 (x1 + x2 ). Thus utility is
given by
u1 (x1 ; x2 ) = 2 (x1 + x2 )
u2 (x2 ; x1 ) = 2 (x1 + x2 )
3x1
3x2
(a) What is the unique Nash equilibrium (x1 ; x2 ) of this one-shot game? What are
the Nash equilibrium utilities (u1 ; u2 )?
(b) If the students can jointly decide on equal e¤ort levels x01 = x02 = x0 , what e¤ort
level would they choose? What are the resulting utilities (u01 ; u02 )?
(c) Assume that the students have agreed on e¤ort levels x01 = x02 . If student 1 does
not stick to the agreement, what is his best response x00 = BR1 (x2 = x0 )? What
is the resulting utility u001 of student 1?
Now suppose that the students engage repeatedly in homework projects, i.e. they
play the above stage game in…nitely often and discount payo¤s at a rate of per
round. We want to study whether e¤ort levels x1 = x2 = x0 on the equilibrium
path can be sustained by the following “grim-trigger”strategies
1. If both students have always exerted e¤ort x1 = x2 = x0 exert e¤ort x0
2. Otherwise exert the Nash equilibrium e¤ort (x1 ; x2 )
(d) Assuming that student 2 is employing this grim-trigger strategy, calculate student
1’s discounted pro…t from sticking to this strategy by exerting e¤ort x0 and the
discounted pro…t from deviating and exerting e¤ort x00 !
(e) Under what condition on is the grim-trigger strategy a subgame perfect Nash
equilibrium? (it is ok if you only check the constraint on the equilibrium path)
(f) Now assume that there is an arbitrary number N of students and utilities are
given by
N
X
ui (xi ; x i ) = 2
xj 3xi
j=1
How do your answers to questions b,c,d,e change? Explain why cooperation is
possible for less patient students as N increases.
5. Raising External Equity: You want to sell a 50% stake in your start-up to an
outside investor to raise capital and grow the start-up. The start-up will succeed with
probability q, which is independent of the sale, but the value of the start-up in case of
success is $800k if the capital is raised, and $700k otherwise. If the start-up does not
succeed it is worth 0. There is an exogenous price p, and you have to choose whether
or not to sell the 50% stake at this price and the investor has to decide whether or not
to buy the stake at this price. Payo¤s in this game are given by:
Investor
B
You S p + q 400, q 400
NS
q 700,
NB
p q 700, 0
0 q 700, 0
3
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(a) Both you and the investor know q. At which prices p will trade happen in the
equilibrium in weakly undominated strategies?
Now assume that only you know the probability of success q, while the investor thinks
that your start-up is either “good”or “bad”and that q is either 50% or 20%, and
that both cases are equally likely.
(b) In this game of incomplete information, what is the set of your strategies, and
what is the set of strategies for the investor?
(c) At which prices p will you (o¤er to) sell the 50% stake if q = 50%?
(d) At which prices p will you (o¤er to) sell the 50% stake if q = 20%?
(e) Given your answers to (c) and (d), what are the prices p at which the investor
is facing adverse selection (i.e. only bad start-ups with q = 20% are sold, while
good start-ups with q = 50% are not)?
(f) For which prices p will the investor (o¤er to) buy the stake in Bayes-NashEquilibrium?
(g) For which prices p is the stake in a bad start-up traded in Bayes-Nash-Equilibrium?
Explain!
(h) Is there any price p at which the stake in a good start-up will be traded (i.e. you
decide to sell and the investor decides to buy)?
4
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