Download ECON 504 Sample Questions for Midterm Exam Levent Koçkesen 1

ECON 504
Sample Questions for Midterm Exam
Levent Koçkesen
1. Find the set of (pure and mixed strategy) Nash equilibria of the following game:
T
M
B
L
4, 3
1, 2
1, 0
C
3, 0
2, 4
4, 3
R
1, 1
0, 3
2, 1
T
M
B
L
3, 1
1, b
c, 2
C
a, 3
2, 0
4, 1
R
0, 4
1, 1
2, d
2. Consider the following game:
(a) Assume a = 5, b = 2, c = 4, d = 2 and find the set of (pure and mixed) Nash equilibria of this game.
(b) Provide values for a, b, c, d such that the game has a strictly dominant strategy equilibrium.
(c) Provide values for a, b, c, d such that there is no strictly dominant strategy equilibrium but there is a
unique pure strategy Nash equilibrium.
3. Consider the strategic form game represented by the following bimatrix.
Player 2
S
H
4, 4 0, 2
2, 0 2, 2
3, 0 1, 0
S
Player 1 H
N
(a) Does any player have a strictly dominated action?
(b) Find the set of all Nash equilibria (pure and mixed).
4. A buyer and a seller simultaneously submit a price, which can be any non-negative number. If the price
chosen by the buyer is at least as large as the price chosen by the seller, i.e., p b ≥ p s , trade occurs and the
buyer pays p b whereas the seller receives p s . The rest, i.e., p b − p s , goes to a charity. If trade occurs, payoff
of the buyer is his value v minus the price he pays and the payoff of the seller is the price she receives minus
her cost, c. Assume that v > c ≥ 0. If trade does not occur, both players receive zero payoff. Therefore, the
payoff functions are given by
ub (p b , p s ) =

0,
u s (p b , p s ) =

0,
pb < ps
v − p , p ≥ p
s
b
b
pb < ps
 p − c, p ≥ p
s
s
b
(a) Show that any (p b , p s ) such that c ≤ p b = p s ≤ v is a Nash equilibrium.
(b) Find the set of all pure strategy Nash equilibria.
5. Player 1 and 2 are trying to share a cake of size 100. They each simultaneously submit a real number in the
interval [0, 100]. If the sum of their numbers is smaller than or equal to 100, then each receives the number
she submitted. If the sum is greater than 100, then the player who submitted the smaller number, say x,
receives that number while the other receives 100 − x. If the sum is greater than 100 and they are equal,
then each receives 50.
Page 1
ECON 504
Sample Questions for Midterm Exam
Levent Koçkesen
(a) Find the set of pure strategy Nash equilibria of this game.
(b) Consider the same game but assume that the numbers must be integers, i.e., 0, 1, 2, . . . , 100. Consider
the following order of eliminating weakly dominated strategies: At each stage all the weakly dominated
strategies of both players are eliminated. Which outcome does that procedure lead to?
6. Let G be a finite 2-player strategic form game.
(a) Prove that if (s1∗ , s2∗ ) is a Nash equilibrium of G, then (s1∗ , s2∗ ) survives iterated elimination of strictly
dominated strategies.
(b) Does a Nash equilibrium of G necessarily survive iterated elimination of weakly dominated strategies?
If so, prove it. If not, give a counter-example.
7. Let G = (N , (S i ), (ui )) be a finite strategic form game. Show that every pure strategy Nash equilibrium is
rationalizable.
8. An incumbent monopolist firm (player 1) is deciding whether to undertake a costly investment to expand
its production capacity. A potential entrant (player 2) is considering entering into this market, which is
profitable only if the incumbent does not invest. The cost of investment is either low or high, which is
known by the incumbent but not by the entrant. The entrant believes that the entrant’s cost is low with
probability q ∈ (0, 1). Payoffs to different action and type profiles are given below:
E
O
I
1, −1
6, 0
N
2, 2
4, 0
Low Cost 〈q〉
E
O
I
0, −1
2, 0
N
2, 2
4, 0
High Cost 〈1 − q〉
where the row player is the incumbent and the column player is the entrant; I denotes invest, N not invest,
E enter, and O stay out. All of this is common knowledge.
(a) Find the set of pooling pure strategy Bayesian equilibria.
(b) Find the set of separating pure strategy Bayesian equilibria.
(c) Is there an equilibrium in which at least one type of player 1 chooses both I and N with positive
probability?
9. A robber decides whether to attack (A) or pass (P ) and a victim simultaneously decides whether to fight (F )
or yield (Y ). Victim is either weak or strong and the payoff matrices (where robber is the row player and
victim is column player) corresponding to the two types are given as follows:
A
P
F
−1, −3
0, 1
Y
2, −2
0, 0
A
P
Weak (1 − q)
F
−1, −1
0, 1
Y
2, −2
0, 0
Strong (q)
Robber does not know the type of the victim but believes that she is strong with probability q ∈ (0, 1). All of
this is common knowledge.
(a) Find the set of pure strategy Bayesian equilibria of this game as a function of q.
(b) Assume that q < 2/3 and find a Bayesian equilibrium in which the robber plays both A and P with
strictly positive probabilities.
Page 2
ECON 504
Sample Questions for Midterm Exam
Levent Koçkesen
10. Two animals, player 1 and player 2, simultaneously decide whether to be hawkish (H ) or dovish (D) when
they are trying to share a prey. If they get into a fight they both suffer injuries, which cost player 1 c 1 and
player 2 c 2 . There is a fight if and only if both of them choose to be hawkish. If both are hawkish or dovish
they share the prey, whereas if only one is hawkish, he gets the entire prey. The value of the prey for each of
them is v > 0. This game, therefore, can be represented by the following bimatrix.
H
D
H
v/2 − c 1 , v/2 − c 2
0, v
D
v, 0
v/2, v/2
It is common knowledge that c 1 > v/2, but c 2 is either 0 (player 2 is Tough) or strictly greater than v/2
(Player 2 is Weak). Player 2 knows c 2 , whereas player 1 believes that c 2 > v/2 with probability q ∈ (0, 1) and
c 2 = 0 with probability 1 − q. All of this is common knowledge.
(a) Find the set of pooling pure strategy Bayesian equilibria.
(b) Find the set of separating pure strategy Bayesian equilibria.
(c) Are there mixed strategy Bayesian equilibria?
11. Two players, who are involved in a competition to win a prize, are considering how much to bribe a bureaucrat, who will award the prize to the highest briber. In case of a tie, each player wins the prize with equal
probability. They choose how much to bribe independently and without knowing the choice of the other
player, and if they don’t win the prize they cannot get the bribe back. The value of the prize to player i is
v i . Player i 6= j knows v i but is uncertain about v j : she believes that v j is a random variable with uniform
distribution over [0, 1]. Therefore, for any bribe profile (b 1 , b 2 ), the payoff function of player i is:



−b i ,

if b i < b j
ui (b 1 , b 2 ) = 12 v i

− b i , if b i = b j


v − b ,
if b i > b j
i
i
where j 6= i . All of this is common knowledge.
Denote the pure strategy of player i of type v by βi (v). Find a pure strategy Bayesian equilibrium of this
game, where β1 (v) = β2 (v) = β(v) for all v. You may assume that β is strictly increasing and differentiable.
12. Consider the following two bidder second-price sealed bid auction. Players receive private and independent signals, t1 and t2 , that have a uniform distribution over [0, 1]. Value of the object to player i is αti +γt j ,
where i 6= j and α > γ ≥ 0.
(a) Show that βi (ti ) = (α + γ)ti , i = 1, 2 is a Bayesian equilibrium of this game.
(b) Characterize all symmetric equilibria in which player i bids β(ti ), where β is strictly increasing and
differentiable.
Page 3