First period cooperation
A longer horizon aects payo from conditional cooperation
Does not aect risk (since youre a sucker for one period at
most)
Consider only two strategies: trigger and always defect
sizeBAD=probability assigned to partner playing
Homework 10
1. For this question, use the extensive form game shown above. Find the behavioral strategy of player 1 that is equivalent to her mixed strategy in which she plays
(B, r) with probability 0.4, (B, l) with probability 0.1, and (A, l) with proba
Homework 8
1. Find all the mixed strategy NE of this game, where 0 < < 1:
A
B
C
A , 1,-1 -1, 1
B -1, 1 , 1, -1
C 1,-1 -1, 1 ,
Solution: Let (p1 , p2 , 1 p1 p2 ) denote the mixed strategy of Player 1 and
(q1 , q2 , 1 q1 q2 ) the mixed strategy of Player 2
Homework 11
1. Find all the sequential equilibria of the game above.
Solution: Let (, , ) denote the strategy of player 1 and (p, 1 p) the strategy
of player 2.
If > , then player 2 chooses L (p = 1), and hence player 1 chooses M ( = 1).
(L,M) is indeed a
MIDTERM 3
Question 1. (a-5 points) We write si Dai to denote that a pure strategy ai is
strictly dominated (D) by a mixed strategy si . Dene what this means (in class,
we showed three equivalent denitions of strict dominance; you can use any of these
in y
Homework 9
1. Consider the following two player extensive form game. Player 1 chooses how
to allocate 2 dollars between himself and Player 2. His three choices are: (2,0), (1,1),
and (0,2). Player 2 then chooses whether he accepts or rejects the allocatio
Homework 7
1. Find all the perfect equilibria of the following game:
A
B
C
A 0,0 0,0 0,0
B 0,0 1,1 2,0
C 0,0 0,2 2,2
Solution: Consider an perturbation of the game above. Player 1s payo from
A is 0. His payo from B is at least 1 2 + 2 2 . Similarly, his p
Homework 15
1. Consider the following strategic situation. Two opposed armies are poised to
seize an island. Each armys general can choose either attack or not attack. In
addition, each army is either strong or weak with equal probability (the draws for
e
MIDTERM 3
Question 1 (Seltens horse). Find all sequential equilibria in the game below
and defend your answer:
Let (r, 1 r), (q, 1 q), (p, 1 p) be a strategy prole. Let (, 1 ) denote
the beliefs of player 3 at his information set.
1 Player 2 chooses c if
MIDTERM 2
Question 1.
(a-5 points) Dene a perfect equilibrium.
(b-5 points) Dene a subgame perfect equilibrium.
Solution: See class notes.
Question 2. For this question, consider the game below.
(a-8 points) Find the unique subgame perfect Nash equilibriu
MIDTERM 1
Your name:
Question 1. Let s S be a Nash Equilibrium strategy prole. Show that any
a Ai s.t. si (ai ) > 0 is rationalizable. (In words, show that any action used with a
positive probability in a mixed strategy Nash Equilibrium is rationalizable.
Homework 12
1. Find the set of feasible, strictly individually rational payos in the following
games:
L
R
T 0,0 2,1
B 1,2 0,0
T
B
L
R
1,-1 -1,1
-1,1 1,-1
L
R
T 2,2 0,3
B 3,0 1,1
T
M
B
L
R
1,1 0,0
1,1 2,1
0,0 2,1
2. Consider the following game:
L
R
T 2,3 1
Homework 15
1. Consider the following strategic situation. Two opposed armies are poised to
seize an island. Each armys general can choose either attack or not attack. In
addition, each army is either strong or weak with equal probability (the draws for
e
Homework 13
1. Consider the game from Example 2 in the notes (this is the prisoners dilemma
with unobserved payos). Let the players common discount factor be equal to .
Can you nd an equivalent game with imperfect monitoring in which the players
observe p
Homework 10
1. For this question, use the extensive form game shown above. Find the behavioral strategy of player 1 that is equivalent to her mixed strategy in which she plays
(B, r) with probability 0.4, (B, l) with probability 0.1, and (A, l) with proba
Homework 14
1. (a) In class, we showed that the incentive constraint for discouraging one
deviation (in the rst period) with AMP block strategies is given by:
T
(1 )5 T q2 (v 2).
(0.1)
Derive this incentive constraint rigorously by nding player is utility
Homework 9
1. Consider the following two player extensive form game. Player 1 chooses how
to allocate 2 dollars between himself and Player 2. His three choices are: (2,0), (1,1),
and (0,2). Player 2 then chooses whether he accepts or rejects the allocatio
Homework 2
1. There are N rms in an industry. Each can try to convince Congress to
give the industry a subsidy. Let hi denote the number of hours of eort put
in by rm i, and let ci (hi ) = wi h2 denote the cost of eort, with wi a positive
i
constant. When
Homework 7
1. Find all the perfect equilibria of the following game:
A
B
C
A 0,0 0,0 0,0
B 0,0 1,1 2,0
C 0,0 0,2 2,2
2.* Show that the set N E(G ) is nonempty for any
i
j j < 1 i.
s.t.
i
j
> 0 i, j and
Hint: Dene a correspondence BRi (s) from S to S and s
Homework 8
1. Find all the mixed strategy NE of this game, where 0 < 1:
A
B
C
A , 1,-1 -1, 1
B -1, 1 , 1, -1
C 1,-1 -1, 1 ,
2. Consider the following game:
A
B
A
2,2
-100,0
B 0,-100
1,1
Find the unique mixed strategy NE, and argue that it is not ESS.
3.*
Homework 6
1.* Dene rows paranoid problem as
min R s.t.
n
j=1
x0,R
aij xj R i cfw_1, ., m
,
n
xj = 1
j=1
and columns paranoid problem as
max C
y0,C
s.t.
m
i=1
yi aij C
m
i=1
j cfw_1, ., n
.
yi = 1
Show that columns paranoid problem is a dual of rows paran
Homework 5
1. Let L(x) denote the Lagrangian for the constrained optimization problem in
the lecture notes. Show that if one of the constraints takes the form of xl 0, then
the FOC for xl changes to L(x,) 0.
xn
2. Consider a zero-sum game with a payo matr
Homework 4
1. Consider a three player game with three rms i = 1, 2, 3. Each rm faces
the demand curve p = ab(q1 +q2 +q3 ) and per-unit costs of production c. Does
iterated elimination of strictly domination actions yield a unique prediction in
this game?
Homework 1
1. Consider the following game. There are two players, Mr. A and Mr. B.
The two players are separated and cannot communicate. They are supposed to
meet in New York City at noon for lunch but have forgotten to specify where.
Each must decide whe