1
Homework #1 solutions / IEOR4407
1.1. (Gibbons 1.2) Elimination of dominated strategies gives us the above table below:
T
M
B
L
(2,0)
(3,4)
(1,3)
C
(1,1)
(1,2)
(0,2)
R
(4,2)
T
(2,3)
M
(3,0)
L
(2,0)
(3,4)
C
(1,1)
(1,2)
R
(4,2) T
(2,3)
M
L
(2,0)
(3,4)
R
(
1
Homework #2 solutions / IEOR4407
1. (Gibbons 1.11) Using Problem 1.7 (Gibbons 1.14), we can eliminate dominated strategies before
computing NE. Hence using solution to 1.1 we start with
L
(2,0)
(3,4)
T
M
R
(4,2)
(2,3)
Let the player 1 and 2s mixed strat
1
Homework #1 solutions / IEOR4407
1.1. (Gibbons 1.2) Elimination of dominated strategies gives us the above table below:
T
M
B
L
(2,0)
(3,4)
(1,3)
C
(1,1)
(1,2)
(0,2)
R
(4,2)
T
(2,3)
M
(3,0)
L
(2,0)
(3,4)
C
(1,1)
(1,2)
R
(4,2) T
(2,3)
M
L
(2,0)
(3,4)
R
1
Homework #2 solutions / IEOR4407
1. (Gibbons 1.11) Using Problem 3 below (Gibbons 1.14), we can eliminate dominated strategies
before computing NE. Hence using solution to 1.2 we start with
L
(2,0)
(3,4)
T
M
R
(4,2)
(2,3)
Let the player 1 and 2s mixed s
1
Homework #5 solutions / IEOR4407
1. (Gibbons 3.6) Following the arguments developed in class. Let bidder i bid bi = Si (vi ), where vi
are drawn from a Uniform(0, 1) distribution, and bidder i sees his own vi , for i = 1, ., n.
Si (vi ) should maximize
Homework #4 solutions / IEOR4407
1
1. (Gibbons 3.2) Firm 1 has two types and has to pick an action for each type. Firm 2 has only one
type and has to pick one action. Hence the strategy space for rm 1 is S1 = Re+ Re+ . The
strategy space for rm 2 is S2 =
1
Homework #3 solutions / IEOR4407
1. (a) What is the probability that the proposition will pass, as a function of and ?
Solution: The proposition passes when: (i) When A votes the proposition passes when
C and V both vote, which happens with probability
Problem Set 2 Solution
17.881/882
October 10, 2004
1
Gibbons 1.10 (p.50)
As a general rule, you should try to minimize the algebraic calculations that
you make. Use the concepts to simplify the problem.
Here, Im going to be making use of the propositions
1
Homework #7 solutions / IEOR4407
1 (a) An ecient allocation assigns the TV to the roommates only if the sum of their values, va +vb
is greater than or equal to vc = 100. To think about this problem, consider that the mechanism
designer takes the TV away
Sketch of sample nal solutions / IEOR4407
1
1. An ecient allocation is the one that maximizes the sum of the utilities. It is a combinatorial
problem, but here its easily seen that giving both items to player 1 maximizes the social welfare.
To calculate t
Midterm solutions / IEOR4407
1
1. (a) The LP formulation for player 1 is
Max z
T
ze pT U
p0
T
p e=1
If we change the problem to U = U c1 eeT , where eeT is a matrix of all ones and c1 is
a constant value. We will have an LP with the inequality constraint
1
Homework #3 solutions / IEOR4407
1. (a) What is the probability that the proposition will pass, as a function of and ?
Solution: The proposition passes when: (i) When A votes the proposition passes when
C and V both vote, which happens with probability
1
Homework #4 solutions / IEOR4407
1. (Gibbons 2.6) Each rm i wants wants to choose qi to maximize (P (Q) c)qi .
First, lets x rm 1s choice q1 . Then we look at rm 2 and 3s problem. We look at rm 3 rst.
Its problem is
max q3 (a q1 q2 q3 c)
q3
aq1 q2 c
.
2
Homework #5 solutions / IEOR4407
1
1. (Gibbons 3.1) Refer to section 3.1 of the text.
2. (Gibbons 3.2) Firm 1 has two types and has to pick an action for each type. Firm 2 has only one
type and has to pick one action. Hence the strategy space for rm 1 is