Econ 2040: Problem set 6
November 13, 2020
1 (a) The prices are (9, 3, 0) and z gets a, y gets b and x gets c. To see this,
notice that the new value net of prices as:
CTR
Slot
Adv
Val p.c.
Val(A)-p
Val(B)-p
Val(C)-p
7
5
2
A
B
C
z
y
x
5
3
1
26
12
-2
22
12
2
10
3
2
1 (b) VCG results in the same matching as market clearing matching as above.
The VCG prices are:
PAz = (21 + 5) − (15 + 2) = 9
PBy = (35 + 5) − (35 + 2) = 3
PCx = 0
2 (a) The matching is x—a, y—b, z—c. and the prices are (17, 5, 0). To see this,
CTR Slot Adv Val p.c. Val(A)-p Val(B)-p Val(C)-p
consider:
5
3
2
A
B
C
x
y
z
8
6
5
23
13
8
19
13
15
16
12
10
2 (b) No it is not a NE.
To see this, if x bids 8, it’s payoff is 10. However, if x bids 4, it gets a
payoff on 16.
2 (c) Yes, it is a NE. The easy way to see this is notice that the prices are market
clearing. With this observation it’s clear why no one wants to deviate as
any deviation is equivalent to picking something other than the slot they
chose at market clearing prices and this is inconsistent with those prices
being market clearing. An explanation that simply notes that the bids
induce market clearing prices and thus that they from a Nash equilibrium
is fine.
1
3 (a) In the second price auction each bidder bids truthfully. So x bids 7, y bids
9 and z bids 6. Bidder y has the highest bid so he wins. The price that y
pays is the second highest bid which is 7.
3 (b) In the VCG procedure y will be assigned item a and the two fictional
items, b and c, will be assigned to bidders x and z. (It does not matter
which of fictional items is assigned to x and to z.) The price that y pays
is the value that x and z could obtain without y but with the item that
y receives, which is 7, minus the value that x and z could obtain without
y and without the item that y receives, which is 0. So y pays 7 just as
in the second price auction. This price is the harm that y creates for the
other bidders as it is the reduction in the maximum value that they could
obtain that is created by y taking item a. Bidders x and z pay nothing.
4
CTR
Slot
Adv
Val p.c.
Val(A)
Val(B)
Val(C)
5
4
3
A
B
C
x
y
z
5
3
2
25
15
10
20
12
8
15
9
6
4 (a) The match is that x gets a, y gets b and z gets c. The VCG prices are:
PAz = (15 + 8) − (12 + 6) = 5
PBy = (25 + 8) − (25 + 6) = 2
PCx = 0
4 (b1)
CTR
Slot
Adv
Val p.c.
Val(A)
Val(B)
Val(C)
Val(D)
5
4
3
v
A
B
C
D
x
y
z
z
5
3
2
0
25
15
10
5v
20
12
8
4v
15
9
6
3v
0
0
0
0
W gets the fictitious slot D.
4 (b2)
PAx = 8 ⇒ (15 + 8 + 3v) − (12 + 6) = 8 ⇒ v = 1
2