# ECON 100C UC San Diego Utility Functions Micro Economics Worksheet

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Econ 100C
Problem 3.
Suppose there are three advertisers (label them 1, 2 and 3) and three advertising slots (label them a, b
and c) on a search result generated by Google (this is called sponsored search). Slot a has the most
prominent position on the page, slot b is second in prominence, and slot c is third in prominence. Each
advertiser only wants one slot. Suppose their values (which are their private information) are the
following:
1
2
3
value for slot a
10
16
15
value for slot b
9
5
8
The VCG mechanism for selling the slots works as follows:
1. (Reporting) Each advertiser reports his/her values to the mechanism.
value for slot c
3
4
1
2. (Allocation) The mechanism matches the advertisers to the slots to maximize the total
(reported) value.
3. (Payment) Each advertiser pays a price for his/her matched slot to the mechanism.
This problem walks you through the allocation and payment suppose the advertisers truthfully report
their values in the above table. (Truthful reporting will be a dominant strategy for the advertisers,
though you are not required to prove that for this problem.)
a. There are six possible matchings that assign advertisers to slots. Complete the following table.
What is the optimal matching that maximizes the total value of all advertisers (second column)?
This is the matching implemented by the VCG mechanism.
Matching
Total value of all
Total value of Total value of Toal value of 1
2 and 3
1 and 3
and 2
1-a, 2-b, 3-c
16
6
11
15
1-a, 2-c, 3-b
1-b, 2-a, 3-c
1-b, 2-c, 3-a
1-c, 2-a, 3-b
1-c, 2-b, 3-a
b. How much does advertiser 1 pay for his/her matched slot in the VCG mechanism?
i.
What is the total value of advertisers 2 and 3 in the optimal matching from a
(implemented by the VCG mechanism)?
ii.
What is the total value of advertisers 2 and 3 if advertiser 1 is not present in the
mechanism? If advertiser 1 is not present, then the VCG mechanism will implement the
optimal matching that maximizes the total value of 2 and 3 (third column):
iii.
What is the externality that advertiser 1 exerts on advertisers 2 and 3 (total value from ii
minus total value from i)? This is what advertiser 1 pays in the VCG mechanism.
c. How much does advertiser 2 pay for his/her matched slot in the VCG mechanism?
d. How much does advertiser 3 pay for his/her matched slot in the VCG mechanism?