Various auctions formats are explored. Perfect matching, optimal assignment and market clearing prices are studied

1. Social and Economic
Network Analysis
UNIT – V
ECONOMIC NETWORK ANALYSIS

2. Overview
Auctions &
Matching Markets
Sponsored Search
Markets
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3. Auctions
NETWORKS, CROWDS AND MARKETS - CHAPTER 9
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4. Types of Auctions
▪Intrinsic value – bidder’s true value for the item
▪Ascending-bid auctions / English auctions
Seller gradually raises the price
▪Descending-bid auctions / Dutch auctions
Seller gradually lowers the price from some high initial value
▪First-price sealed-bid auctions
The highest bidder wins the object and pays the value of her bid
▪Second-price sealed-bid auctions / Vickery auctions
The highest bidder wins the object and pays the value of second highest bid

5. Descending-Bid and First-Price Auctions

6. Ascending-Bid and Second-Price Auctions
How long to stay in auction?

7. Second-Price
Auctions

8. Second price auctions as game
▪Players - Bidders
▪Let vi be bidder i’s true value for the object and bi be i’s strategy to
bid a object
▪Payoffi = 0 if bi is not winning bid
= vi-bj if bi is a winning bid and bj is the second placed bid
▪The bidders don’t know each other’s values
▪True value is a dominant strategy (vi = bi)

9. • Consider i as a loser
Decreasing bid – still lose & receive nothing
(b’i < vi )
Increasing bid - still lose & receive nothing
(b’i > vi ) - win but pays more than i values the
object
Payoff to i from deviating would be vi − bj ≤ 0
• Consider i as a winner
Increasing bid – still win & pays second prize
(b”i > vi )
Decreasing bid - still win & pays second prize
(b”i < vi ) - lose & receive nothing
Payoff to i from deviating would be 0

10. Matching markets
NETWORKS, CROWDS AND MARKETS – CHAPTER 10
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11. Matching
markets
Bipartite matching problem
Markets in which
you can’t just
choose what you
want even if you
can afford it – you
also have to be
chosen
Task: assigning
each student a
room that they’d
be happy to accept
Perfect Matching

12. Perfect Matching
▪When there are an equal number of
nodes on each side of a bipartite graph,
then in a perfect matching assignment
I. each node is connected by an edge to
the node it is assigned to
II. no two nodes on the left are assigned
to the same node on the right

13. Perfect matching
Constricted set

14. Perfect matching
▪Let S be set of nodes on the right hand side of bipartite graph
▪N(S) – neighbors set of S is a set of nodes on the left hand side of
bipartite graph which has an edge to a node in S
▪S is a constricted set if S > N(S)
▪Constricted set No perfect matching
Matching Theorem: If a bipartite graph (with equal numbers of nodes on
the left and right) has no perfect matching, then it must contain a
constricted set

15. Valuation & Optimal Assignment
▪Valuation each individual expresses how much they’d like each object
▪Quality of an assignment = sum of all individuals valuations for what
they get
Assignment
Quality = 12+6+5
=23
Optimal
Assignment:
Max. Quality

16. Valuation & Optimal Assignment
▪Valuation maximizes happiness of everyone for what they get
Finding
optimal
assignment
Bipartite
matching
problem
Perfect matching exists:
Optimal Assignment =
# students

17. Prices and Market Clearing Property
Preferred seller graph
Payoff
Vij – Pi
Market clearing prices

18. Prices and Market Clearing Property
Is the set of prices Market clearing?
Yes, if preferred-seller has perfect matching

19. Prices and Market Clearing Property
For any set of market-clearing prices, a perfect matching
in the resulting preferred-seller graph has the maximum
total valuation of any assignment of sellers to buyers
Let M be perfect matching
Total payoff = Sum of all buyers payoff
Total Payoff of M = Total Valuation of M − Sum of all prices

20. Market clearing prices

21. Market clearing prices