1. The Structure of Networks
with emphasis on information and social networks
Chapter 15
Ýmir Vigfússon

6. Sponsored search How should we set prices?
Lots of different keywords
Should we post prices (like a store)?
Too many keywords and advertisers
What if there were just one ad slot?
Just a single-item auction!
Sealed-bid second-price auction appealing
But we have multiple ad slots
How do we deal with that?

7. Sponsored search Ad slots have different characteristics
Diverse clickthrough rates
Some slots more valuable than others
What if we knew the buyers‘ valuations?
Assign buyers to slots
Known as a matching market (Chapter 10)
Let‘s review the properties

8. Matchings

9. Matchings
Def: A perfect matching (i) assigns everybody on the left with someone on the right, and (ii) no two nodes on the left are assigned to the same node on the right
Def: If the neighbors N(S) of some set S of nodes are fewer than the nodes in S, then S is constricted set

10. Matchings

11. Matchings
Thm: A bipartite graph has a perfect matching if and only if there is no constricted set
Constricted sets are the only obstacles to perfect matchings

12. Matching markets
Right nodes have a valuation for every node on the left
How much do you want each item?
Quality of matching assignment = Sum of the valuations for items people get

13. Matching markets Buyer‘s payoff = valuation – price paid
Thm: We can always set prices so that if buyers buy the item they most want, all items are sold (market clearing prices)

14. Matching markets How does this work?
We connect each buyer to the most preferred seller
Raise prices for items that are in high demand
Repeat, stop when there is a perfect matching

15. Matching markets
Thm: Market clearing prices always produce socially optimal outcome
Maximum total valuation of any matching
What about sponsored search?
Use matching markets to handle sponsored search when valuations are known

16. Sponsored search If there is just one ad slot
Run an auction
If we knew the advertisers‘ valuations
Run a matching market (Chapter 10)
What if we don‘t know the valuations?
Want to encourage truthful bidding
Like in the second-price auctions we looked at
How do we do that?