CSV 886 Social Economic and Information Networks. Lecture 5: Matching Markets, Sponsored Search. R Ravi

Transcription

1 CSV 886 Social Economic and Information Networks Lecture 5: Matching Markets, Sponsored Search R Ravi ravi+iitd@andrew.cmu.edu

2 Simple Models of Trade Decentralized Buyers and sellers have to find each other (search costs) and agree on price (negotiation costs) Central exchange with competition Auctions set prices Centralized exchange Intermediaries setting fixed prices Faster search and clearing 2

3 Perfect Matching Market 3

4 Is there a perfect assignment? 4

5 One reason for no perfect assignment A constricted set : A set of students whose wish list is smaller than the number of students 5

6 The ONLY reason for no perfect assignment If there is a constricted set there is no perfect matching If there is no constricted set, there is always a perfect matching! 6

7 General Matching Market Valuations for different items, only want one Buyers X, Y, Z for items A, B, C A B C X Y Z

8 Alternate view A B C X Y Z

9 Prices Set a price per item and let each buyer choose the item with most payoff = (value - price). Try (1,3,4) A B C X Y Z

10 Market-clearing price Prices such that the graph of edges from buyers to preferred sellers has a perfect matching. Try (4,3,1) A B C X Y Z

11 Two amazing facts For ANY set of buyer valuations, marketclearing prices exist For ANY set of market-clearing prices, the resulting assignment has maximum total valuation (over all assignment of items to buyers) 11

12 Generalized auction round Current prices, smallest at zero Check preferred seller graph for matching If none, find a constricted set and raise prices of the items wished by the set by one unit Rescale prices by subtracting smallest from all to make the smallest zero 12

13 Example start of Round 1 13

14 Example start of Round 2 14

15 Example start of Round 3 15

16 Example start of Round 4 16

17 Why does auction end? Energy of buyer = maximum payoff Energy of item/seller = price Energy of auction = sum of above All energies non-negative at all times Δ Energy in one round? 17

18 Basic Models of Internet Advertising CPM CPC CPA 18

19 Sponsored Search

20 Charging for ads First generation CPI or cost-per-impression (also called CPM or cost-per-mille) Second generation Overture: CPC or cost-per-click Key question: How much to charge per click?

21 First Take First price auction Results (Overture, circa )

22 Second Take Generalized Second Price Auction Two generalizations One related to maintaining truthful bidding of valuations (VCG) Other, more superficial, related to mechanics of second price (GSP)

23 Setting Sellers: descending click-thru rates Buyers: descending revenue per click

24 Market Clearing Prices?

25 Problem Don t know everyone s values to announce these clearing prices Need a pricing mechanism that encourages everyone to announce their true valuations

26 VCG Principle Allocate items so as to maximize total valuation (not payoff to buyers, which subtracts the price) Winner is charged a price equal to the harm (decrease) he causes to all other bidders valuations by getting his item

27 Example: Maximum value allocation

28 Example: Price charged to x Reduction in y s and z s value due to x =

29 Example: Price charged to y Reduction in x s and z s value due to y =

30 Example: Max value allocation of VCG A B C X Y Z

31 Price charged to X A B C X Y Z

32 Price charged to Y A B C X Y Z

33 Price charged to Z A B C X Y Z

34 General Setting S = set of sellers (items) B = set of buyers V S B = maximum valuation matching of S to B Suppose item i is given to j in this allocation. Charge it the VCG price p ij = V S B-j V S-i B-j (Value if this seller j was not there Value if seller j got item i)

35 Why does VCG induce truth telling? Suppose item i goes to buyer j in VCG mechanism for a price p ij By lying about value, suppose j can get item h instead. For this to be of no use to j, we need for every h other than i, v ij - p ij v hj - p hj

36 First is overall maximum, second fixes h to j

37 Reality is not as nice GSP or generalized second price auction Rank slots in decreasing order of CTR Rank buyers in decreasing order of bids Give 1 st slot to 1 st buyer at price of 2 nd buyer s bid per click Give 2 nd slot to 2 nd buyer at price of 3 rd buyer s bid per click Give i th slot to ith buyer at price of (i+1) st bid per click (total price CTR i times bid i+1 )

38 Example: all bid true values

39 Example: x underbids at 5

40 Truth-telling may not be equilibrium x has incentive to underbid from 7 to 5 Payoff bidding 7 = Payoff bidding 5 =

41 Redeeming fact GSP always has a Nash equilibrium where the prices correspond to market clearing prices

42 Reality Assumption of decreasing CTRs may not hold if ads placed higher up has bad quality Fudge quality factor q in GSP: Sort buyers by order of q j b j rather than b j Each buyer pays minimum amount to keep its current position ( = q j+1 b j+1 / q j ) Expressiveness of keywords at both sides (concern of SEM)

43 Advanced Material: Sec 15.9 VCG prices treated as posted prices are the unique market clearing with minimum total sum

44 HOMEWORK 1. Give the fastest algorithm to find, for any edge of a graph, the number of pairs of nodes whose shortest paths go through this node. (Extra: Use the algorithm to compute the fraction of the total number of shortest s-t paths that go through this edge, for a given pair s,t). 2. Give a polynomial time algorithm for finding the smallest affiliation network explaining a given social network edges, or show that it is NP-hard. 44

45 HOMEWORK CONTD 3. Exercise of the text on adding tolls in traffic networks 4. Exercise 10 from Chapter 9 (on entry price for auctions) 5. Exercise 11 from Chapter 10 (on matching markets) 6. Exercise 5 from Chapter 15 (on sponsored search markets) DUE in class MONDAY, February 2 nd. OK to to ravi+iitd@andrew.cmu.edu 45

46 OTHER WORK Second Homework: one problem per day, for a total of four, due over by next Monday, February 9 th. Exam: 2 hours, 4 questions. What day? 46