II-5. Intermediaries. Outline. Introduction. SponSearch. Introduction. II-5. Intermediaries. Introduction. SponSearch. II-5.

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1 Outline (Part II: Security of Economics) Lecture 5: Market with intermediaries and advertising Sponsored search Market with intermediaries Spring 2013 Outline Market is a system of exchange protocols Sponsored search compute the prices Market with intermediaries regulate the exchange Market is a system of exchange protocols An auction is a market organized by compute the prices regulate the exchange aseller:supplyauction abuyer:procurementauction We focus on computing the prices.

2 Markets in general are organized by Markets in general are organized by universal buyers/sellers universal buyers/sellers merchants, traders, dealers, entrepreneurs, advertisers (push), solicitors (pull) who mediate among the buyers and the sellers merchants, traders, dealers, entrepreneurs, advertisers (push), solicitors (pull) who mediate among the buyers and the sellers just like the universal goods money securities (bonds, equity, derivatives) mediate among the goods In this lecture Outline Multi-item auctions example: sponsored search problem of incentive compatibility Later: What is the value of advertising? Market with intermediaries traders strategies trading profits and social benefits Sponsored search Sponsored search setting Market vs auction Generalized Second Price auction Market with intermediaries Sponsored search setting Sponsored search as a matching problem

3 Sponsored search as a market Market mechanism n buyers, n item take n = {0, 1,...,n 1} Market mechanism Market mechanism n buyers, n item take n = {0, 1,...,n 1} n buyers, n item take n = {0, 1,...,n 1} buyers valuations per item v =(v ij ) n n buyers valuations per item v =(v ij ) n n item prices p =(p i ) n Market mechanism Market mechanism n buyers, n item take n = {0, 1,...,n 1} n buyers, n item take n = {0, 1,...,n 1} buyers valuations per item v =(v ij ) n n buyers valuations per item v =(v ij ) n n item prices p =(p i ) n item prices p =(p i ) n matching σ vp : n n assigns item σ vp (i) to i matching σ vp : n n assigns item σ vp (i) to i i s utility u i R is u i = v iσvp(i) p σvp(i)

4 Goal of the market mechanism Markets respect preference Maximize social welfare, i.e. buyers total payoff U(v, p) = = = u i i n v iσvp(i) p σvp(i) i n v iσ(i,v) P i n To maximize utility, σ vp : n n maximizes valuations v iσ(i,v) v ij where P = i<n p i Position auction mechanism n bidders, n positions Position auction mechanism n bidders, n positions bidders valuations v ij = w i r j where Position auction mechanism n bidders, n positions Position auction mechanism n bidders, n positions bidders valuations v ij = w i r j where bidders valuations v ij = w i r j where bidders bid b =(b i ) n bidders bid b =(b i ) n price per position π ij (b) =p i (b) r j where price per click p(b) = ( pi(b) ) n

5 Position auction mechanism n bidders, n positions Position auction mechanism n bidders, n positions bidders valuations v ij = w i r j where bidders valuations v ij = w i r j where bidders bid b =(b i ) n bidders bid b =(b i ) n price per position π ij (b) =p i (b) r j where price per position π ij (b) =p i (b) r j where price per click p(b) = ( pi(b) ) n price per click p(b) = ( pi(b) ) n matching τ : n R n n assigns item τ(i, b) to i matching τ : n R n n assigns item τ(i, b) to i i s utility u i : R n R is u i (b) = v iτ(i,b) π iτ(i,b) (b) = ( w i p i (b) ) r τ(i,b) Goal of the position auction mechanism Position auctions respect preference Maximize seller s revenue P(b) = = π iτ(i,b) (b) i<n p i (b) r τ(i,b) i<n To maximize p i (b) with u i (b) always use τ(i, b) < τ(j, b) = b i b j,i.e. where τ(i, b) =j if b i is j-th largest entry in b all p i grow with b bidder i bids b i to maximize u i (b). Assumption Generalized Second Price Auction n bidders, n positions The bidders are ordered by their bids bidders valuations v ij = w i r j where b 1 b 2 b 3 b n The positions are ordered by click-through rates bidders bid b =(b i ) n r 1 r 2 r 3 r n

6 Generalized Second Price Auction Generalized Second Price Auction n bidders, n positions n bidders, n positions bidders valuations v ij = w i r j where bidders valuations v ij = w i r j where bidders bid b =(b i ) n bidders bid b =(b i ) n price per click p i (b) =b i+1 price per click p i (b) =b i+1 i s utility u i : R n R is u i (b) = ( w i b i+1 ) ri Does GSPA encourage truthful bidding? Does GSPA encourage truthful bidding? with truthful bid: u x (7, 6, 1) = (7 6) 10 = 10 with untruthful bid: u x (5, 6, 1) = (7 1) 4 = 24 Position auction example Matching problem view

7 Idea Idea How much does x subtract from social welfare? How much does y subtract from social welfare? Idea: Vickrey, Clarke, Groves Notation B setofbidders S setofsellers(items) Each bidder should pay the cost that their bid incurs on social welfare i.e., the sum of the losses that they cause to other bidders v =(v ij ) B S bidders valuations V S B maximaltotalvaluation Notation B setofbidders Remember the assumption S setofsellers(items) v =(v ij ) B S bidders valuations The bidders are ordered by their bids V S B maximaltotalvaluation b 1 b 2 b 3 b n Remark The positions are ordered by click-through rates If #B < #S, thenadd#s #B bidders with all valuations 0 r 1 r 2 r 3 r n If #B > #S, thenadd#b #S sellers valued 0 by all.

8 n bidders, n positions n bidders, n positions bidders valuations v ij = w i r j where bidders valuations v ij = w i r j where bidders bid b =(b i ) n bidders bid b =(b i ) n price per item π ij (b) =V S B\i V S\j B\i n bidders, n positions bidders valuations v ij = w i r j where Theorem bidders bid b =(b i ) n The VCG auction is incentive compatible: truthful bidding is the unique Nash equilibrium for all players. price per item π ij (b) =V S B\i V S\j B\i i s utility u i : R n R is u i (b) = v ii π ii (b) Problem Homework For the sponsored search market Corollary The VCG auction maximizes social wellfare, i.e. the total utility of bidders. compute seller s revenue (i.e. the total of the prices charged for all items) if the positions are auctioned by a and by a VCG auction Show that neither of these mechanisms maximizes seller s revenue.

9 Billion $ problem Outline Design an auction mechanism that maximizes seller s revenue. Sponsored search Market with intermediaries Toy market Toy market buyers B = {B 1, B 2,...B n } have valuations v i There is just one type of goods. sellers S = {S 1, S 2,...S n } have valuations w j Every buyer needs to buy one item. Every seller needs to sell one item. Toy market Toy market buyers B = {B 1, B 2,...B n } have valuations v i sellers S = {S 1, S 2,...S n } have valuations w j Goal of the market Find a bijection σ : B S that maximizes social benefit Remark If the numbers are different, then add buyers with the valuation 0, or sellers with the valuation 1. SB σ = n v i w σi i=1

10 Market with intermediaries Market with intermediaries Just like the goods are compared through universal goods The intermediaries mediate the flows money, securities merchants buy, move and sell goods the buyers and the sellers are connected through universal buyers/sellers traders buy and sell goods without moving them advertisers and solicitors move information merchants, traders, advertisers Market with intermediaries Market with intermediaries as a game buyers B = {B 1, B 2, B 3 } their reserve prices (valuations) v1 = v 2 = v 3 = 1 sellers S = {S 1, S 2, S 3 } their reserve price (valuations) w1 = w 2 = w 3 = 0 traders T = {T 1, T 2 } ask relation T1 a B1, T 1 a B2, T 2 a B2, T 2 a B3 T 1 s buyers B 1 = {B 1, B 2} T 2 s buyers B 2 = {B 2, B 3} bid relation S1 b T1, S 2 b T 1, S2 b T2, S 3 b T2 T 1 s sellers S 1 = {S 1, S 2} T 2 s sellers S 2 = {S 2, S 3} Market with intermediaries as a game Market with intermediaries as a game Game buyers B = {B 1,...,B n } Bi s reserve price (valuation) is v i players: moves: traders T 1,...,T m for the trader T k s the set of moves is sellers S = {S 1,...,S n } P k = Pb k Pa k, where Sj s reserve price (valuation) is w j Pb k = R p with p =#S k traders T = {T 1,...T m } Pa k = R q with q =#B k ask relation a T B where T k s buyers B k = {B i B T k a Bi} b k = b k1, b k2,...,b kp Pb k are T k s b bid relation S T b T k s sellers S k = {S j S S j Tk} bid prices for all S j S k a k = a k1, a k2,...,a kq Pa k aret k s ask prices for all B i B k

11 Market with intermediaries as a game Play Each T k announces its bid and ask prices p k = b k, a k Market with intermediaries as a game Trader T k s utility If #MB k #MS k (sufficient supplies) then ( ) u k p = a ki Bi MBk Sj MSk b kj Each S j agrees to sell to a T k with a maximal b kj If #MB k > #MS k (insufficient supplies) then ( ) u k p = a ki b kj a ki Each B i agrees to buy from a T k with a minimal a ki Bi MB + k Sj MSk Bi MB k Each T k thus forms the sets of suppliers MSk = { S j S k l. b lj b kj } customers MBk = {B i B k l. a ki a li} where MB k = MB + k MB k,and MB + is the set of #MSk buyers who accepted k the highest ask prices MB k are the remaining #MBk #MSk buyers with the lowest ask prices Distribution of social benefit Distribution of social benefit If the bijection σ : B S that maximizes social benefit If the bijection σ : B S that maximizes social benefit SB σ = n v i w σi i=1 SB σ = n v i w σi i=1 is found through the traders κ : B T,thenthebenefitis distributed is found through the traders κ : B T,thenthebenefitis distributed SB σ = n (v i a κ(i)i ) +(a κ(i)i b κ(i)σ(i) ) +(b κ(i)σ(i) w σi) } {{ }} {{ }} {{ } i=1 UB UT US SB σ = n (v i a κ(i)i ) +(a κ(i)i b κ(i)σ(i) ) +(b κ(i)σ(i) w σi) } {{ }} {{ }} {{ } i=1 UB UT US where where UB is the utility of the buyer UB is the utility of the buyer UT is the utility of the trader UT is the utility of the trader US is the utility of the seller US is the utility of the seller The traders maximize UT. Distribution of social benefit Implicit perfect competition But how do the traders achieve their payoffs? What are the equilibria in the trading game?

12 Indifference principle Indifference principle At equilibrium At equilibrium All bid prices offered to a seller must be equal All bid prices offered to a seller must be equal The seller will accept the bid from the trader who has access to the highest paying buyers because that trader can increase the bid by ε The seller will accept the bid from the trader who has access to the highest paying buyers because that trader can increase the bid by ε All ask prices offered to a buyer must be equal The buyer will accept the offer from the trader who has access to the lowest charging sellers because that trader can undercut the offer by ε Ripple effects Ripple effects 0 x 2 1 y 2 1 z 3