Sponsored Search Auctions Introduction Web search engines - PowerPoint PPT Presentation

Sponsored Search Auctions του Κυριάκου Σέργη

Introduction � Web search engines like Google and Yahoo! �������� their service �� ���������� off advertising ����� ���� �� ����� �������� ����������� ������ ������� .

Introduction � For example, Apple or Best Buy may bid to appear among the advertisements – usually ������� ����� or ������������ of the algorithmic results

Introduction � These sponsored results are displayed in a format similar to algorithmic results: � as a list of items each containing � title, � text description � hyperlink to the advertiser’s Web page.

Introduction � We call each position in the list a ���� .

Introduction � ������������� of ��������� visit a ������������� ��������� � Americans conduct roughly �������������� ������������������ � �������� � ���� �� to commercial sites is ��������������������������� � �����!���� �������� �������� on the Web are ���������������������������� .

Introduction � Today, Internet giants "����� and #����$ boast a combined ���%����������������� of �����&����������� , largely on the strength of sponsored search. � Roughly '�� of "�����(��&!)��������� and roughly !�� of #����$(��&�)*�������� in +����������� is likely attributable to sponsored search.

Introduction � Advertisers specify: � List of pairs of keywords � Bids � Total maximum daily or weekly budget. � Every time a user searches for a keyword, an auction takes place among the set of interested advertisers who have not exhausted their budgets.

Existing Models � Static � Vickrey Clarke Grooves Mechanism (VCG) � Generalized First Price (GFP) � Generalized Second Price (GSP) � Dynamic � On@line Allocation Problem

Static � n bidders/advertisers � k slots (k is fixed apriori – k<n) α as a click through rate (CTR) of � ij the bidder j if placed in slot i v is the value of the bidder j for a � j click

Static � Αssumptions Bidders prefer a higher slot to a lower slot � α ≥ α − i 1, j for i=1,2,..., k 1 + ij v is independent of the slot position ( ������ ) � i CTR for a slot does not depend on the identity of � other bidders. CTRs are assumed to be common knowledge � ( ������ nature) � not the reality @ CTRs can fluctuate dramatically over small periods)

Static � Revenue Maximization � Allocative Efficiency

Revenue Maximization � Result of Myerson � The generalized Vickrey auction is applied v not to the actual values but to the j corresponding virtual values � Generalized Vickrey auction with reserve prices

Revenue Maximization � Maximization bidder payments: n ∑ m ax p j = j 1

Revenue Maximization � Surplus Allocation: x (b) : expected CTR of n ∑ j max x (b)v bidder j who j j = j 1 bids b � Virtual Surplus Allocation: v n : drawn ind/ntly ∑ ϕ j max x (b) (v ) j j j from continuous = j 1 prob. − 1 F (v ) where: � ϕ = − j j (v ) v distribution j j j f (v ) j j d   = ≤ = F(z) Pr v z , f (z) F(z)   j j j j dz

Revenue Maximization � Expected ,�� �� of a Truthful Mechanism � ,is equal to the Expected Virtual Surplus:   ∑ = ϕ E (M(t)) E (v )x (t)   t t j j j   j � Proof: h ∫   = = = ϕ E (p (b)) p (b)f(b)db ... E (b)x (b)   b j j j j = b 0 � Mechanism Truthful in Expectation: x (b) Monotone non@decreasing � j b − ∫ � = p (b) b x (b) x (z)dz j j j j 0

Revenue Maximization � Thus, Virtual surplus is truthful if and only if v ϕ is monotone non@decreasing in (v ) j j j � Myerson Mechanism: � Given bids b and F (here Bayesian – Nash ′ = ϕ distribution), compute ‘virtual bids’: b (b ) i i i � Run VCG on b’ to get x’ and p’ ′ = ϕ − 1 � Output x=x’ and p with p (p ) i i i

Revenue Maximization � F is the Bayesian – Nash distribution of of the generalized Vickrey (second price) auction (second price) with reserve prices � Proof similar with the Vickrey (second price) auction (second price) with reserve price for 1 item

Revenue Maximization � Revenue without reserve price: 1 = R 0 3 � Revenue with reserve price r: 1 5 = = r , R 12 2 12

Revenue Maximization � Revenue without reserve price: � Given V A , B’s valuation is likely to lie anywhere between 0 and V A � On average V B = V A /2 � On average, V B halfway between 0 and V A � On average, V A halfway between V B and 1

Revenue Maximization � Revenue without reserve price: � E[ V B ] = 1/3 and E[ V A ] = 2/3 � E[ V B ] = E[ V A ]/2 = 1/3

Revenue Maximization � Revenue with reserve price r: � It may be the case that a bidder has positive valuation but negative virtual valuation. � Thus, for allocating a single item, the optimal mechanism finds the bidder with the largest nonnegative virtual valuation if there is one, and allocates to that bidder

Revenue Maximization Revenue with reserve price r: � bidder 1 (same for bidder 2) wins precisely when: � { } ϕ ≥ ϕ ⇒ (b ) max (b ),0 1 1 2 2 { } = ϕ ≥ ϕ ∧ϕ ≥ p inf b: (b) (b ) (b) 0 1 1 2 2 1 ϕ = ϕ = ϕ Since � 1 2 { } = ϕ − = ϕ − 1 1 p min b , (0) (0) 1 1 For � 1 = = ⇒ ϕ = − ⇒ #1 F(z) z , f (z) 1 (z) 2z 1 φ (0)= 2

Revenue Maximization � Revenue with reserve price r: For r=1/2: � � Pr[both below 1/2]=1/2*1/2=1/4 � Pr[both above 1/2]=1/2*1/2=1/4 � Pr[one above 1/2]=1/2 � Est. payoff both below = 0 � Est. payoff both above = 4/6 � Est. payoff one above = 1/2 1 1 4 1 1 5 = ⋅ + ⋅ + ⋅ = R 0 12 4 4 6 2 2 12

Allocative Efficiency = � Let if bidder j is assigned slot i x 1 ij = x 0 otherwise � ij

VCG � Solution of LP: k n ∑∑ α max v x ij j ij = = i 1 j 1 n ∑ ≤ ∀ s.t. x 1 , i=1,2,...,k ij = j 1 k ∑ ≤ ∀ x 1 , j=1,2,...,n ij = i 1 ≥ ∀ ∀ x 0 , i=1,2,...,k , j=1,2,...,n ij

VCG � Dual: k n ∑ ∑ + min p q i j = = i 1 j 1 + ≥ α ∀ ∀ s.t. p q v , i=1,2,...,k , j=1,2,...,n i j ij j ≥ ∀ ∀ p ,q 0 , i=1,2,...,k , j=1,2,...,n i j p : expected payment bidder i q : expected profit bidder j

VCG � Special Case: � CTRs bidder independent: α = � ij i � Simple algorithm Northwest Corner Rule: � Assign bidder with highest value top slot, second highest value second slot e.t.c � -���������� assignment

VCG � Cons � requires solving a computational problem which needs to be done online for every search and is expensive � Other mechanisms better revenues than VCG

GFP � Let b1,…,bn be the bids. The GFP mechanism is as follows: � Sorts bidders according to the bids b1,…,bn. � Assigns slots according to the order (assign top slot to the highest bidder and so on). � Charge bidder i according to his bid. � Yahoo! used a GFP auction until 2004.

GSP � Let w1,…,wn be the weights on bidders which are static and independent of the bids b1,…,bn. The GSP mechanism is as follows: = s w b Sort bidders by � i i i ≥ ≥ ≥ � (assume s s ... s ) 1 2 n Allocate slots to bidders 1 ,…,k in that order � ≤ (i.e., bidder i gets the ith slot if ). i k Charge i the mininum bid he needs to retain his � s slot (i.e., ). = + p i 1 i w i

GSP = � Overture model: For every i, w 1 i (bidders ordered according to the bids only). � Google model: Google assigns weights α based on the CTR at the top slot . w ≃ i i1 α The assumption here is that is static (or i1 slow changing) � This ordering is also called ‘revenue order’ = α s b since is the expected revenue if i i1 i i is put in slot 1 and there is only one slot.

GFP not truthful � Payoff in general:

GSP not truthful � Payoff in general:

GSP not truthful � Payoff in general:

GSP not truthful � Payoff in general:

VCG Payoff � Payoff in general: eachbidder j would be made to pay the sum of � − − (c c )b i 1 i i for every I below him

GSP vs VCG � Search engines revenues under GSP better than VCG: − = − ≤ − = − VCG VCG c p c p (c c )b c b c b c p c p + + + + + + + + + i i i 1 i 1 i i 1 i 1 i i 1 i 1 i 2 i i i 1 i 1