Ad Auctions: Game-Theoretic Perspectives

Moshe Tennenholtz

Technion—Israel Institute of Technologyand

Microsoft Research

Position (Ad) Auction

Ad Auctionsk positions, n players (bidders) n ≥ kvi - player i’s valuation per-click, vi drawn from F~[0,1] (i.i.d)

j- position j’s click-through rate 1 ≥ 2 ≥ ≥ k>0

Allocation rule – si(b1,…,bn) – player i’s position

Payments – pi(b1,...,bn) - i’s payment per click

Position payments - qj(b1,…,bn) payment per-click for player in position j

Quasi-linear utilities: j(vi-pi) if player i is in position j and pays pi

Reserve price: r ≥ 0

Special Ad Auctions

• Allocation rule – jth highest bid to jth highest position • Tie breaks - fixed order priority rule• Payment scheme• Self price (first price): every player pays his bid

qj(b1,…,bn)=b(j)

• Next–price (GSP): a player who got slot j pays

the bid of the player who

submitted the j+1 highest bid

qj(b1,…,bn)=b(j+1)

Generalized Second Price Auction (GSP)

Every player pays the bid of the next player: ps=bs+1.

position

value bid price click-rate

1 5 4 3 3

2 6 3 2 2

3 3 2 1 1

4 4 1 0 0

Utility for player (in position) 2 is: 2(6-2)=8

Remarks1. To move up one position – beat the bid of the player above

2. To move one position down – beat the price of the player below

Remarks1. To move up one position – beat the bid of the player above

2. To move one position down – beat the price of the player below

3. GSP is not truthful

1=2 2=1.9 v1=5, v2=4, v3=1

Suppose 2 and 3 bid truthfully

If 1 bids truthfully: u1=2(5-4)=2

If 1 bids 2: u1=1.9(5-1)=7.6>2

Nash Equilibrium

Definition: A Nash Equilibrium (NE) is a profile of bids by the participants such that unilateral deviations are not beneficial.

For GSP: Varian showed the existence of NE for

GSP, and characterized explicitly a set of them – termed SNEs.

GSP and the Vickrey auction

• In the case of only one position GSP coincides with the famous Vickrey auction:

Each player submits his bid, the winner is the player who submitted the highest bid, and he pays the second highest bid.

In the Vickrey auction it is a dominant strategy to submit the actual valuation as the bid: you can only gain by being truthful, regardless of the others do,

This is an instance of the general VCG auction.

Vickrey-Clarke-Groves (VCG)Allocation rule: chooses an allocation that maximizes social welfare: assign players in decreasing order of their bids.

A player pays for the difference between other players welfare when not participating to other players welfare when participating.

Example: v=(10,5,2)

payment for player 1:

1-2)+2(2-0)=7

payment for player 2:

10(1-1)+2(2-0)=2

10 2 7

5 1 2

2

vs s

VCG and GSP

VCG is truthful: it is a dominant strategy to bid your valuation as your bid.

Basic Theorems about GSP (Varian):

VCG payments coincide with the worst possible SNE payments of GSP.

In Varian’s setting: GSP is preferable to VCG from the revenue perspective.

Some celebrated results• There exist pure strategy equilibria for the next-price

position auction with complete information.• A set of equilibria of that kind has been characterized,

and the best equilibrium in this set (from an agent’s utility perspective) coincides with the VCG outcome.

Research highly influenced by: Position Auctions (Varian 2005)

Internet Advertising and the Generalized Second Price Auction (Edelman, Ostrovsky and Schwarz 2005)

Quality FactorsGoogle/Yahoo/MSFT assign advertisers according to

quality × bid.Each advertiser pays the minimum price to retain

her position. The click-rate an advertiser experiences in position

s is ess where es is player’s s quality factor.

qst – price for advertiser s in t

qstes=bt+1et+1 qst=bt+1et+1/es

The analysis remains as before when is replaced by

sv

ssve

A game-theorist wish list• The system perspective: Deal with the general situation

where more than one search engine conducts an auction for a given keyword. Who will make more money?

• The agent (advertiser) perspective: can we provide an advice for agent bidding in a standard ad auction such as GSP; how much should I actually bid?

• The user (surfer) perspective: re-visit the Varian’s model when explicit user models are taken into account: would this effect the basic results obtained about GSP/VCG?

• The mediator perspective: can a mediator coordinate advertisers’ bidding, in order that they will be able to ignore uncertainty about each other bids, while yielding them high utility?

A game-theorist wish listSerious challenges: • We wish to deal with the (more realistic but more

challenging) incomplete information setting. • The game-theoretic analysis of the system perspective,

addressing competition among auctions, has been tackled without success in the past.

• The user perspective introduces the need to consider dependence between ads.

• Can positive results be obtained for the agent perspective?

• Can positive results be obtained for the mediator perspective?

The System Perspective: Competing Ad Auctions

A basic story

It is not a monopolist market after all:

the system perspective

Which auction will gain more revenue?

Two Ad Auctions – The Setting

Auction A Auction B

No. of positions kA kB

Click-through rates 1¸ 2¸ ¸kA 1¸ 2¸ ¸ kB

Reserve prices rA rB

and B are regular

1 1

Definition:

A is stronger than B if j j for every position j

Incomplete Information:Symmetric Bayesian Equilibrium

Equilibrium analysis in the case of incomplete information is rather detailed.

Roughly speaking, given an auction, an equilibrium is a profile of bidding strategies (mapping from valuations to bids), such that each player strategy is the best response to the others, given the probabilistic assumptions. This is termed Bayesian Equilibrium.

In a symmetric Bayesian equlibrium all players use the same bidding strategies.

Regular ad auctions In a regular ad auction symmetric Bayesian

equilibrium exists under some assumptions. Our results do hold when the individual auctions are regular.

VCG ad auctions are regular, and for ease of exposition we will refer to them.

This is done in order to concentrate on the complexity arising from the fact we have two ad auctions rather than only one, and not in open questions regarding a single ad auction.

Symmetric Bayesian Equilibrium

Valuations are assumed to be selected from [0,1] according to distribution F with density f.

Strategies - bi:[0,1]→ [0,1], bi(vi) – player i’s bid when her value is vi

Utilities - Ui(vi,bi,b-i) – expected utility for player i in the strategy profile (b1,…,bn) given that her valuation is vi.

Bayesian equilibrium – a profile (b1,…,bn) such that for every player i, every vi

Ui(vi,bi(vi),b-i) ≥ Ui(vi,bi’(vi),b-i) for every strategy bi’.

Symmetric Bayesian equilibrium – if in addition b1=b2=…=bn

Regular Ad Auctions An ad auction is regular if there exists a symmetric equilibrium which is increasing

in types above the reservation price for any identical independent distribution F with the following structure:

let

F(0) can be positive

for every

for every

]1,(],0( 21 zzv 0)( vf0)( vf ],( 21 zzv

10 21 zz

Regular Ad Auctions

• All the results presented for competing ad auctions are applicable for the case where A and B are regular ad auctions.

• The VCG ad auction is regular.

• For simplicity of exposition, we assume both A and B are VCG ad auctions, where agents use the truth-revealing dominant strategy equilibrium.

Two Ad Auctions

Non-Competing every bidder participates in both auctions.

Competing every bidder chooses to participate only in a single auction.

Non-Competing AuctionsClaim: There exist a non-competing setting and

a distribution F, such that

(i) rA=rB=0,

(ii) A is stronger than B, but Rev(A) < Rev(B).

(Rev(A)–expected revenue in A)

The claim follows from the following…

Non-Competing AuctionsProposition: Let A and B be VCG ad auctions and

suppose A is stronger than B where j>j for only a single slot j.

If j=1 Rev(A)>Rev(B). If j > 1 Rev(A) > ( ) Rev(B) if and only if

jv(j+1) > ( ) (j-1)v(j)

where v(j) is the expected jth highest bid.

Non-Competing Auctions - Intuition

2 players and 2 positions:

A B

1=10 1=10

2=9.98 2=5

Position 1 is less attractive in auction A

More generally: Increasing number of click-rates may yield less competition for attractive click-rates, which may result in lower revenues.

Two Ad Auctions

Non-Competing every bidder participates in both auctions.

Competing every bidder chooses to participate only in a single auction.

Competing AuctionsThe game H=H(A,B):

Strategy for i, i=(qiA,qi

B,biA, bi

B), consists of:

qiA(vi), qi

B(vi), - probabilities attending A and B given vi (participation function).

qiB(vi)=1- qi

A(vi) ;

biA(vi) – the bid in A

biB(vi) – the bid in B

Utility for i - UiH(vi,i,-i).

To compare revenues we need to do some equilibrium analysis…

Equilibrium in H(A,B)A profile of strategies (1,…,n) such that:

(1) In both auctions, A and B, the strategies induce a Bayesian equilibrium (according to the induced value distributions in each auction).

(2) Given -i player i does not want to change her participation function.

Does there exist an equilibrium?

Equilibrium - ResultTheorem: There exists a unique symmetric

equilibrium in H(A,B). (for any number of positions, any click-through rates, and any reserve prices). Uniqueness is with regard to participation function.

Two things to prove:

(1) Existence

(2) Uniqueness

Elaborated proof employing the implicit function theorem…

Equilibrium - Examples

kA=kB=1 - single positiongeneral structure

Revenues in Competing Auctions: Main ResultTheorem: Suppose A is stronger than B, and rA =

rB.

Then Rev(A) Rev(B).

The SellersLet the sellers choose reserve prices:

There is no pure strategy equilibrium in the 2-stage

“sellers game”!Let Rev(A, rA, rB) (resp. Rev(B, rA, rB)) be the expected revenue

in A (resp. B) given reserve prices rA and rB in A and B respectively.

Proposition: Let A and B be VCG auctions. Suppose A is stronger than B and rA=rB.

Then for every rB’>rB such that

Rev(B,rA,rB) < Rev(B,rA,rB’)

we have that

Rev(A,rA,rB’)-Rev(A,rA,rB) > Rev(B,rA, rB’)-Rev(B,rA,rB)

Intermediate conclusions: the effects of competition

The revenue in a “more visited” auction can be lower than the revenue in a “less visited” auction.

In the competing setting the revenue of the stronger auction is always higher.

Mechanisms can have different properties in a competing and non-competing settings:

Ad revenue is not a direct implication of search engine success; advertisers’ acquisition / mechanism selection may be a winning strategy!

The Agent Perspective:

Competitive Safety Analysis

in Ad Auctions

But what should I do? The agent perspective

$7.6?

An Agent Centric Approach to Games

Given a game, what is a useful strategy for an agent in this game? The criteria for the evaluation of the strategy are:

The utility of the agent. The assumptions about other agents. Safety-level strategy guarantees a payoff

regardless of what others do, but is it good enough?

An Agent Centric Approach to Games

Example (Aumann 1985):

Pr(U)=3/4 is the safety-level strategy for the row player.Pr(U)=Pr(D)=1/2 ; Pr(L)=Pr(R)=1/2 is the unique NEThe row player expected payoff in the safety-level

strategy is as in the Nash equilibrium!

L R

U (2,6) (4,2)

D (6,0) (0,4)

Competitive Safety Analysis

A c-competitive safety level strategy guarantees a payoff which is at least 1/c of what is obtained in a (best) Nash equilibrium. C-competitive strategies extend upon Aumann observation, and enables it to be applicable as a design paradigm.

C-competitive strategies for small C exist for interesting classes of congestion games.

Competitive Safety Analysis in Position Auctions

In Varian’s (complete information) model we prove that for GSP:

With exponentially decreasing click-rate functions, assuming N bidders, the competitive safety ratio can be arbitrary close to N.

With linearly decreasing click-rate functions the ratio can not be greater than 1+ln(n); this bound is tight.

Click-Rates

Exponential click rate: for slot i

Linear click rate: for slot i

qx i 1

)1( iNd

An incomplete information settingEach advertiser knows his valuation per-click, but not the ones of others. The agents’ valuations are assumed to be taken from a known distribution. We consider an agent’s expected payoff under the VCG position auction, for a given valuation, when taking expectation over other agents’ valuations.

This coincides with the expectation of the agent’s payoff assuming the system converges to the best SNE for the agents in the GSP auction for the actual valuations.

An incomplete information settingWe consider safety-level strategies, where the other agents can see the agent’s strategy and valuation, and minimize its payoff in purely adversarial manner, as long they do not overbid. This gives a highly demanding setting. We present our results for the uniform distribution over the agents’ valuation

Competitive Safety Analysis in Position Auctions (revisited for incomplete information)

Results:The competitive safety ratio for exponentially decreasing click-rates is e. The competitive safety ratio for linearly decreasing click-rates is 2. Close to 1 competitive safety ratio if the agent’s valuation is “low”.

Intermediate Conclusions: providing advice to the bidder

Given the GSP auction, competitive safety strategies exist for the (more realistic) incomplete information setting!

The User Perspective: Revisiting the basic (complete information) model

We have users and not only advertisers!

I will try the first Is ad and see

An Alternative Model (Kuminov and T, Muthukrishnan et. al.):

A user considers advertisers according to their order on the page.Each advertiser has a fixed probability

0 < < 1 that a user will consume its ad, given that he is considering it. There is a fixed probability q > 0 that after considering an ad, and deciding against it, the user will continue to the next ad.

ip

An Alternative Model

• If advertisers are numbered according to their slots, the probability the user will consume the ad in position i is:

• The efficient allocation in this model is that the agents are allocated to slots according to where

iiv

1

1)1(

i

jji pqp

)1(1 i

ii

pq

p

An Alternative Model

Main theorems: Full characterization of a class of Nash

Equilibria (SNEs) for GSP in this new model.

The best SNE from the organizer perspective gives the VCG allocation / payments.

VCG leads to higher revenue than GSP!

An Alternative Model

Comments: The models are not reducible to one

another. The KT model is especially

appealing for PPA.

Intermediate Conclusions: incorporating a user model

• Users’ behavior should be incorporated

(Google -- Muthukrishnan et. al.,

Yahoo – Madian et. al.

Kuminov and T

suggest similar alternative model

The perspective advocated by Athey-Ellison at Harvard/MSR too)

• Basic results are reversed when considering the alternative model, asking to re-consider the use of GSP vs. VCG!

Mediators in Position Auctions

If help is needed then someone will provide it: the mediator perspective

Why mediators?

Advertisers aim at simple behavior leading to high utility. Mediators can offer such services.

Search engines aim at long-term revenues, and as a result encourage the use of mediators, even in the price of losing on short-term revenue.

Who may be the mediator?

• A 3rd party. Notice that this party can not change the auction rules.

• The search engine itself. This party does not want to change the auction’s basic rule/interface.

Mediators and the VCG outcome

In general, a mediator asks the agents for their valuations, and bid in the auction as a function of the reported valuations.

Agents may decide to participate in the auction directly.

We care about mediators who can generate welfare maximizing output in equilibrium, which is independent of probabilistic information, in which advertisers decide to approach the mediator and report their true valuations.

• Only the VCG outcome can be implemented! (Ashlagi, 2007)• The question is in which ad auctions, we can construct mediators

implementing the VCG outcome.

Exampleself-price, single slot auction 1=1, n=2

c-mediatorv1

v2

v2

0v1 v2

c>1 can lead to negative utilities for players who trust the mediator.

Exampleself-price, single slot auction 1=1, n=2

c-mediatorv1

v2

v2

0v1 v2

c-mediatorvi cvi

For every c 1 vcg can be implemented in the single-slot self-price auction.

c>1 can lead to negative utilities for players who trust the mediator.

Exampleself-price, single slot auction 1=1, n=2

c-mediatorv1

v2

v2

0v1≥v2

c-mediatorvi cvi

For every c 1 vcg can be implemented in the single-slot self-price auction.

Valid Mediators – players who trust the mediator never lose moneyThe c-mediator is valid for c=1

Self-Price Position Auctions

n=3, k=2

v1=5, v2=5, v3=10

The VCG outcome function can not be implemented in the self-price position auction unless k=1.

Self-Price Position Auctions

n=3, k=2

v1=5, v2=5, v3=10

The VCG outcome function can not be implemented in the self-price position auction unless k=1.

VCG

player 3, pays 5

player 1, pays 5

player 2, pays 0

Self-Price Position Auctions

n=3, k=2

v1=5, v2=5, v3=10

The mediator must submit 5 on behalf of both players 1 and 3. But then player 3 will not be assigned to the first position!

VCG can not be implemented in the self-price position auction unless k=1.

VCG

player 3, pays 5

player 1, pays 5

player 2, pays 0

Theorem: There exists a valid mediator that implements vcg in the next-price position auction

Next-price Position Auctions

Edelman, Ostrovsky and Schwarz provided a mechanism that can be viewed as a “simplified” form of a mediatorwhere participation is mandatory.

1+p1vcg(v)

p2vcg(v)

p1vcg(v)

pk-1vcg(v)

pkvcg(v)/2 pk

vcg(v)/2 pkvcg(v)/2

Positions according to v

If all players choose the mediator:

MN(v}=

Mediator for the next-price auction

1+p1vcg(v)

p2vcg(v)

p1vcg(v)

pk-1vcg(v)

pkvcg(v)/2 pk

vcg(v)/2 pkvcg(v)/2

Positions according to v

If some players play directly: MS(vS)=vS

If all players choose the mediator:

MN(v}=

Mediator for the next-price auction

Existence of Valid Mediators for Position AuctionsTheorem: Let G be a position auction. If the following

conditions hold then there exists a valid mediator that implements vcg in G.C1: position payment depends only on lower

positions’ bids.C2: VCG cover – any VCG outcome can be

obtained by some bid profile.C3: G is monotone – an increase in the bid

profile can not decrease payments at any slot.

Each one of these conditions is necessary.

Existence of Valid Mediators for Position Auctions (cont.)Corollaries (examples): Valid mediators exist in k-price

position auctions Quality factors

Valid mediators exist in the existing (Google, Yahoo, MSN) position auctions, where the click-through rate for player i in position j is j

where is player i’s quality factor

ie

ie

Intermediate conclusions: mediators

Introduced the study of mediators in games with incomplete information.Applied mediators to the context of position auctions.Characterization of the position auctions in which the VCG outcome function can be implemented.

Conclusions -- the wish list revisited:

• The system perspective: first game-theoretic analysis of competing ad auctions; great search does not imply great revenue in sponsored search.

• The agent perspective: competitive safety strategies exist for non-trivial ad auctions setting.

• The user perspective: re-considering user models brings new insights; basic results are reversed.

• The mediator perspective: general results/protocols for the context of position auctions.

Thank you