Peaches, Lemons, and Cookies: Designing Auction Markets with - - PowerPoint PPT Presentation

Peaches, Lemons, and Cookies: Designing Auction Markets with Dispersed Information

Moshe Babaioff (Microsoft Research)

Joint work with Ittai Abraham (Microsoft Research), Susan Athey (Harvard & Microsoft) Michael Grubb (MIT)

Innovations in Algorithmic Game Theory Hebrew University, Jerusalem, Israel

May 25, 2011

Advertiser/ DSP/bidder

Real Time Market for Impressions*

Publisher Seller/Auctioneer

impression

Advertiser/ DSP/bidder Advertiser/ DSP/bidder

Impression data Compute value based on user cookies Real time bid

*A stylized model

• ffline bid

impression

2nd Price Auction (SPA) random tie breaking

Market Characteristics

• Real Time allocation of impressions to

advertisers

• Asymmetric information about the

impressions’ value, due to cookies

• Examples:

– Ad Exchanges:

Advertiser’s Valuations

• Advertisers derive value from presenting display

advertisements to users (buying impressions).

• Realistically: a correlated value setting
• First approximation: a common value setting

– High value users - spend much money online. – Low value users – bots (non-human impressions), people that do no spend much.

• Our focus: information asymmetry in common

value SPA

A Simple Example

A Simple Setting

Common Value: Impressions have same value (quality) q to all bidders

• Item is either a Peach (value of H), or a Lemon (value of

L<H), value is unobserved

• Known prior: Pr[q=H]=Pr[q=L]=½
• Expected value is E=(L+H)/2

Asymmetric information:

• Informed bidder: with some (small) probability knows

something about the value

– Bidder has cookies on some user’s machines

• Uninformed bidder: Knows only the prior

Does not have a cookie, or is bidding in advance (not in real time)

Equilibrium Selection Problem

• 1 is perfectly informed, 2 is uninformed
• For any X the following is a NE:

– The informed is bidding the posterior value – The uninformed is bidding X.

• For any X, the uninformed has 0 utility.
• Revenue prediction?

H L X

Our Refinement (informal): Tremble Robust Equilibrium (TRE)

• Refinement of Nash Equilibrium (NE)
• Models some (tiny) uncertainty about the

environment.

• (ϵ,R)-Tremble of the game: An additional

bidder enters the auction with probability ϵ>0, bids according to a “standard” distribution R.

• Tremble Robust Equilibrium: the limit of a

sequence of NE in the (ϵ,R)-tremble of the game, as ϵ0.

Our Refinement (informal): Tremble Robust Equilibrium (TRE)

• Refinement of Nash Equilibrium (NE)
• Models some (tiny) uncertainty about the

environment.

• (ϵ,R)-Tremble of the game: An additional

bidder enters the auction with probability ϵ>0, bids according to a “standard” distribution R.

• Tremble Robust Equilibrium: the limit of a

sequence of NE in the (ϵ,R)-tremble of the game, as ϵ0.

We suggest this new refinement as, even in the simple setting, the standard refinement of “Tremble Hand Perfect Equilibrium”(Selten’75) is not sufficient to provide the unique (intuitive) solution.

Back to the Equilibrium Selection Problem

• Recall: For any X the following is a Nash Eq.:

– The informed is bidding the posterior value. – The uninformed is bidding X.

• With the random bidder any bid X>L yields

negative utility.

• In the unique TRE in undominated strategies

the uninformed bids L.

• Revenue prediction: L

H L X b

Peaches vs. Lemons

• 3 signals: L, H, D (Default)
• If value is H, gets signal H with probability ϵH
• If value is L, gets signal L with probability ϵL
• Otherwise he gets signal D

Cream Skimming ϵH>0, ϵL=0 H E- E Avoiding Lemons ϵH=0, ϵL>0 L E+ Assortment of Cookies ϵH>0, ϵL>0 H ED L

Revenue = expected value given the lowest signal

Markets for Lemons

• SPA market collapse in the spirit of Akerlof’s

Markets for Lemons.

• Few differences:

– The uninformed bidder can actually win high quality items in NE (by bidding H+>H). – Multiplicity of equilibria due to 2nd price payment. Any bid: the uninformed pays a fair price.

• Our TRE refinement retrieves the intuitive
• utcome as a unique prediction.

Outline

• Model and Refinement
• A Single Informed Bidder
• Two Bidders, Binary Signals (2x2)
• Mechanism Design: Discussion
• Summary

Model and Refinement

Model

• Unobserved state of the world ω in Ω.
• For each bidder i, finite set of signals Si.
• Each bidder i gets a private signal si in Si about

the state.

• Common prior.
• Common value v(ω) in [0,1], determined by

the unknown state ω.

Tremble of the Game

• (ϵ,R)-Tremble of a SPA game λ(ϵ, R): the SPA

game with an additional “random bidder” entering with probability ϵ, and bidding according to a standard distribution R

Definition: A distribution R is standard if

• 1. the support of R is [0,1],
• 2. R is continuous, strictly increasing and differentiable,
• 3. its density r is continuous and positive on the interval [0,1].

Tremble Robust Equilibrium (TRE)

Definition: A Nash Equilibrium μ is a Tremble Robust Equilibrium (TRE) of the game, if for every standard distribution R the following holds.

• For every decreasing sequence of positives ϵ1, ϵ2 ,ϵ3,…

such that

1. limj∞ ϵj = 0 2. μ(ϵj) is a Nash equilibrium of the (ϵj, R)-Tremble

It holds that {μi(ϵj)(si)}j=1

∞ convergence in distribution to

μi(si) for every bidder i and signal si . Moreover, a TRE μ is a strong Tremble Robust Equilibrium if for every sequence satisfying (1) and (2) above there exists k such that for every j>k, in (2) it holds that μ(ϵj)=μ.

A Single Informed Bidder

Single Informed Bidder: TRE

Theorem: In any domain with a single informed bidder, a strong TRE (in pure strategies) of the SPA game is the profile of strategies μ in which:

• The informed bidder bids his posterior
• Each of the uniformed bidders bids the

expected value conditional on the informed lowest signal. Moreover, this profile is the unique TRE in undominated strategies.

Single Informed Bidder: Revenue

Corollary: in the unique TRE in undominated strategies the revenue equals to the expected value conditional on the informed lowest signal. Revenue collapse: independent of the probability that the informed bidder gets the lowest signal. What happens in more complex information structures?

Two Bidders, Binary Signals (2x2)

The Setting

• 2 bidders, bidder i gets a signal in {Li, Hi},

1>Pr(Hi)>0 and Pr(H1,H2)>0

• v(L1,L2)= 0, v(H1,H2)=1
• v(H1,L2)=v1, v(L1,H2)=v2
• Monotonic domain: v1,v2 in [0,1]
• Special case covered by Milgrom-Weber (‘82)

– Symmetric bidders: v1 = v2 & Pr (H1,L2) = Pr(L1,H2)

H1 H2

v2 v1 1

H1 H2

V2 V1 1

Special Cases: Case 1

• Both bidders perfectly informed about the value
• f 1: v(H1) = v(H2) = v(H1 ,H2) =1.

– bidder i: bid 1 on Hi, 0 on Li – a strong TRE, unique in undominated strategies (US). Pr(H1,L2) (1-v1)= Pr(L1,H2) (1-v2)=0

H1 H2

1

H1 H2

V2=1 V1=1 1

H1 H2

1 v2=1

Special Cases: Case 2

• Only bidder 1 is perfectly informed about

value 1: v(H2) < v(H1) = v(H1 ,H2) =1, v2<1

– Bidder 1: bids 1 on H1, 0 on L1 – Bidder 2: bids v2 on H2, 0 on L2 – a strong TRE, unique in US. 0=Pr(H1,L2) (1-v1)< Pr(L1,H2) (1-v2)

H1 H2

1 v2

H1 H2

v2 v1=1 1

The General Case

0<Pr(H1,L2) (1-v1)≤ Pr(L1,H2) (1-v2)<1

(in particular, max{v(H1),v(H2)}<v(H1,H2)=1 )

Which equilibrium should be picked? Say v1=v2=0:

• 1. Bidder 1: bids 1 on H1. Bidder 2: bids 0 on H2
• 2. Bidder 1: bids 1 on H1. Bidder 2: bids 1 on H2
• 3. Maybe a mixed NE? Which one?

H1 H2

v2 v1 1

2 Bidders, Binary signals: Main Result

Theorem: Assume that 0<Pr(H1,L2) (1-v1)≤ Pr(L1,H2) (1-v2)<1. The unique TRE of the SPA game is the profile of strategies μ in which:

• Every bidder i bids 0 when getting signal Li.
• Bidder 1 with signal H1 always bids 1.
• Bidder 2 with signal H2

– bids 1 with probability

Pr⁡ 𝐼1,𝑀2 (1−𝑤1) Pr⁡ 𝑀1,𝐼2 (1−𝑤2)

– bids v2 with the remaining probability.

2 Bidders, Binary signals: Main Result

Theorem: Assume that 0<Pr(H1,L2) (1-v1)≤ Pr(L1,H2) (1-v2)<1. The unique TRE of the SPA game is the profile of strategies μ in which:

• Every bidder i bids 0 when getting signal Li.
• Bidder 1 with signal H1 always bids 1.
• Bidder 2 with signal H2

– bids 1 with probability

Pr⁡ 𝐼1,𝑀2 (1−𝑤1) Pr⁡ 𝑀1,𝐼2 (1−𝑤2)

– bids v2 with the remaining probability.

Ratio of bidders’ “strength”. The strength of bidder i is Pr(Hi,Lj) (1-vi). (The strength equals to the loss of bidder i with signal Hi that is bidding 1, if he pays his bid (tie with the random bidder) ). Bidder 1 is (weakly) stronger.

Relation to Milgrom-Weber (1982)

• A special case also covered by MW’82:

– Symmetric bidders: v1 = v2 & Pr (H1,L2) = Pr(L1,H2) Symmetric NE: bidder i with signal Hi bids 1 (bidding as if the other bidder got the same signal)

• This is also the unique TRE

– A justification for picking this specific equilibrium

• The TRE refinement also gives a unique

prediction in asymmetric cases.

Revenue Prediction

Consider the case that v1=v2=0.

• In the unique TRE: revenue is Pr⁡ 𝐼1,𝑀2

Pr⁡ 𝑀1,𝐼2 ⁡-fraction of

the surplus.

• Proportional to the “exclusive lemon information

ratio”

• Revenue=welfare in the symmetric case (MW’82).

H1 H2

1

Proof idea

• Characterize NE η in any (ϵ,R)-Tremble of the

game, when ϵ is small. Show existence.

• Show that the limit of any sequence of such

NE must be μ, when ϵ goes to 0.

Necessary conditions for NE η in an (ϵ,R)-Tremble, small ϵ

bmin>max{vi,vj} bmax

𝑐𝑛𝑗𝑜 = Pr 𝐼

𝑘 𝐼𝑗 𝐻 𝑘 𝑤𝑘 + 𝑤𝑗 Pr 𝑀𝑘 𝐼𝑗

Pr 𝐼

𝑘 𝐼𝑗 𝐻 𝑘 𝑤𝑘 + Pr 𝑀𝑘 𝐼𝑗

𝐻𝑗 𝑐𝑛𝑗𝑜 = ε Pr 𝑀𝑗 𝐼

𝑘

Pr 𝐼𝑗 𝐼

𝑘

𝑦 − 𝑤𝑘 1 − 𝑐𝑛𝑗𝑜 𝑠 𝑦 𝑆 𝑐𝑛𝑗𝑜 𝑒𝑦

𝑐𝑛𝑗𝑜 𝑤𝑘

i j

Strictly increasing on this joint range, no atoms bmin=vj>vi

bmax

i j 1 If and only if bidders are symmetric

vj

𝑆 𝑐 =1-ε(1-𝑆 𝑐 ), 𝑠 𝑐 =εr 𝑐

bmin=v1=v2

bmax

CDFs on (bmin, bmax)

• Using First Order Conditions (indifference over

all bids in the support) we show that for every bidder i and every bid b in (bmin, bmax):

𝐻𝑗 𝑐 = ⁡𝜁

Pr 𝑀𝑗 𝐼𝑘 Pr 𝐼𝑗 𝐼𝑘 𝑦−𝑤𝑘 1−𝑦 𝑠 𝑦 𝑆 𝑐 𝑒𝑦 𝑐 𝑐𝑛𝑗𝑜

+

𝑆 𝑐𝑛𝑗𝑜 𝑆 𝑐

𝐻𝑗 𝑐𝑛𝑗𝑜

Atoms

𝐻𝑗 𝑐 =⁡

Pr 𝑀𝑗 𝐼𝑘 Pr 𝐼𝑗 𝐼𝑘 𝑦−𝑤𝑘 1−𝑦 𝑠 𝑦 𝑆 𝑐 𝑒𝑦 𝑐 𝑐𝑛𝑗𝑜

+

𝑆 𝑐𝑛𝑗𝑜 𝑆 𝑐

𝐻𝑗 𝑐𝑛𝑗𝑜 1−

𝑆 𝑐𝑛𝑗𝑜 𝑆 𝑐𝑛𝑏𝑦 𝐻𝑗 𝑐𝑛𝑗𝑜 =⁡ Pr 𝑀𝑗 𝐼𝑘 Pr 𝐼𝑗 𝐼𝑘 𝑦−𝑤𝑘 1−𝑦 𝑠 𝑦 𝑆 𝑐𝑛𝑏𝑦 𝑒𝑦 𝑐𝑛𝑏𝑦 𝑐𝑛𝑗𝑜

1−𝛾𝐻2 𝑐𝑛𝑗𝑜 1−𝛾𝐻1 𝑐𝑛𝑗𝑜 = Pr 𝐼1, 𝑀2 Pr 𝑀1, 𝐼2 (1−𝑈𝑤1) (1−𝑈𝑤2) 𝐻2 𝑤2 = 𝐻2 𝑐𝑛𝑗𝑜

ε→0 1 −⁡Pr 𝐼1, 𝑀2

Pr 𝑀1, 𝐼2 (1−𝑤1) (1−𝑤2)

“Strength Ratio”, at most 1. (strength of i=Pr(Hi,Lj) (1-vi) )

Uniform noise, ϵ=1/10 v1=1/8, v2=1/4, Pr[H1,H2]=Pr[H1,L2]=1/5, Pr[L1,H2]= Pr[L1,L2]= 3/10, bmin=v2=1/4, G1(1/4)=0, G2(1/4)=2/9, G2(b) G1(b) bmin=v2=1/4

Mechanism Design: Discussion

Maximizing Revenue

• As observed, SPA has low revenue with

asymmetrically informed bidders.

• A (trivial,) optimal, dominant strategy, ex-ante

individually rational, mechanism: always sell to the uninformed bidder at the price of E.

• This misses the fact that in reality additional

value is created by serving the right ad to the user.

Ad Targeting

• Simple setting: if an informed advertiser gets

his highest signal, he can tailor the ad to the user and get a Bonus B>0.

SPA revenue collapse results, as well as the uniqueness of TRE in undominated strategies extend to this setting.

Ad Targeting

• Simple setting: if an informed advertiser gets his

highest signal, he can tailor the ad to the user and get a Bonus B>0.

• Assume that a random advertiser is informed.
• Ex Ante allocation is a bad idea!
• We must allocate the impressions in real time to

get maximal welfare.

• SPA for every impression gets high welfare but

poor revenue.

SPA with Entry Fees

Entry fees can be used for full revenue extraction by an ex-ante individually rational (IR) mechanism. The TRE refinement is crucial: provides a unique prediction. Practical drawback: Seller must know the prior.

An Interim IR Mechanism

• A randomly chosen bidder is exclusively

informed, with a binary signal.

• A mechanism in the spirit of Myerson’s private

value optimal auction:

– SPA on all bids above r (set optimally), – pooling at a floor price f if no bid above r.

• A dominant strategy, interim-IR mechanism,

(1-1/n)-fraction of the welfare as revenue.

Interim IR Mechanism – extension

• What if the seller does not know the

conditional expectations?

• A two stage mechanism

– Stage 1:

• bidders are bidding to set the floor and reserve prices.
• Each set by the lowest bid, this bidder is excluded.

– Stage 2: run the previous mechanism.

• interim-IR mechanism, (1-1/n)2-fraction of the

welfare as revenue in subgame-perfect eq.

Summary

• Slight asymmetry in information has major effect
• n common value SPA auctions.

– Prediction based on a new refinement of NE. – Revenue collapses with a single informed bidder.

• Our new refinement gives a unique prediction in

the 2 bidders, binary signals setting

– Revenue relates to the “strength ratio”

• Initial results for n bidders with finitely many

signals each (in the paper).

• Mechanism design: ex-ante IR, interim IR.

Thanks!