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

Real Time Market for Impressions* 2 nd Price Auction (SPA) random tie breaking Publisher Seller/Auctioneer impression impression Real time bid Compute value Impression offline bid based on user data cookies Advertiser/ Advertiser/ Advertiser/ DSP/bidder DSP/bidder DSP/bidder *A stylized model

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

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 We suggest this new refinement as, even in the simple setting, the standard refinement of “Tremble environment. Hand Perfect Equilibrium”(Selten’75) is not sufficient • ( ϵ ,R)- Tremble of the game: An additional to provide the unique (intuitive) solution. 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.

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 . X L b H • 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

Peaches vs. Lemons • 3 signals: L, H, D (Default) • If value is H , gets signal H with probability ϵ H • If value is L , gets signal L with probability ϵ L • Otherwise he gets signal D E Cream Skimming ϵ H >0, ϵ L =0 E - H Avoiding Lemons ϵ H =0, ϵ L >0 E + L Assortment of Cookies ϵ H >0, ϵ L >0 E D L H 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 2 nd price payment. Any bid: the uninformed pays a fair price. • Our TRE refinement retrieves the intuitive outcome 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 S i . • Each bidder i gets a private signal s i in S i 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 ϵ 1 , ϵ 2 , ϵ 3 ,… such that lim j  ∞ ϵ j = 0 1. μ( ϵ j ) is a Nash equilibrium of the ( ϵ j , R ) -Tremble 2. It holds that { μ i ( ϵ j )(s i )} j=1 ∞ convergence in distribution to μ i (s i ) for every bidder i and signal s i . 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( ϵ j ) = 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 {L i , H i }, 1>Pr(H i )>0 and Pr(H 1 ,H 2 )>0 • v(L 1 ,L 2 )= 0, v(H 1 ,H 2 )=1 0 V 2 V 1 H 1 1 H 2 • v(H 1 ,L 2 )=v 1 , v(L 1 ,H 2 )=v 2 • Monotonic domain: v 1 ,v 2 in [0,1] • Special case covered by Milgrom- Weber (‘82) – Symmetric bidders: v 1 = v 2 & Pr (H 1 ,L 2 ) = Pr(L 1 ,H 2 ) 0 H 1 H 2 v 2 v 1 1

Special Cases: Case 1 • Both bidders perfectly informed about the value of 1 : v(H 1 ) = v(H 2 ) = v(H 1 ,H 2 ) =1. – bidder i : bid 1 on H i , 0 on L i – a strong TRE, unique in undominated strategies (US). 0 H 1 0 0 1 V 2 =1 V 1 =1 H 1 1 H 2 H 1 v 2 =1 1 H 2 H 2 Pr(H 1 ,L 2 ) (1-v 1 )= Pr(L 1 ,H 2 ) (1-v 2 )=0

Special Cases: Case 2 • Only bidder 1 is perfectly informed about value 1: v(H 2 ) < v(H 1 ) = v(H 1 ,H 2 ) =1, v 2 <1 – Bidder 1: bids 1 on H 1 , 0 on L 1 – Bidder 2 : bids v 2 on H 2 , 0 on L 2 – a strong TRE, unique in US. 0 1 0 H 1 v 2 v 2 v 1 =1 H 1 1 H 2 H 2 0=Pr(H 1 ,L 2 ) (1-v 1 )< Pr(L 1 ,H 2 ) (1-v 2 )

The General Case 0<Pr(H 1 ,L 2 ) (1-v 1 )≤ Pr(L 1 ,H 2 ) (1-v 2 )<1 (in particular, max{v(H 1 ),v(H 2 )}<v(H 1 ,H 2 )=1 ) 0 H 1 H 2 v 2 v 1 1 Which equilibrium should be picked? Say v 1 =v 2 =0 : 1. Bidder 1: bids 1 on H 1 . Bidder 2: bids 0 on H 2 2. Bidder 1: bids 1 on H 1 . Bidder 2: bids 1 on H 2 3. Maybe a mixed NE? Which one?

2 Bidders, Binary signals: Main Result Theorem: Assume that 0<Pr(H 1 ,L 2 ) (1-v 1 )≤ Pr(L 1 ,H 2 ) (1-v 2 )<1. The unique TRE of the SPA game is the profile of strategies μ in which: • Every bidder i bids 0 when getting signal L i . • Bidder 1 with signal H 1 always bids 1. • Bidder 2 with signal H 2 Pr⁡ 𝐼 1 ,𝑀 2 (1−𝑤 1 ) – bids 1 with probability Pr⁡ 𝑀 1 ,𝐼 2 (1−𝑤 2 ) – bids v 2 with the remaining probability.

2 Bidders, Binary signals: Main Result Theorem: Assume that 0<Pr(H 1 ,L 2 ) (1-v 1 )≤ Pr(L 1 ,H 2 ) (1-v 2 )<1. Ratio of bidders’ “strength”. The unique TRE of the SPA game is the profile of The strength of bidder i is Pr(H i ,L j ) (1-v i ). strategies μ in which: (The strength equals to the loss of bidder i with signal H i that • Every bidder i bids 0 when getting signal L i . is bidding 1 , if he pays his bid (tie with the random bidder) ). • Bidder 1 with signal H 1 always bids 1. Bidder 1 is (weakly) stronger. • Bidder 2 with signal H 2 Pr⁡ 𝐼 1 ,𝑀 2 (1−𝑤 1 ) – bids 1 with probability Pr⁡ 𝑀 1 ,𝐼 2 (1−𝑤 2 ) – bids v 2 with the remaining probability.

Relation to Milgrom-Weber (1982) • A special case also covered by MW’82 : – Symmetric bidders: v 1 = v 2 & Pr (H 1 ,L 2 ) = Pr(L 1 ,H 2 ) Symmetric NE: bidder i with signal H i 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.