Revenue Maximization in Probabilistic Single-Item Auctions by means of Signaling Joint work with: Yuval Emek (ETH) Iftah Gamzu (Microsoft Israel) Moshe Tennenholtz (Microsoft Israel & Technion) Michal Feldman Hebrew University & Microsoft Israel

Asymmetry of information  Asymmetry of information is prevalent in auction settings  Specifically, the auctioneer possesses an informational superiority over the bidders  The problem: how best to exploit the informational superiority to generate higher revenue?

Ad auctions – market for impressions

 The goods: end users (“impressions”) (navigate through web pages)  The bidders: advertisers (wish to target ads at the right end users, and usually have very limited knowledge for who is behind the impression)  The auctioneer: publisher (controls and generates web pages content, typically has a much more accurate information about the site visitors) Market for impressions

Valuation matrix... … … 1 … ………10100 i … n Bidders (advertisers) Items (impressions)

Probabilistic single-item auction (PSIA)  A single item is sold in an auction with n bidders  The auctioned item is one of m possible items  V i,j : valuation of bidder i  [n] for item j  [m]  The bidders know the probability distribution p  (m) over the items  The auctioneer knows the actual realization of the item  The item is sold in a second price auction  Winner: bidder with highest bid  Payment: second highest bid  An instance of a PSIA is denoted A = (n,m,p,V)

Probabilistic single-item auction Good m … Good j … Good 1 Bidder # 1 … i … n V i,j p(1)p(j)p(m) E p [v 1,j ] E p [v i,j ] E p [v n,j ]  Observation: it’s a dominant strategy (in second price auction) to reveal one’s true expected value (same logic as in the deterministic case)  Expected revenue = max2 i  [n] { E p [V i,j ] } max1 max2 Bidders

Market for impressions  Various business models have been proposed and used in the market for impression, varying in  Mechanism used to sell impressions (e.g., auction, fixed price)  How much information is revealed to the advertisers  We propose a “signaling scheme” technique that can significantly increase the auctioneer’s revenue  The publisher partitions the impressions into segments, and once an impression is realized, the segment that contains it is revealed to the advertisers

Signaling scheme  Given a PSIA A = (n,m,p,V)  Auctioneer partitions goods into (pairwise disjoint) clusters C 1 U  U C k = [m]  Once a good j is chosen (with probability p(j)), the bidders are signaled cluster C l that contains j, which induces a new probability distribution: p(j | C l ) = p(j) / p(C l ) for every good j  C l (and 0 for j  C l )  The Revenue Maximization by Signaling (RMS) problem: what is the signaling scheme that maximizes the auctioneer’s revenue?  Recall: 2 nd price auction --- each bidder i submits bid b i, and highest bidder wins and pays max2 i  n {b i }

Signaling schemes Female/ Arizona p(4) Female/ California p(3) Male/ Arizona p(2) Male/ California p(1) Bidder # 1 2 V i,j  Single cluster (reveal no information)  Singletons (reveal actual realization)  Other signaling schemes:  Male / Female  California / Arizona C1C1 C1C1 C2C2 C3C3 C4C4 C1C1 C2C2 C1C1 C2C2 Bidders

Is it worthwhile to reveal info?  Revealing: 1  Not revealing: 1/2 Good 4 1/4 Good 3 1/4 Good 2 1/4 Good 1 1/4 Bidder # Good 2 1/2 Good 1 1/2 Bidder #  Revealing: 0  Not revealing: 1/4

Other structures  Single cluster: expected revenue = 1/m  Singletons: expected revenue = 0  Clusters of size 2: expected revenue = 1/2 m …… … 1 i 1 … n 1/m Bidders Goods 0 0

Revenue Maximization (RMS)  Given a signaling scheme C, the expected revenue of the auctioneer is  RMS problem: design signaling scheme C that maximizes R(C)

Revenue Maximization (RMS)  Given a signaling scheme C, the expected revenue of the auctioneer is  RMS problem: design signaling scheme C that maximizes R(C)

Revenue Maximization (RMS)  Given a signaling scheme C, the expected revenue of the auctioneer is  RMS problem: design signaling scheme C that maximizes R(C) 1 n P(j) j i

Revenue Maximization (RMS)  Given a signaling scheme C, the expected revenue of the auctioneer is  SRMS problem (simplified RMS): design signaling scheme C that maximizes last expression 1 n j i

Female/ Arizona Female/ California Male/ Arizona p(2) Male/ California Bidder # 1 n  i,j C1C1 C2C2 max1 max2max1 max2 + =R(C) Revenue maximization by signaling i j

RMS hardness  Theorem: given a fixed-value matrix  nxm and some integer , it is strongly NP-hard to determine if SRMS on  admits a signaling scheme with revenue at least   Proof: reduction from 3-partition  Corollary: RMS admits no FPTAS (unless P=NP)  Remarks:  Problem remains hard even if every good is desired by at most a single bidder, and even if there are only 3 bidders  Yet, some cases are easy; e.g., if all values are binary, then the problem is polynomial

Aproximation  Constant factor approximation:  Step 1: greedy matching -- match sets that are “close” to each other  Step 2: choose the best of (i) a single cluster of the rest, or (ii) singleton clusters of the rest m n 11 22 44 nn  n-1

Bayesian setting  Practically, the auctioneer does not know the exact valuation of each bidder  Bidder valuations V i,j (and consequently  i,j ) are random variables  Auctioneer revenue is given by

Bayesian setting  Theorem: if the (valuation) random variables are sufficiently concentrated around the expectation, then the problem possesses constant approximation to the RMS problem  By running the algorithm on the matrix of expectations  Open problem: can our algorithm work for a more extensive family of valuation matrix distributions?

Summary  We study auction settings with asymmetric information between auctioneer and bidders  A well-designed signaling scheme can significantly enhance the auctioneer’s revenue  Maximizing revenue is a hard problem  Yet, a constant factor approximation exists for some families of valuations  Future / ongoing directions:  Existence of PTAS  Approximation for general distributions  Asymmetric signaling schemes Thank you.