Revenue Maximization in Probabilistic Single-Item Auctions by means of Signaling

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

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

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)

Valuation matrix Items (impressions) Bidders (advertisers)

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 • Vi,j: valuation of bidder i[n] for item j[m] • The bidders know the probability distribution pD(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 p(1) p(j) p(m) Ep[v1,j] max2 Bidders Vi,j Ep[vi,j] max1 Ep[vn,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] { Ep[Vi,j] }

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) clustersC1 U  U Ck = [m] • Once a good j is chosen (with probability p(j)), the bidders are signaled cluster Cl that contains j, which induces a new probability distribution: p(j | Cl) = p(j) / p(Cl)for every good j Cl(and 0for jCl) • The Revenue Maximization by Signaling (RMS) problem: what is the signaling scheme that maximizes the auctioneer’s revenue? • Recall: 2nd price auction --- each bidder i submits bid bi, and highest bidder wins and pays max2in{bi}

Signaling schemes C1 C2 C1 C1 C2 C1 C2 C3 C4 Bidders • Single cluster (reveal no information) • Singletons (reveal actual realization) • Other signaling schemes: • Male / Female • California / Arizona

Is it worthwhile to reveal info? • Revealing: 1 • Not revealing: 1/2 • Revealing: 0 • Not revealing: 1/4

Other structures Goods • Single cluster: expected revenue = 1/m • Singletons: expected revenue = 0 • Clusters of size 2: expected revenue = 1/2 1/m 1/m 0 Bidders 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) j P(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 j i

Revenue maximization by signaling C1 C2 j max2 max2 max1 • i,j + i max2 max2 max1 =R(C)

RMS hardness • Theorem:given a fixed-value matrix YZnxm and some integer a, it is strongly NP-hard to determine if SRMS on Y admits a signaling scheme with revenue at least a • 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 g1 • 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 g2 g4 gn-1 gn

Bayesian setting • Practically, the auctioneer does not know the exact valuation of each bidder • Bidder valuations Vi,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 asymmetricinformation 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.

Revenue Maximization in Probabilistic Single-Item Auctions by means of Signaling