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

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

…

… … … 10 100i

…

nBidd

ers (

adve

rtise

rs)

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 Vi,j: valuation of bidder i[n] for item j[m] The bidders know the probability distribution

pD(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 auctionGood m … Good j … Good 1 Bidder

#

1

…

i

…n

Vi,j

p(1) p(j) p(m)

Ep[v1,j]

Ep[vi,j]

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] }

max1

max2

Bidd

ers

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 C1 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 0 for jCl )

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 max2in{bi}

Signaling schemesFemale/Arizona p(4)

Female/California

p(3)

Male/Arizona p(2)

Male/California

p(1)

Bidder #

1

2

Vi,j 3

45

Single cluster (reveal no information) Singletons (reveal actual realization) Other signaling schemes:

Male / Female California / Arizona

C1C1 C2 C3 C4C1 C2

C1 C2

Bidd

ers

Is it worthwhile to reveal info?

Revealing: 1 Not revealing: 1/2

Good 41/4

Good 31/4

Good 21/4

Good 11/4

Bidder #

0 0 0 1 1

0 0 1 0 2

0 1 0 0 3

1 0 0 0 4

Good 21/2

Good 11/2

Bidder #

0 1 1

0 1 2

1 0 3

1 0 4

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

11

1…

1i

1…

11

11 n

1/m 1/mBi

dder

s

Goods

00

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)

][

,][ )|(2max)()(kl Cj

jilnill

VCjpCpCR

)(]|[E ,p lbCV ilji




][,][

][,][

][,][

2max

)(2max

)|(2max)()(

kl Cjjini

kl Cjjini

kl Cjjilnil

l

l

l

Vjp

VCjpCpCR




][,][

][,][

][,][

2max

)(2max

)|(2max)()(

kl Cjjini

kl Cjjini

kl Cjjilnil

l

l

l

Vjp

VCjpCpCR

1

n

P(j)j

i



SRMS problem (simplified RMS): design signaling scheme C that maximizes last expression

][,][

][,][

][,][

2max

)(2max

)|(2max)()(

kl Cjjini

kl Cjjini

kl Cjjilnil

l

l

l

Vjp

VCjpCpCR

1

n

j

i

Female/Arizona

Female/California

Male/Arizona p(2)

Male/California

Bidder #

1

n

i,j

C1 C2

max1

max2 max1

max2

max2

max2

][

,][2max)(kl Cj

jinil

CR

+

=R(C)

Revenue maximization by signaling

i

j

RMS hardness Theorem: given a fixed-value matrix YZnxm 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

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 2 1

1

2

4

n

g1g2

g4

gn

gn-1

Bayesian setting Practically, the auctioneer does not

know the exact valuation of each bidder

Bidder valuations Vi,j (and consequently Yi,j) are random variables

Auctioneer revenue is given by

Y

][

,][2max)(,

kl CjjiniA

l

jiECR

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.

Documents

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