Post on 23-Feb-2016
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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
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
)(2max
)|(2max)()(
kl Cjjini
kl Cjjini
kl Cjjilnil
l
l
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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)
][,][
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2max
)(2max
)|(2max)()(
kl Cjjini
kl Cjjini
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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
][,][
][,][
][,][
2max
)(2max
)|(2max)()(
kl Cjjini
kl Cjjini
kl Cjjilnil
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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.