Revenue Maximization in
Probabilistic Single-Item Auctions
by means of Signaling
Michal Feldman
Hebrew University & Microsoft Israel
Joint work with:
Yuval Emek (ETH)
Iftah Gamzu (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
Bidders (advertisers)
Items (impressions)
…
…
...
…
…
…
1
…
i
…
n
100
10
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
Bidder Good 1
p(1)
#
…
Good j
p(j)
Good m
p(m)
Ep[v1,j] max2
1
Bidders
…
…
i
Vi,j
Ep[vi,j] max1
…
n
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)
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 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 C3 C2 C4
Bidder
#
Male/
California
Male/
Arizona
Female/
California
Female/
Arizona
p(1)
p(2)
p(3)
p(4)
Bidders
1
2
3
Vi,j
4
5
 Single cluster (reveal no information)
 Singletons (reveal actual realization)
 Other signaling schemes:
 Male / Female
 California / Arizona
Is it worthwhile to reveal info?
Bidder Good 1 Good 2 Good 3 Good 4
#
1/4
1/4
1/4
1/4
1
2
3
4
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
 Revealing: 0
 Not revealing: 1/4
Bidder Good 1 Good 2
#
1/2
1/2
1
2
3
4
1
1
0
0
0
0
1
1
 Revealing: 1
 Not revealing: 1/2
Other structures
Goods
Bidders
1
1
…
i
…
n
1/m
1
…
m
…
1/m
0
1
1
1
0
1
1
 Single cluster: expected revenue = 1/m
 Singletons: expected revenue = 0
 Clusters of size 2: expected revenue = 1/2
1
1
Revenue Maximization (RMS)
 Given a signaling scheme C, the expected revenue of
the auctioneer is


R (C )   p (Cl )  max 2i[ n ]   p ( j | Cl )  Vi , j 
l[ k ]
 jCl

 E p [Vi , j | Cl ]  bi (l )
 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


R(C )   p (Cl )  max 2i[ n ]   p( j | Cl ) Vi , j 
l[ k ]
 jCl



  max 2i[ n ]   p( j ) Vi , j 
l[ k ]
 jCl



  max 2i[ n ]   i , j 
l[ k ]
 jCl

 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


R(C )   p (Cl )  max 2i[ n ]   p( j | Cl ) Vi , j 
l[ k ]
 jCl



  max 2i[ n ]   p( j ) Vi , j 
l[ k ]
 jCl



  max 2i[ n ]   i , j 
l[ k ]
 jCl

j
P(j)
1
i
n
 RMS problem: design signaling scheme C that
maximizes R(C)
𝑉𝑖,𝑗
Revenue Maximization (RMS)
 Given a signaling scheme C, the expected revenue of
the auctioneer is


R(C )   p (Cl )  max 2i[ n ]   p( j | Cl ) Vi , j 
l[ k ]
 jCl



  max 2i[ n ]   p( j ) Vi , j 
l[ k ]
 jCl



  max 2i[ n ]   i , j 
l[ k ]
 jCl

j
1
i
𝜓𝑖,𝑗
n
 SRMS problem (simplified RMS): design signaling
scheme C that maximizes last expression
Revenue maximization by signaling
C1
Bidder
#
Male/
California
C2
Male/
Arizona
Female/
California
p(2)
j
Female/
Arizona
1
max2
i
max1
i,j
+
max2
n
max1


R (C )   max 2i[ n ]   i , j 
l[ k ]
 jCl

=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
1
1
2
4
n
g1
2
m
g2
g4
gn
gn-1
 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
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



RA (C )   EYi , j max 2i[ n ]   i , j 
l[ k ]

 jCl

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.