Item Pricing for Revenue Maximization in
Combinatorial Auctions
Maria-Florina Balcan,
Carnegie Mellon University
Joint with Avrim Blum and Yishay Mansour
Outline of the Talk
Revenue Maximization in Combinatorial Auctions
• Item Pricing in Unlimited Supply Combinatorial Auctions
• Single-minded bidders.
[Balcan-Blum’06]
• General bidders. [Balcan-Blum-Mansour’07]
• Item Pricing in Limited Supply Combinatorial Auctions
• Bidders with subadditive valuations.
[Balcan-Blum-Mansour’07]
Supermarket Pricing Problem
• A supermarket trying to decide on how to price the goods.
Seller’s Goal: set prices to maximize revenue.
• Simple case: customers make separate decisions on each item.
• Harder case: customers buy everything or nothing based on
sum of prices in list.
• Or could be even more complex.
“Unlimited supply combinatorial auction with additive /
single-minded /unit-demand/ general bidders”
Supermarket Pricing Problem
Various recent results have been focused on single minded
and unit demand consumers.
Algorithmic
• Seller knows the market well.
Incentive Compatible Auction
• Must be in customers’ interest (dominant strategy) to
report truthfully.
Online Pricing
• Customers arrive one at a time, buy what they want at
current prices. Seller modifies prices over time.
Algorithmic Problem, Single-minded Bidders [BB’06]
• n item types (coffee, cups, sugar, apples), with unlimited
supply of each.
• m customers.
• Customer i has a shopping list Li and will only shop if the
total cost of items in Li is at most some amount wi
• All marginal costs are 0, and we know all the (Li, wi).
What prices on the items will make you the most money?
• Easy if all Li are of size 1.
• What happens if all Li are of size 2?
Algorithmic Problem, Single-minded Bidders [BB’06]
• A multigraph G with values we on edges e.
•
Goal: assign prices on vertices
to maximize total profit, where:
5
15
10
30
10
40
20
5
Unlimited supply
• APX hard [GHKKKM’05].
A Simple 2-Approx. in the Bipartite Case
• Given a multigraph G with values we on edges e.
• Goal: assign prices on vertices
where:
to maximize total profit,
Algorithm
L
• Set prices in R to 0 and separately fix
prices for each node on L.
• Set prices in L to 0 and separately fix
prices for each node on R.
• Take the best of both options.
Proof
simple
!
R
15
25
35
15
25
5
40
OPT=OPTL+OPTR
A 4-Approx. for Graph Vertex Pricing
• Given a multigraph G with values we on edges e.
• Goal: assign prices on vertices
maximize total profit, where:
to
5
15
10
30
10
40
20
5
Algorithm
• Randomly partition the vertices into two sets L and R.
• Ignore the edges whose endpoints are on the same side
and run the alg. for the bipartite case.
Proof
simple
!
In expectation half of OPT’s profit
is from edges with one endpoint in L
and one endpoint in R.
Algorithmic Pricing, Single-minded Bidders,
k-hypergraph Problem
15
What about lists of size · k?
10
Algorithm
20
– Put each node in L with probability 1/k, in R with
probability 1 – 1/k.
– Let GOOD = set of edges with exactly one endpoint in
L. Set prices in R to 0 and optimize L wrt GOOD.
• Let OPTj,e be revenue OPT makes selling item j to customer
e. Let Xj,e be indicator RV for j 2 L & e 2 GOOD.
• Our expected profit at least:
Algorithmic Problem, Single-minded Bidders [BB’06]
Summary:
• 4 approx for graph case.
• O(k) approx for k-hypergraph case.
Improves the O(k2) approximation [BK’06].
Can also apply the [B-B-Hartline-M’05] reductions to
obtain good truthful mechanisms.
Can be naturally adapted to the online setting.
Algorithmic Problem, Single-minded Bidders [BB’06]
Other known results:
•
O(log mn) approx. by picking the best single price
[GHKKKM05].
• (log n) hardness for general case [DFHS06].
What about the most general case?
20$
100$
5$
25$
30$
1$
20$
30$
General Bidders
Can we say anything at all??
Can extend [GHKKKM05] and get a log-factor approx for
general bidders by an item pricing.
Theorem
There exists a price a p which gives a log(m) +log (n)
approximation to the total social welfare.
General Bidders
• Can extend [GHKKKM05] and get a log-factor approx
for general bidders by an item pricing.
Note: if bundle pricing is allowed, can do it easily.
– Pick a random admission fee from {1,2,4,8,…,h} to charge everyone.
– Once you get in, can get all items for free.
For any bidder, have 1/log chance of getting within factor of 2 of
its max valuation.
Can we do this via Item Pricing?
Unlimited Supply, General Bidders
Focus on a single customer. Analyze demand curve.
# items
n0
n1-
nL p0=0
p1
p2
pL-1
pL
price
• Claim 1: # is monotone non-increasing with p.
Unlimited Supply, General Bidders
Focus on a single customer. Analyze demand curve.
# items
n0
n1-
nL p0=0
p1
p2
pL-1
pL
price
• Claim 2: customer’s max valuation · integral of this curve.
Unlimited Supply, General Bidders
Focus on a single customer. Analyze demand curve.
# items
n0
n1-
nL p0=0
p1
p2
pL-1
pL
price
• Claim 2: customer’s max valuation · integral of this curve.
Unlimited Supply, General Bidders
Focus on a single customer. Analyze demand curve.
# items
n0
n1-
nL p0=0
p1
p2
pL-1
pL
price
• Claim 2: customer’s max valuation · integral of this curve.
Unlimited Supply, General Bidders
Focus on a single customer. Analyze demand curve.
# items
n0
n1-
nL p0=0
p1
p2
pL-1
pL
price
• Claim 2: customer’s max valuation · integral of this curve.
Unlimited Supply, General Bidders
Focus on a single customer. Analyze demand curve.
# items
n0
n1-
nL p0=0
p1
p2
pL-1
pL
price
• Claim 2: customer’s max valuation · integral of this curve.
Unlimited Supply, General Bidders
Focus on a single customer. Analyze demand curve.
# items
n0
n1-
nL 0
h/4
h/2
h
price
• Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a
log(n)-factor approx.
What about the limited
supply setting?
What about Limited Supply?
Assume one copy of each item.
Goal: Profit Maximization
Fixed Price (p):
Set R=J. For each bidder i, in some arbitrary order:
• Let Si be the demanded set of bidder i given the
following prices: p for each item in R and
for each
item in J\R.
• Allocate Si to bidder i and set R=R \ Si.
Assume bidders with subadditive valuations.
Limited Supply, Subadditive Valuations
There exists a single price mechanism whose profit is a
approximation to the social welfare.
Can show a
lower bound, for =1/4.
Other known results:
welfare & revenue
[DNS’06] show a
approximation to the total welfare
for bidders with general valuations.
welfare
[DNS’06], [D’07] show that a single price mechanism provides a
logarithmic approx. for social welfare in the submodular,
subadditive case.
A Property of Subadditive Valuations
Lemma 1 Assume vi subadditive.
Let (T1, …, Tm) be feasible allocation. There exists
(L1, …, Lm) and a price p such that :
(1)
(2) (L1, …, Lm) is supported at price p.
Li the subset that bidder i buys in a store where he sees only
Ti and every item is priced at p.
Subadditive Valuations, Limited Supply
Lemma 1
Let (T1, …, Tm) be feasible allocation. 9 (L1, …, Lm) and
price p such that :
and (L1, …, Lm) is supported at price p.
Lemma 2
Assume (L1, …, Lm) is supported at p and let (S1, …, Sm)
be the allocation produced by FixedPrice (p/2). Then:
Theorem
• There exists a single price mechanism whose profit is a
approximation to the social welfare.
Conclusions and Open Problems
Summary:
• Item Pricing mechanism for limited supply setting.
• Matching upper and lower bounds.
Open Problems
• Better revenue maximizing mechanisms for the limited
supply?