Item Pricing for Revenue Maximization in Combinatorial Auctions

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Item Pricing for Revenue Maximization in
Combinatorial Auctions
Maria-Florina Balcan, Carnegie Mellon University
Joint with Avrim Blum and Yishay Mansour
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Outline of the Talk
Revenue Maximization in Combinatorial Auctions

• Item Pricing in Unlimited Supply Combinatorial
  Auctions

Balcan-Blum06

• Single-minded bidders.

• General bidders.

Balcan-Blum-Mansour07

• Item Pricing in Limited Supply Combinatorial
  Auctions

• Bidders with subadditive valuations.

Balcan-Blum-Mansour07
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Supermarket Pricing Problem

• A supermarket trying to decide on how to price
  the goods.

Sellers 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
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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.

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Algorithmic Problem, Single-minded Bidders BB06

• 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?

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Algorithmic Problem, Single-minded Bidders BB06
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• A multigraph G with values we on edges e.

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• Goal assign prices on vertices
• to maximize total profit, where

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30
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Unlimited supply

• APX hard GHKKKM05.

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A Simple 2-Approx. in the Bipartite Case

• Given a multigraph G with values we on edges e.

• Goal assign prices on vertices to
  maximize total profit, where

Algorithm

• 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.

simple!
Proof
OPTOPTLOPTR
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A 4-Approx. for Graph Vertex Pricing

• Given a multigraph G with values we on edges e.

• Goal assign prices on vertices to
  maximize total profit, where

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 OPTs profit is from
edges with one endpoint in L and one endpoint in
R.
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Algorithmic Pricing, Single-minded
Bidders,k-hypergraph Problem
What about lists of size k?
Algorithm

• 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

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Algorithmic Problem, Single-minded Bidders BB06
Summary

• 4 approx for graph case.
• O(k) approx for k-hypergraph case.

• 4 approx for graph case.
• O(k) approx for k-hypergraph case.

Improves the O(k2) approximation BK06.
Can also apply the B-B-Hartline-M05
reductions to obtain good truthful mechanisms.
Can be naturally adapted to the online setting.
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Algorithmic Problem, Single-minded Bidders BB06
Other known results

• O(log mn) approx. by picking the best single
  price GHKKKM05.

• ?(log? n) hardness for general case DFHS06.

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What about the most general case?
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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.

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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?

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Unlimited Supply, General Bidders

• Focus on a single customer. Analyze demand curve.

• Claim 1 is monotone non-increasing with p.

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Unlimited Supply, General Bidders

• Focus on a single customer. Analyze demand curve.

items
n0
n1
-
nL
-
p00
pL-1
pL
p1
p2
price

• Claim 2 customers max valuation integral of
  this curve.

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Unlimited Supply, General Bidders

• Focus on a single customer. Analyze demand curve.

items
n0
n1
-
nL
-
p00
pL-1
pL
p1
p2
price

• Claim 2 customers max valuation integral of
  this curve.

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Unlimited Supply, General Bidders

• Focus on a single customer. Analyze demand curve.

items
n0
n1
-
nL
-
p00
pL-1
pL
p1
p2
price

• Claim 2 customers max valuation integral of
  this curve.

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Unlimited Supply, General Bidders

• Focus on a single customer. Analyze demand curve.

items
n0
n1
-
nL
-
p00
pL-1
pL
p1
p2
price

• Claim 2 customers max valuation integral of
  this curve.

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Unlimited Supply, General Bidders

• Focus on a single customer. Analyze demand curve.

items
n0
n1
-
nL
-
p00
pL-1
pL
p1
p2
price

• Claim 2 customers max valuation integral of
  this curve.

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Unlimited Supply, General Bidders

• Focus on a single customer. Analyze demand curve.

items
n0
n1
-
nL
-
0
h/2
h/4
h
price

• Claim 3 random price in h, h/2, h/4,, h/(2n)
  gets a
• log(n)-factor approx.

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Unlimited Supply, General Bidders

• Focus on a single customer. Analyze demand curve.

items
n0
n1
-
nL
-
0
h/2
h/4
h
price

• Claim 3 random price in h, h/2, h/4,, h/(2n)
  gets a
• log(n)-factor approx.

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• What about the limited
• supply setting?

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What about Limited Supply?
Assume one copy of each item.
Goal Profit Maximization
Fixed Price (p)
Set RJ. 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 RR \ Si.

Assume bidders with subadditive valuations.
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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

• DNS06 show a approximation to the
  total welfare for bidders with general
  valuations.

welfare
DNS06, D07 show that a single price
mechanism provides a logarithmic approx. for
social welfare in the submodular, subadditive
case.
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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.
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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.
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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?

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Thank you !