Approximation Algorithms and Online
Mechanisms for Item Pricing
Maria-Florina Balcan & Avrim Blum
CMU, CSD
Maria-Florina Balcan
Revenue Maximization in Unlimited Supply
Comb. Auctions with Single-minded Bidders
• Imagine a supermarket trying to decide on how to price the
goods it is selling (coffee, milk, sugar, oranges, apples...).
• Customers have shopping lists. Decide what to buy or
whether to shop at all based on prices of items in list.
– Customers buy everything or nothing based on sum of prices in list.
• Seller’s Goal: set prices to maximize revenue.
Maria-Florina Balcan
Supermarket Pricing Problem
Three versions of the problem (easiest to hardest)
Algorithmic
• Seller knows the market well.
Incentive Compatible Auction
• Customers submit lists / valuations to mechanism, which
decides who gets what for how much. 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.
Maria-Florina Balcan
Algorithmic Problem, Single-minded Bidders
• n item types (coffee, cups, sugar, apples), with unlimited
supply of each.
• m customers.
• Each 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
(otherwise he will go elsewhere).
• Say all marginal costs to you are 0 [revisit this in a bit],
and you know all the (Li, wi) pairs.
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?
Maria-Florina Balcan
Algorithmic Pricing, Single-minded Bidders
• A multigraph G with values we on edges e.
• Goal: assign prices on vertices pv¸ 0 to
maximize total profit, where:
5
15
10
30
10
40
20
5
• APX hard [GHKKKM’05].
Maria-Florina Balcan
A Simple 2-Approx. in the Bipartite Case
• Given a multigraph G with values we on edges e.
• Goal: assign prices on vertices pv ¸ 0 as 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.
Proof
simple
!
L
R
15
25
35
15
25
5
40
OPT=OPTL+OPTR
Maria-Florina Balcan
A 4-Approx. for Graph Vertex Pricing
• Given a multigraph G with values we on edges e.
• Goal: assign prices on vertices pv¸ 0 to
maximize total profit, where:
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.
Maria-Florina Balcan
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:
Maria-Florina Balcan
Algorithmic Pricing, Single-minded Bidders
• What if items have constant marginal cost to us?
• We can subtract these from each edge (view
edge as amount willing to pay above our cost).
5
2
10
• Reduce to previous problem.
3
3
15
7
40
32
20
8
5
But, one difference:
• Can now imagine selling some items below cost in
order to make more profit overall.
Maria-Florina Balcan
Algorithmic Pricing, Single-minded Bidders
What if items have constant marginal cost to us?
• We can subtract these from each edge (view edge
as amount willing to pay above our cost).
• Reduce to previous problem.
But, one difference:
• Can now imagine selling some items below cost in
order to make more profit overall.
2
1
4
1
2
8
4
1
0 4
2
4 8
2
4 4
1
0
• Previous results only give good approximation wrt
best “non-money-losing” prices.
• Can actually give a log(n) gap between the two benchmarks.
Maria-Florina Balcan
Algorithmic Problem, Single-minded Bidders
Summary:
• 4 approx for graph case.
• O(k) approx for k-hypergraph case.
Improves the O(k2) approximation of Briest and Krysta,
SODA’06.
Coming
– Also simpler and
up!
can be naturally adapted to the online setting.
• Can de-randomize algorithms if desired.
Other known results:
• General O(log mn) approx. by picking the best single
price [GHKKKM05].
• (log n) hardness for general case [DFHS06].
Maria-Florina Balcan
Incentive Compatible Auction Problem
Same setup, but we don’t know lists or valuations.
Goal: incentive compatible auction.
• Customers submit valuation information.
• Auction mechanism determines who buys what for how much.
• Must be in customers’ self-interest to submit their true
valuations.
Maria-Florina Balcan
Incentive Compatible Auction Problem
Generic randomized reductions to alg problem [BBHM’05].
E.g.:
Basic Reduction: Random Sampling Auction
• Bidders submit bids.
• Randomly split the bidders into S1 and S2.
• Run A on Si to get (nearly optimal) gi 2 G w.r.t. Si.
• Apply g1 over S2 and g2 over S1.
S1
S
g1=OPT(S1)
g2=OPT(S2)
S2
Maria-Florina Balcan
Incentive Compatible Auction Problem
Basic Reduction: Random Sampling Auction
• Bidders submit bids.
• Randomly split the bidders into S1 and S2.
• Run A on Si to get (nearly optimal) gi 2 G w.r.t. Si.
• Apply g1 over S2 and g2 over S1.
Example Guarantee:
If all valuations are between 1 and h, then Õ(hn/2) bidders
are sufficient so that whp this loses only factor of (1+) in
revenue.
Maria-Florina Balcan
On-line Pricing
Customers arrive one at a time, buy or don’t buy at current
prices.
In (full information) auction model, we know valuation info
for customers 1,…,i-1 when customer i arrives.
In posted-price model, only know who bought what for how
much.
Goal - do well compared to best fixed set of item prices.
Maria-Florina Balcan
On-line Pricing
Our O(k)-approx. alg. can be naturally adapted to the online
setting, by using results of [BH’05] and [BKRW’03] for the
online digital good auction.
Can run separate online auctions over items in L, customers in
GOOD (customers who want exactly one item in L).
Guarantee: perform comparably to best fixed set of item prices
(for pts in L, people in GOOD).
Let OPTi be the best profit achievable (from item i) using a
fixed price for item i from customers in GOOD whose bundle
contain item i.
Can use [BH’05] auction -- the expected profit of the online
auction for item i is
Maria-Florina Balcan
On-line Pricing
Can run separate online auctions over items in L, customers in
GOOD (customers that who want exactly one item in L).
Let OPTi be the best profit achievable (from item i) using a
fixed price for item i from customers in GOOD whose bundle
contain item i.
Using the [BH’05] auction, the expected profit of the online
auction for item i is:
Overall, we achieve profit at least:
Profit of the offline approx. alg.
Maria-Florina Balcan
Conclusions and Open Questions
• Simple randomized alg achieving factor 4 for graphvertex pricing problem.
• Factor O(k) for k-hypergraph vertex pricing.
• Can then use generic technique to apply in auction
setting. Use online learning methods to apply in online
setting.
Open Problems
• 4 - , o(k).
• How well can you do if pricing below cost is allowed?
Maria-Florina Balcan