Mechanism Design, Machine Learning and
Pricing Problems
Maria-Florina Balcan
Joint work with Avrim Blum,
Jason Hartline, and Yishay Mansour
Maria-Florina Balcan
Outline of the Talk
• Reduce problems of incentive-compatible mechanism
design to standard algorithmic questions. [BBHM05]
• Focus on revenue-maximization, unlimited supply.
- Digital Good Auction
- Attribute Auctions
- Combinatorial Auctions
• Use ideas from Machine Learning.
–Sample Complexity techniques in MLT for analysis.
• Approximation Algorithms for Item Pricing. [BB06]
• Revenue maximization, unlimited supply combinatorial auctions
with single-minded consumers
Maria-Florina Balcan
MP3 Selling Problem
• We are seller/producer of some digital good (or
any item of fixed marginal cost), e.g. MP3 files.
Goal: Profit Maximization
Maria-Florina Balcan
MP3 Selling Problem
• We are seller/producer of some digital good, e.g.
MP3 files.
Goal: Profit Maximization
Digital Good Auction (e.g., [GHW01])
• Compete with fixed price.
or…
• Use bidders’ attributes:
• country, language, ZIP code, etc.
• Compete with best “simple” function.
Attribute Auctions [BH05]
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Example 2, Boutique Selling Problem
20$
100$
5$
25$
30$
1$
30$
20$
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Example 2, Boutique Selling Problem
20$
100$
5$
25$
30$
1$
30$
20$
Combinatorial Auctions
Goal: Profit Maximization
• Compete with best item pricing [GH01].
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Generic Setting (I)
• S set of n bidders.
• Bidder i:
– privi (e.g., how much is willing to pay for the MP3 file)
– pubi (e.g., ZIP code)
• Space of legal offers/pricing functions.
• g maps the pubi to pricing over the outcome space.
• g(i) – profit obtained from making offer g to bidder i
Digital Good g=“ take the good for p, or leave it”
Goal: Profit Maximization
g(i)= p if p · privi
g(i)= 0 if p>privi
• G - pricing functions.
• Goal: IC mech to do nearly as well as the best g 2 G.
Profit of g: ig(i)
Unlimited supply
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Attribute Auctions
• one item for sale in unlimited supply (e.g. MP3 files).
• bidder i has public attribute ai 2 X
Attr. space
• G - a class of ‘’natural’’ pricing functions.
Example:
X=R2, G - linear functions over X
valuations
attributes
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Generic Setting (II)
• Our results: reduce IC to AD.
• Algorithm Design: given (privi, pubi), for all i 2 S, find
pricing function g 2 G of highest total profit.
• Incentive Compatible mechanism: offer for bidder i
based on the public information of S and private info
of S n{i}.
Maria-Florina Balcan
Main Results [BBHM05]
• Generic Reductions, unified analysis.
• General Analysis of Attribute Auctions:
– not just 1-dimensional
• Combinatorial Auctions:
– First results for competing against opt item-pricing in
general case (prev results only for “unit-demand”[GH01])
– Unit demand case: improve prev bound by a factor of m.
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Basic Reduction: Random Sampling Auction
RSOPF(G,A) Reduction
• 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
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Basic Analysis, RSOPF(G, A)
Theorem 1
Proof sketch
1) Consider a fixed g and profit level p. Use McDiarmid ineq. to
show:
Lemma 1
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Basic Analysis, RSOPF(G,A), cont
2) Let gi be the best over Si. Know gi(Si) ¸ gOPT(Si)/.
In particular,
Using also OPTG ¸  n, get that our profit g1(S2) +g2(S1) is
at least (1-)OPTG/.
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Attribute Auctions, RSOPF(Gk, A)
Gk : k markets defined by Voronoi cells around k
bidders & fixed price within each market.
Assume we discretize prices to powers of (1+).
attributes
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Attribute Auctions, RSOPF(Gk, A)
Gk : k markets defined by Voronoi cells around k
bidders & fixed price within each market.
Assume we discretize prices to powers of (1+).
Corollary (roughly)
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Structural Risk Minimization Reduction
What if we have different functions at different levels of
complexity?
Don’t know best complexity level in advance.
SRM Reduction
Let
• Randomly split the bidders into S1 and S2.
• Compute gi to maximize
• Apply g1 over S2 and g2 over S1.
Theorem
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Attribute Auctions, Linear Pricing Functions
Assume X=Rd. N= (n+1)(1/) ln h.
d+1
|G’| · N
x
x
valuations
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
attributes
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Covering Arguments
•What if G is infinite w.r.t S?
valuations
Use covering arguments:
attributes
•find G’ that covers G ,
•show that all functions in G’ behave well
Definition:
G’ -covers G wrt to S if for 8 g 9 g’ 2 G’ s.t.
8 i |g(i)-g’(i)| ·  g(i).
Theorem (roughly)
If G’ is -cover of G, then the
with |G| replaced by |G’|.
Analysis
Techniqu
e
previous
theorems
hold
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Conclusions and Open Problems [BBHM05]
• Explicit connection between machine learning and
mechanism design.
• Use of ideas in MLT for both design and analysis
in auction/pricing problems.
• Unique challenges & particularities:
• Loss function discontinuous and asymmetric.
• Range of valuations large.
Open Problems
• Apply similar techniques to limited supply.
• Study Online Setting.
Maria-Florina Balcan
Outline of the Talk
• Reduce problems of incentive-compatible mechanism
design to standard algorithmic questions. [BBHM05]
• Focus on revenue-maximization, unlimited supply.
• Use ideas from Machine Learning.
–Sample Complexity techniques in MLT for analysis.
• Approximation Algorithms for Item Pricing. [BB06]
• Revenue maximization, unlimited supply combinatorial auctions
with single-minded bidders
Maria-Florina Balcan
Algorithmic Problem, Single-minded Customers
• m item types (coffee, cups, sugar, apples, …), with
unlimited supply of each.
• n 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 Customers
• 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].
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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 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
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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 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.
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Algorithmic Pricing, Single-minded Customers,
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:
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Algorithmic Pricing, Single-minded Customers
• 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 Customers
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(m) gap between the two benchmarks.
Maria-Florina Balcan
Conclusions and Open Problems [BB06]
Summary:
• 4 approx for graph case.
• O(k) approx for k-hypergraph case.
Improves the O(k2) approximation of Briest and Krysta,
SODA’06.
– Also simpler and
can be naturally adapted to the online setting.
Open Problems
• 4 - , o(k).
• How well can you do if pricing below cost is allowed?
Maria-Florina Balcan