Mechanism Design, Machine Learning, and Pricing Problems
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Transcript of Mechanism Design, Machine Learning, and Pricing Problems
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Maria-Florina Balcan
Mechanism Design, Machine Learning, and Pricing Problems
Maria-Florina Balcan
11/13/2007
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Maria-Florina Balcan
Outline• Reduce problems of incentive-compatible mechanism design to standard algorithmic questions.
• 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.• Revenue maximization, unlimited supply combinatorial auctions with single-minded consumers
[BBHM05]
[BB06]
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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
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Maria-Florina Balcan
MP3 Selling Problem• We are seller/producer of some digital good,
e.g. MP3 files.
• Compete with fixed price.
or…
• Use bidders’ attributes: • country, language, ZIP code, etc.
Goal: Profit Maximization
Digital Good Auction (e.g., [GHW01])
Attribute Auctions [BH05]• Compete with best “simple” function.
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Maria-Florina Balcan
Example 2, Boutique Selling Problem
20$
30$30$
5$
25$
20$
100$
1$
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Maria-Florina Balcan
Example 2, Boutique Selling Problem
Goal: Profit Maximization
Combinatorial Auctions
• Compete with best item pricing [GH01].
20$
30$30$
5$
25$
20$
100$
1$
(unit demand consumers)
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Maria-Florina Balcan
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)
Unlimited supply
Goal: Profit Maximization
Profit of g: ig(i)
• Space of legal offers/pricing functions.
• G - pricing functions.• Goal: IC mech to do nearly as well as the best g 2 G.
• g(i) – profit obtained from making offer g to bidder i
g(i)= p if p · privig(i)= 0 if p>privi
Digital Good
• g maps the pubi to pricing over the outcome space.
g=“ take the good for p, or leave it”
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Maria-Florina Balcan
Attribute Auctions• one item for sale in unlimited supply (e.g. MP3 files).• bidder i has public attribute ai 2 X
Example: X=R2, G - linear functions over X
• G - a class of ‘’natural’’ pricing functions.
Attr. space
attributes
valuations
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Maria-Florina Balcan
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}.
• Focus on one-shot mechanisms, off-line setting
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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.
could offer great improvement over single price
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Maria-Florina Balcan
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.
SS1
S2
g1=OPT(S1)g2=OPT(S2)
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Maria-Florina Balcan
Basic Analysis, RSOPF(G, A)
Theorem 1
1) Consider a fixed g and profit level p. Use McDiarmid ineq. to show:
Proof sketch
Lemma 1
h - maximum valuation, G - finite
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Maria-Florina Balcan
Basic Analysis, RSOPF(G,A), cont2) Let gi be the best over Si. Know gi(Si) ¸ g
OPT(Si)/.
In particular,
Using
get that our profit g1(S2) +g2(S1) is at least (1-)OPTG/.
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Maria-Florina Balcan
Basic Analysis, RSOPF(G, A)
Theorem 2
Theorem 1
h - maximum valuation, G - finite
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Maria-Florina Balcan
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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Maria-Florina Balcan
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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Maria-Florina Balcan
Structural Risk Minimization Reduction
SRM Reduction Let
• Randomly split the bidders into S1 and S2.• Compute gi to maximize • Apply g1 over S2 and g2 over S1.
What if we have different functions at different levels of complexity?Don’t know best complexity level in advance.
Theorem
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Maria-Florina Balcan
Attribute Auctions, Linear Pricing Functions
Assume X=Rd. N= (n+1)(1/) ln h.|G’| · Nd+1
attributes
valuations
xx
xx
xx
xx
x
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
x xx
xx
xx
x
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Maria-Florina Balcan
Covering Arguments
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).
•What if G is infinite w.r.t S?
Use covering arguments: •find G’ that covers G , •show that all functions in G’ behave well
Theorem (roughly)If G’ is -cover of G, then the previous theorems hold with |G| replaced by |G’|.
attributes
valuations
Analysis Techniq
ue
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Maria-Florina Balcan
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.
• Apply similar techniques to limited supply.• Study Online Setting.
Open ProblemsOpen Problems
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Maria-Florina Balcan
Outline• Reduce problems of incentive-compatible mechanism design to standard algorithmic questions.
• Focus on revenue-maximization, unlimited supply.• Use ideas from Machine Learning.
–Sample Complexity techniques in MLT for analysis.
• Approximation Algorithms for Item Pricing.• Revenue maximization, unlimited supply combinatorial auctions with single-minded bidders
[BBHM05]
[BB06]
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Maria-Florina Balcan
Algorithmic Problem, Single-minded Customers
• m item types (coffee, cups, sugar, apples, …), with unlimited supply of each.
• n customers.
• Say all marginal costs to you are 0 [revisit this in a bit], and you know all the (Li, wi) pairs.
• 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).
What prices on the items will make you the most money?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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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:
• APX hard [GHKKKM’05].
10
40
15
2030
5
10
5
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Maria-Florina Balcan
A Simple 2-Approx. in the Bipartite Case
• Goal: assign prices on vertices pv ¸ 0 as to maximize total profit, where:
• 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.
AlgorithmAlgorithm
• Given a multigraph G with values we on edges e.
ProofProof simple!
OPT=OPTL+OPTR
40
152535
15255
L R
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Maria-Florina Balcan
A 4-Approx. for Graph Vertex Pricing
• Goal: assign prices on vertices pv¸ 0 to maximize total profit, where:
• 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.
AlgorithmAlgorithm
ProofProof In expectation half of OPT’s profit is from edges with one endpoint in L and one endpoint in R.
• Given a multigraph G with values we on edges e.
simple!
10
40
15
2030
5
10
5
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Maria-Florina Balcan
Algorithmic Pricing, Single-minded Customers,k-hypergraph Problem
What about lists of size · k?
– 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:
AlgorithmAlgorithm10
15
20
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Maria-Florina Balcan
Algorithmic Pricing, Single-minded Customers
• What if items have constant marginal cost to us?
10
40
15
203
5
7
5• We can subtract these from each edge (view
edge as amount willing to pay above our cost).
But, one difference:• Can now imagine selling some items below cost
in order to make more profit overall.
23
32 8• Reduce to previous problem.
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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.
• Previous results only give good approximation wrt best “non-money-losing” prices.
• Can actually give a log(m) gap between the two benchmarks.
4 48
1 12 2
4 48
1 12 2
0 4 4 0
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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 andcan be naturally adapted to the online setting.
• 4 - , o(k).• How well can you do if pricing below cost is allowed?
Open ProblemsOpen Problems
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Maria-Florina Balcan
More On Revenue Maximization in Combinatorial Auctions
• Item Pricing in Unlimited Supply Combinatorial Auctions• General bidders.
• Item Pricing in Limited Supply Combinatorial Auctions• Bidders with subadditive valuations.
[Balcan-Blum-Mansour’07]
[Balcan-Blum-Mansour’07]
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Maria-Florina Balcan
General Bidders
Can extend [GHKKKM05] and get a log-factor approx for general bidders by an item pricing.
There exists a price a p which gives a log(m) +log (n) approximation to the total social welfare.
TheoremTheorem
Can we say anything at all??
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Maria-Florina Balcan
General Bidders
Can we do this via Item Pricing?
• 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.
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Maria-Florina Balcan
Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.
• Claim 1: # is monotone non-increasing with p.
# items
price
n0
p0=0 p1 p2 pL-1 pL
n1
nL -
-
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Maria-Florina Balcan
Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.
price
# items n0
p0=0 p1 p2 pL-1 pL
n1
nL -
-
• Claim 2: customer’s max valuation · integral of this curve.
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Maria-Florina Balcan
Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.
price
n0
p0=0 p1 p2 pL-1 pL
n1
nL -
-
• Claim 2: customer’s max valuation · integral of this curve.
# items
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Maria-Florina Balcan
Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.
price
n0
p0=0 p1 p2 pL-1 pL
n1
nL -
-
• Claim 2: customer’s max valuation · integral of this curve.
# items
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Maria-Florina Balcan
Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.
price
n0
p0=0 p1 p2 pL-1 pL
n1
nL -
-
• Claim 2: customer’s max valuation · integral of this curve.
# items
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Maria-Florina Balcan
Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.
price
n0
p0=0 p1 p2 pL-1 pL
n1
nL -
-
• Claim 2: customer’s max valuation · integral of this curve.
# items
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Maria-Florina Balcan
Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.
price
n0
0 h/4 h/2 h
n1
nL -
-
• Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a log(n)-factor approx.
# items
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Maria-Florina Balcan
What about the limited supply setting?
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Maria-Florina Balcan
What about Limited Supply?Assume one copy of each item.
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.
Goal: Profit Maximization
Assume bidders with subadditive valuations.
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Maria-Florina Balcan
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.
[DNS’06], [D’07] show that a single price mechanism provides a logarithmic approx. for social welfare in the submodular, subadditive case.
[DNS’06] show a approximation to the total welfare for bidders with general valuations.
welfare & revenue
Other known results:
welfare
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Maria-Florina Balcan
A Property of Subadditive ValuationsLemma 1Lemma 1
Let (T1, …, Tm) be feasible allocation. There exists (L1, …, Lm) and a price p such that :
(2) (L1, …, Lm) is supported at price p.
(1)
Assume vi subadditive.
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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Maria-Florina Balcan
Subadditive Valuations, Limited SupplyLemma 1Lemma 1price p such that :
and (L1, …, Lm) is supported at price p.
Lemma 2Lemma 2be the allocation produced by FixedPrice (p/2). Then:
Let (T1, …, Tm) be feasible allocation. 9 (L1, …, Lm) and
Assume (L1, …, Lm) is supported at p and let (S1, …, Sm)
• There exists a single price mechanism whose profit is a
TheoremTheorem
approximation to the social welfare.
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Maria-Florina Balcan
Summary
• Item Pricing mechanism for limited supply setting.
• Matching upper and lower bounds.