Mechanism Design, Machine Learning, and Pricing Problems

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Mechanism Design, Machine Learning, and Pricing
Problems
Maria-Florina Balcan
11/13/2007
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Outline

• 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

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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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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
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Example 2, Boutique Selling Problem
Combinatorial Auctions
Goal Profit Maximization

• Compete with best item pricing GH01.

(unit demand consumers)
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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
g(i) p if p privi g(i) 0 if pgtprivi
Goal Profit Maximization

• 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

• G - a class of natural pricing functions.

Example
XR2, G - linear functions over X
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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 ni.

• Focus on one-shot mechanisms, off-line setting

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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-demandGH01)
• Unit demand case improve prev bound by a factor
  of m.

could offer great improvement over single price
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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.

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Basic Analysis, RSOPF(G, A)
h - maximum valuation, G - finite
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
get that our profit g1(S2) g2(S1) is at least
(1-?)OPTG/?.
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Basic Analysis, RSOPF(G, A)
h - maximum valuation, G - finite
Theorem 1
Theorem 2
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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? Dont 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 XRd.
N (n1)(1/?) ln h.
G Nd1
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Covering Arguments

• 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

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)
Analysis Technique
If G is ?-cover of G, then the previous theorems
hold with G replaced by G.
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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.

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Outline

• 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

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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?

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Algorithmic Pricing, Single-minded Customers
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• A multigraph G with values we on edges e.

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• Goal assign prices on vertices pv 0 to maximize
  total profit, where

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30
5

• 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 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.

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 pv 0 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
Customers,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 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
3
2
15
10
7
40
20
3

• Reduce to previous problem.

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5

• But, one difference
• Can now imagine selling some items below cost in
  order to make more profit overall.

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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).

• 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.

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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, SODA06.

• 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?

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More On Revenue Maximization in Combinatorial
Auctions

• Item Pricing in Unlimited Supply Combinatorial
  Auctions

• General bidders.

Balcan-Blum-Mansour07

• Item Pricing in Limited Supply Combinatorial
  Auctions

• Bidders with subadditive valuations.

Balcan-Blum-Mansour07
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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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Summary

• Item Pricing mechanism for limited supply setting.

• Matching upper and lower bounds.