Mechanism Design, Machine Learning, and Pricing Problems

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Mechanism Design, Machine Learning, and Pricing Problems

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Title: Mechanism Design, Machine Learning, and Pricing Problems

1
Mechanism Design, Machine Learning, and Pricing
Problems
Maria-Florina Balcan
11/13/2007
2
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

3
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
4
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.

Use bidders attributes
country, language, ZIP code, etc.

Compete with best simple function.

Attribute Auctions BH05
5
Example 2, Boutique Selling Problem
6
Example 2, Boutique Selling Problem
Combinatorial Auctions
Goal Profit Maximization

Compete with best item pricing GH01.

(unit demand consumers)
7
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
8
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
9
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

10
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
11
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.

12
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
13
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/?.
14
Basic Analysis, RSOPF(G, A)
h - maximum valuation, G - finite
Theorem 1
Theorem 2
15
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
16
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)
17
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
18
Attribute Auctions, Linear Pricing Functions
Assume XRd.
N (n1)(1/?) ln h.
G Nd1
19
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.
20
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.

21
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

22
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?

23
Algorithmic Pricing, Single-minded Customers
5

A multigraph G with values we on edges e.

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

20
30
5

APX hard GHKKKM05.

24
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
25
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.
26
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

27
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.

8
32
5

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

28
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.

29
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?

30
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
31
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.

32
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?

33
Unlimited Supply, General Bidders

Focus on a single customer. Analyze demand curve.

Claim 1 is monotone non-increasing with p.

34
Unlimited Supply, General Bidders

Focus on a single customer. Analyze demand curve.