Machine Learning for Mechanism Design and Pricing Problems

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Machine Learning for Mechanism Design and Pricing
Problems
Avrim Blum
Carnegie Mellon University
Joint work with Maria-Florina Balcan, Jason
Hartline, and Yishay Mansour
Informs 2009
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Auctions/pricing
Designing auction/pricing mechanisms esp for
complex markets challenging problems at the
intersection of CS and Economics
Auction mechanisms for selling digital goods
Software, movies, information access
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Auctions/pricing
Designing auction/pricing mechanisms esp for
complex markets challenging problems at the
intersection of CS and Economics
Ad-auctions
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Auctions/pricing
Designing auction/pricing mechanisms esp for
complex markets challenging problems at the
intersection of CS and Economics
Combinatorial Auctions
Selling many different kinds of items. Buyers
with complex preferences over bundles I only
want the hotel room if I get the flight
too Some items or services that overlap,
others only good if have something else too. How
should you set prices to make the most profit?
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Auctions/pricing
Designing auction/pricing mechanisms esp for
complex markets challenging problems at the
intersection of CS and Economics

• Even if all customers preference information,
  how much they would be willing to pay, etc. is
  known up-front, setting prices to maximize
  revenue can be a challenging algorithmic problem.
• But in addition, incentive constraints
  customers wont give you the (correct)
  information if (possibly) not in their best
  interest.

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Auction/Pricing Problems
One Seller, Multiple Buyers with Complex
Preferences.
Sellers Goal maximize profit.
CS / optimization
Economics
Version 2 values given by selfish agents.
Version 1 Seller knows the true values.
Algorithm Design Problem (AD)
Incentive Compatible Auction (IC)
Previous Work on IC specific mechanisms for
restricted settings.
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Auction/Pricing Problems
One Seller, Multiple Buyers with Complex
Preferences.
Sellers Goal maximize profit.
CS / optimization
Economics
Version 2 values given by selfish agents.
Version 1 Seller knows the true values.
Algorithm Design Problem (AD)
Incentive Compatible Auction (IC)
Our Work
Generic Reduction using ML
Previous Work on IC specific mechanisms for
restricted settings.
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How is this related to Machine Learning?
Simple version basic digital good auction
problem.
Youve developed a cool new software tool want
to sell it.
- n potential buyers. Buyer i has valuation
vi. - Can potentially sell to all of them, but
buyer i will only purchase if priced below vi. -
Unfortunately, you dont know the vi.
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How is this related to Machine Learning?
Simple version basic digital good auction
problem.
Youve developed a cool new software tool want
to sell it.
- n potential buyers. Buyer i has valuation
vi. - Can potentially sell to all of them, but
buyer i will only purchase if priced below vi. -
Unfortunately, you dont know the vi.

• Classic econ model buyers types (valuations)
  chosen iid from known distribution D. In this
  case, just set sales price pD to maximize
  expected profit.
• But what if dont want to assume this?

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How is this related to Machine Learning?
Simple version basic digital good auction
problem.
Youve developed a cool new software tool want
to sell it.
- n potential buyers. Buyer i has valuation
vi. - Can potentially sell to all of them, but
buyer i will only purchase if priced below vi. -
Unfortunately, you dont know the vi.

• Could ask people for their valuations and use
  this to set a price as before, but people will
  low-ball (not incentive-compatible)

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How is this related to Machine Learning?
Simple version basic digital good auction
problem.
Youve developed a cool new software tool want
to sell it.
- n potential buyers. Buyer i has valuation
vi. - Can potentially sell to all of them, but
buyer i will only purchase if priced below vi. -
Unfortunately, you dont know the vi.
Random sampling auction

• Ask buyers to submit bids bi.
• Randomly partition bidders into two sets
  S1, S2.
• Find best price over bids in S1 and use it
  as offer price on S2! ( vice versa).

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How is this related to Machine Learning?
More interesting version combinatorial auctions
Youre Sperizon-mobile. Want to price various
services.

• Basic service
• Extra lines
• Data package
• TV features,

• People have potentially nonlinear valuations
  over subsets.
• Might also have known info about customers
  (current usage, demographics,).
• Want to perform nearly as well as best (simple)
  pricing function over known info.

(Combinatorial Attribute Auction)
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How is this related to Machine Learning?
More interesting version combinatorial auctions
Youre Sperizon-mobile. Want to price various
services.

• Basic service
• Extra lines
• Data package
• TV features,

• Random sampling auction
• Split randomly into S1, S2.
• Apply optimization alg A on S1, perhaps with
  penalty term.
• Use A(S1) on S2 and vice-versa.

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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• Interesting issues
• What quantities to use for ,
  ?
• What kind of regularization makes sense?

• Random sampling auction
• Split randomly into S1, S2.
• Apply optimization alg A on S1, perhaps with
  penalty term.
• Use A(S1) on S2 and vice-versa.

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Generic Setting

• S set of n bidders.

• Bidder i

privi
, pubi
, bidi

• Space of legal offers/pricing functions G.

• g 2 G maps the pubi to pricing over the outcome
  space.

• g is take it or leave it offer, so any fixed
  g is IC.

• Goal Incentive Compatible mechanism to do
  nearly as well as the best g 2 G.
• Assume max profit h per bidder.

Unlimited supply
Profit of g sum over bidders.
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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• Interesting issues
• What quantities to use for ,
  ?
• What kind of regularization makes sense?

• Random sampling auction
• Split randomly into S1, S2.
• Apply optimization alg A on S1, perhaps with
  penalty term.
• Use A(S1) on S2 and vice-versa.

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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• What should be large?
• bidders? But bidders of valuation 0 dont
  help very much.
• Instead OPT profit.

Even if assume all valuations 1, bounds will be
loose.
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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• What should be large?
• bidders? But bidders of valuation 0 dont
  help very much.
• Instead OPT profit.
• As a function of what?
• functions in G.

Even if assume all valuations 1, bounds will be
loose.
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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• What should be large?
• bidders? But bidders of valuation 0 dont
  help very much.
• Instead (OPT profit)/h.
• As a function of what?
• functions in G.

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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• What should be large?
• bidders? But bidders of valuation 0 dont
  help very much.
• Instead (OPT profit)/h.
• As a function of what?
• functions in G.
• functions in G the alg could possibly output
  over splits S1,S2 1.

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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• As a function of what?
• functions in G.
• functions in G the alg could possibly output
  over splits S1,S2 1.
• Multiplicative L1 cover size.

• E.g., digital-good auction. Algorithm uses S1
  to choose price to offer for S2 and vice-versa.
• Can discretize to powers of (1?). Get G
  (log h)/?.
• Or use fact that alg will only output a bid
  value. G n1.

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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• functions in G.
• functions in G the alg could possibly output
  over splits S1,S2 1.
• Multiplicative L1 cover size.

What if hard to directly bound possible
outputs

• Use covering arguments
• find G that covers G ,
• show that all functions in G behave well

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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• functions in G.
• functions in G the alg could possibly output
  over splits S1,S2 1.
• Multiplicative L1 cover size.

G ?-covers G wrt to S if for all g exists g 2
G s.t. ?i g(i)-g(i) ? g(S). g(i)
profit made from bidder i
Theorem (roughly)
If G is ?-cover of G, then the previous bounds
hold with G replaced by G.
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Attribute Auctions, Linear Pricing Functions
Assume XRd.
N (n1)(1/?) ln h.
G Nd1
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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• functions in G.
• functions in G the alg could possibly output
  over splits S1,S2 1.
• Multiplicative L1 cover size.

• For combinatorial auctions with m items, G
  class of item-pricings, to get (1-?)OPT,
  sufficient to have
• OPT Õ(hm2/?2) for general valuation functions.
• OPT Õ(hm/?2) for unit-demand valuations.
• First results for general case, factor m savings
  over GH01 for unit-demand valuations.

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Goal

• If is large as a function of ,
  then the random sampling auction (perhaps
  regularized) performs nearly as well as best
  pricing function in class G.
• Regularization/SRM
• Can do SRM as usual, penalizing
  higher-complexity function classes.
• But even individual functions can have different
  complexity levels!

• E.g., digital-good auction. Say S1 has 1 bid of
  value h and h-1 bids of value 1.
• So, 1,h are both optimal prices. But much
  better stats for 1.
• Allows to replace h with price used by OPT
  in previous bounds.

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Summary

• Explicit connection between machine learning and
  mechanism design. Use ideas of MLT to analyze
  when random sampling auction will do well.
• This application brings out interesting twists on
  usual ML issues. What has to be large as a
  function of what? SRM.
• Challenges
• Loss function discontinuous and asymmetric.
• Range of valuations large.

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Challenges/Future Directions

• Apply similar techniques to limited supply.
• Online Setting.
• How big a focus group do you need for other
  kinds of pricing/allocation/decision-making
  problems.