Mechanism Design via Machine Learning

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Mechanism Design via Machine Learning
Maria-Florina Balcan
Joint work with Avrim Blum, Jason Hartline, and
Yishay Mansour
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An Overview of the Results

• 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
• Often, improve previous known bounds.

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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.

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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.

• A mapping ?

Digital Good
?(p,privi) p if p privi ?(p,privi) 0 if
pgtprivi
?(offer, privi) ! profiti
Goal Profit Maximization

• G - pricing functions, g 2 G maps the pubi to
  an offer.
• Goal IC mech to do nearly as well as the best
  g 2 G.

• Profit of g ?i?(g(pubi),privi)

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 info of S and private info
  of S ni.

Try to compete with best g 2 G.
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Our Contributions

• 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.

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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)
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?).
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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
x
valuations
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attributes
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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)
If G is ?-cover of G, then the previous theorems
hold with G replaced by G.
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Conclusions

• 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.

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Future Directions

• Apply similar techniques to limited supply.
• Improve existing bounds. Online Setting.
• General cost functions on outcomes.