1. Mechanism Design via Machine Learning
Maria-Florina Balcan
Joint work with Avrim Blum,
Jason Hartline, and Yishay Mansour
Maria-Florina Balcan

2. 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.
Maria-Florina Balcan

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
Maria-Florina Balcan

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.
or…
Use bidders’ attributes:
country, language, ZIP code, etc.
Compete with best “simple” function.
Attribute Auctions [BH05]
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5. Example 2, Boutique Selling Problem
20$
30$ 5$
25$
100$ 1$
Very complicated preferences over subsets
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6. Example 2, Boutique Selling Problem
20$
30$ 5$
25$
100$ 1$
Combinatorial Auctions
Goal: Profit Maximization
Very complicated preferences over subsets
Compete with best item pricing [GH01].
Maria-Florina Balcan

7. Generic Setting (I) A mapping : (offer, privi) ! profiti
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 p>privi
(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
Maria-Florina Balcan

8. Attribute Auctions
one item for sale in unlimited supply (e.g. MP3 files).
bidder i has public attribute ai 2 X
Attr. space
G - a class of ‘’natural’’ pricing functions.
Example:
X=R2, G - linear functions over X
attributes
valuations
Maria-Florina Balcan

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 info of S and private info of S n{i}.
Try to compete with best g 2 G.
Maria-Florina Balcan

10. Our Contributions Generic Reductions, unified analysis.
25$
20$
30$ 5$ 2$
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.
--- The simplest type of reduction: If the number of agents is sufficiently large, then our reduction produces only a (1+) loss in solution quality. An algorithm for the standard algorithmic problem can be converted to a (1+)-approximation for the incentive-compatible design problem. Just a -approximation for the standard algorithmic problem can be converted to a (1+)-approximation for the incentive-compatible design problem.
Maria-Florina Balcan

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. S S1 S2
g1=OPT(S1)
g2=OPT(S2)
optimal
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12. 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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13. 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/.
Maria-Florina Balcan

14. 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+).
Maria-Florina Balcan

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+).
Corollary (roughly)
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16. Structural Risk Minimization Reduction
What if we have different functions at different levels of
complexity?
Don’t 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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17. Attribute Auctions, Linear Pricing Functions
Assume X=Rd.
N= (n+1)(1/) ln h.
|G’| · Nd+1 x
valuations x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
attributes
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18. Covering Arguments What if G is infinite w.r.t S?
attributes
valuations
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’|.
Maria-Florina Balcan

19. 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.
Maria-Florina Balcan