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

2. Maria-Florina Balcan Outline of the Talk 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. Approximation Algorithms for Item Pricing. Revenue maximization, unlimited supply combinatorial auctions with single-minded consumers [BBHM05] [BB06]

3. Maria-Florina Balcan 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. Maria-Florina Balcan MP3 Selling Problem We are seller/producer of some digital good, e.g. MP3 files. Compete with fixed price. or… Use bidders’ attributes: country, language, ZIP code, etc. Goal: Profit Maximization Digital Good Auction (e.g., [GHW01]) Attribute Auctions [BH05] Compete with best “simple” function.

5. Maria-Florina Balcan Example 2, Boutique Selling Problem 20$ 30$ 5$ 25$ 20$ 100$ 1$

6. Maria-Florina Balcan Example 2, Boutique Selling Problem Goal: Profit Maximization Combinatorial Auctions Compete with best item pricing [GH01]. 20$ 30$ 5$ 25$ 20$ 100$ 1$

7. Maria-Florina Balcan Generic Setting (I) S set of n bidders. Bidder i: –priv i (e.g., how much is willing to pay for the MP3 file) –pub i (e.g., ZIP code) Unlimited supply Goal: Profit Maximization Profit of g:  i g(i) Space of legal offers/pricing functions. G - pricing functions. Goal: IC mech to do nearly as well as the best g 2 G. g(i) – profit obtained from making offer g to bidder i g(i)= p if p · priv i g(i)= 0 if p>priv i Digital Good g maps the pub i to pricing over the outcome space. g=“ take the good for p, or leave it”

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

9. Maria-Florina Balcan Generic Setting (II) Our results: reduce IC to AD. Algorithm Design: given (priv i, pub i ), 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 n {i}.

10. Maria-Florina Balcan 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-demand”[GH01]) –Unit demand case: improve prev bound by a factor of m.

11. Maria-Florina Balcan Basic Reduction: Random Sampling Auction RSOPF (G,A) Reduction Bidders submit bids. Randomly split the bidders into S 1 and S 2. Run A on S i to get (nearly optimal) g i 2 G w.r.t. S i. Apply g 1 over S 2 and g 2 over S 1. S S1S1 S2S2 g 1 =OPT(S 1 ) g 2 =OPT(S 2 )

12. Maria-Florina Balcan Basic Analysis, RSOPF (G, A) Theorem 1 1) Consider a fixed g and profit level p. Use McDiarmid ineq. to show: Proof sketch Lemma 1

13. Maria-Florina Balcan Basic Analysis, RSOPF (G,A), cont 2) Let g i be the best over S i. Know g i (S i ) ¸ g OPT (S i )/ . In particular, Using also OPT G ¸  n, get that our profit g 1 (S 2 ) +g 2 (S 1 ) is at least (1-  )OPT G / .

14. Maria-Florina Balcan Attribute Auctions, RSOPF (G k, A) G k : k markets defined by Voronoi cells around k bidders & fixed price within each market. Assume we discretize prices to powers of (1+  ). attributes

15. Maria-Florina Balcan Attribute Auctions, RSOPF (G k, A) G k : k markets defined by Voronoi cells around k bidders & fixed price within each market. Assume we discretize prices to powers of (1+  ). Corollary (roughly)

16. Maria-Florina Balcan Structural Risk Minimization Reduction SRM Reduction Let Randomly split the bidders into S 1 and S 2. Compute g i to maximize Apply g 1 over S 2 and g 2 over S 1. What if we have different functions at different levels of complexity? Don’t know best complexity level in advance. Theorem

17. Maria-Florina Balcan Attribute Auctions, Linear Pricing Functions Assume X=R d. N= (n+1)(1/  ) ln h. |G’| · N d+1 attributes 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 x

18. Maria-Florina Balcan Covering Arguments 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). 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 Theorem (roughly) If G’ is  -cover of G, then the previous theorems hold with |G| replaced by |G’|. attributes valuations Analysis Techniqu e

19. Maria-Florina Balcan 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. Apply similar techniques to limited supply. Study Online Setting. Open Problems

20. Maria-Florina Balcan Outline of the Talk Reduce problems of incentive-compatible mechanism design to standard algorithmic questions. Focus on revenue-maximization, unlimited supply. Use ideas from Machine Learning. –Sample Complexity techniques in MLT for analysis. Approximation Algorithms for Item Pricing. Revenue maximization, unlimited supply combinatorial auctions with single-minded bidders [BBHM05] [BB06]

21. Maria-Florina Balcan Algorithmic Problem, Single-minded Customers m item types (coffee, cups, sugar, apples, …), with unlimited supply of each. n customers. Say all marginal costs to you are 0 [revisit this in a bit], and you know all the (L i, w i ) pairs. Each customer i has a shopping list L i and will only shop if the total cost of items in L i is at most some amount w i (otherwise he will go elsewhere). What prices on the items will make you the most money? Easy if all L i are of size 1. What happens if all L i are of size 2?

22. Maria-Florina Balcan Algorithmic Pricing, Single-minded Customers A multigraph G with values w e on edges e. Goal: assign prices on vertices p v ¸ 0 to maximize total profit, where: APX hard [GHKKKM’05]. 10 40 15 20 30 5 10 5

23. Maria-Florina Balcan A Simple 2-Approx. in the Bipartite Case Goal: assign prices on vertices p v ¸ 0 as to maximize total profit, where: 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. Algorithm Given a multigraph G with values w e on edges e. Proof simple ! OPT=OPT L +OPT R 40 15 25 35 15 25 5 LR

24. Maria-Florina Balcan A 4-Approx. for Graph Vertex Pricing Goal: assign prices on vertices p v ¸ 0 to maximize total profit, where: 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. Algorithm Proof In expectation half of OPT’s profit is from edges with one endpoint in L and one endpoint in R. Given a multigraph G with values w e on edges e. simple ! 10 40 15 20 30 5 10 5

25. Maria-Florina Balcan Algorithmic Pricing, Single-minded Customers, k-hypergraph Problem What about lists of size · k? –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 OPT j,e be revenue OPT makes selling item j to customer e. Let X j,e be indicator RV for j 2 L & e 2 GOOD. Our expected profit at least: Algorithm 10 15 20

26. Maria-Florina Balcan Algorithmic Pricing, Single-minded Customers What if items have constant marginal cost to us? 10 40 15 20 3 5 7 5 We can subtract these from each edge (view edge as amount willing to pay above our cost). But, one difference: Can now imagine selling some items below cost in order to make more profit overall. 2 3 32 8 Reduce to previous problem.

27. Maria-Florina Balcan 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. 4 48 1 12 2 4 48 1 12 2 0 44 0

28. Maria-Florina Balcan Conclusions and Open Problems [BB06] Summary: 4 approx for graph case. O(k) approx for k-hypergraph case. Improves the O(k 2 ) approximation of Briest and Krysta, SODA’06. –Also simpler and can be naturally adapted to the online setting. 4 - , o(k). How well can you do if pricing below cost is allowed? Open Problems