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Mechanism Design, Machine Learning, and Pricing Problems Maria-Florina Balcan

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Overview Software Pricing Digital Music Problems at the intersection of CS and Economics Pricing and Revenue Maximization

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Overview Advertising Problems at the intersection of CS and Economics Pricing and Revenue Maximization

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Pricing Problems Version 1: Seller knows the true values. Version 2: values given by selfish agents. One Seller, Multiple Buyers with Complex Preferences. Computer Science Economics Algorithm Design Problem (AD) Incentive Compatible Auction (IC) Previous Work on IC : very specific mechanisms for restricted settings. Seller’s Goal: maximize profit.

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Pricing Problems Version 1: Seller knows the true values. Version 2: values given by selfish agents. One Seller, Multiple Buyers with Complex Preferences. Computer Science Economics Algorithm Design Problem (AD) Incentive Compatible Auction (IC) Previous Work on IC : very specific mechanisms for restricted settings. Seller’s Goal: maximize profit. BBHM (FOCS 2005) : Generic Reduction

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Reduce IC to AD Generic Framework for reducing problems of incentive- compatible mechanism design to standard algorithmic questions. [Balcan-Blum-Hartline-Mansour, FOCS 2005, JCSS 2007] Focus on revenue-maximization, unlimited supply. - Digital Good Auction - Attribute Auctions - Combinatorial Auctions Use ideas from Machine Learning. –Sample Complexity techniques in ML both for design and analysis.

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Outline Part I: Generic Framework for reducing problems of incentive-compatible mechanism design to standard algorithmic questions. [Balcan-Blum-Hartline-Mansour, FOCS 2005, JCSS 2007] Part II: Approximation Algorithms for Item Pricing. E.g, [Balcan-Blum, EC 2006, TCS 2007] Revenue maximization in combinatorial auctions with single-minded consumers.

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MP3 Selling Problem Seller 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 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.

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Example 2, Boutique Selling Problem $20 30$ $30 $5 $25 $20 $100 $1

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Example 2, Boutique Selling Problem Goal: Profit Maximization Combinatorial Auctions Compete with best item pricing [GH01]. $20 30$ $30 $5 $25 $20 $100 $1 (unit demand consumers)

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Generic Setting (I) S set of n bidders. Space of legal offers/pricing functions. g(i) – profit obtained from making offer g to bidder i g maps the pub i to pricing over the outcome space. Bidder i: –priv i (e.g., how much i is willing to pay for the MP3 file) –pub i (e.g., ZIP code) –bid i ( reported priv i ) Digital Goodg=“ take the good for p, or leave it” g(i)= p if p · bid i g(i)= 0 if p>bid i O outcome space. Incentive Compatible: bid i =priv i

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Generic Setting (I) S set of n bidders. Space of legal offers/pricing functions. g(i) – profit obtained from making offer g to bidder i g maps the pub i to pricing over the outcome space. Bidder i:priv i, pub i, bid i Goal: Profit Maximization G - pricing functions. Goal: Incentive Compatible mechanism to do nearly as well as the best g 2 G. Unlimited supply Profit of g: i g(i)

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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

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Generic Setting (II) Focus on one-shot mechanisms, off-line setting. 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: bid i =priv i –offer for bidder i based on the public information of S and reported private info of S n {i}.

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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.

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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 )

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Basic Analysis, RSOPF (G, A) Theorem 1 Proof sketch Lemma 1 1) Fixed g and profit level p. Use a tail ineq. show: h - maximum valuation, G - finite

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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 / .

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Attribute Auctions, RSOPF (G k, A) G k : k markets defined by Voronoi cells around k bidders & fixed price within each market. Discretize prices to powers of (1+ ). attributes

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Attribute Auctions, RSOPF (G k, A) G k : k markets defined by Voronoi cells around k bidders & fixed price within each market. Corollary (roughly) Discretize prices to powers of (1+ ).

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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 different functions at different levels of complexity? Don’t know best complexity level in advance. Theorem

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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

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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 Technique

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Summary [BBHM05] Explicit connection between machine learning and mechanism design. Use MLT both for design and analysis in auction/pricing problems. Unique challenges & particularities: Loss function discontinuous and asymmetric. Range of valuations large.

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