1 / 33

Recall Myerson Optimal Auction

Recall Myerson Optimal Auction. Myerson: Bayesian Single-item Auction problem: Single item for sale (can be extended to ``service”) bidders Distribution from which bidder values are drawn VCG on virtual values. TexPoint fonts used in EMF.

honey
Download Presentation

Recall Myerson Optimal Auction

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Recall Myerson Optimal Auction • Myerson: Bayesian Single-item Auction problem: • Single item for sale (can be extended to ``service”) • bidders • Distribution from which bidder values are drawn • VCG on virtual values TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

  2. Quick Question (to make sure everyone is awake): • What about Myerson when bidders are correlated? (Not a product distribution) • Answer: Can beat Myerson revenue (by cheating somewhat) • Exp. Utility to agent if value = 10 is zero • Exp. Utility to agent if value = 100 is 30

  3. Joint distribution • Run VCG AND charge extra payments: • P(b) – extra payment if OTHER GUY bids b • Both bid 100: both get utility -60 • Both bid 10: both get utility +30 Better than Reserve price of 100 Revenue = TOTAL “surplus”

  4. Papers today (very partially) • On Profit-Maximizing Envy-Free Pricing • Guruswami, Hartline, Karlin, Kempe, Kenyon, McSherry • SODA 05 • Algorithmic Pricing via Virtual Valuations • Chawla, Hartline, Kleinberg • arXiv 2008 • Pricing Randomized Allocations • Briest, Chawla, Kleinberg, Weinberg • arXiv, 2009 • Approximate Revenue Maximization with Multiple Items • Hart, Nisan • arXiv 2012

  5. Bayesian Unit Demand Pricing Problem • Myerson: Bayesian Single-item Auction problem: • Single item for sale (can be extended to ``service”) • bidders • Distribution from which bidder values are drawn • VCG on virtual values • Bayesian Unit-demand Pricing Problem: • Single unit-demand bidder • items for sale • Distribution from which the bidder valuation for items is drawn

  6. What’s the connection • How do n bidders single item relate to single bidder n items?

  7. For the Bayesian Single item Auction and the Bayesian Unit-demand Pricing problem are the same problem. • Offer the item at a price of where is the virtual valuation function for distribution (For regular distributions, Use ironed virtual values if irregular)

  8. Theorem: For any price vector p, the revenue of Myerson (when bidder value of the single item comes from is the revenue when the single bidder gets value for item from • Given price vector p, new mechanism M: • Allocate the item to the bidder with that maximizes • Have bidder pay the critical price (the lowest bid at which bidder would win. • The allocation function is monotone and so this is a truthful mechanism. • Myerson is optimal amongst all Bayes Nash IC mechanisms, • Myerson gets more revenue than M

  9. Myerson Revenue is Opt revenue for Single bidder Bayesian values • Allocate the item to the bidder with that maximizes • Have bidder pay the critical price (the lowest bid at which bidder would win. • The minimum bid for bidder to win is This is the revenue to M when wins • The revenue from a single bidder who buys item with value at a price of is • QED

  10. Reg distributions: almost tight (??) • Let - the probability that no item is sold at pricing vector • Use prices , where is chosen as follows: • Choose such that and • Theorem (Chawla, Hartline, B. Kleinberg): Gives no less than Myerson revenue / 3 • Algorithmic Pricing via Virtual Valuations

  11. Discrete Distributions

  12. Distribution vs. set of Bidders The following two problems are equivalent: • Given n items and m “value vectors” for m bidders compute the revenue-optimal item pricing • Given a single bidder whose valuations are chosen uniformly at random from compute the revenue-optimal item pricing

  13. APX hardness of optimal (deterministic) item pricing • Reduction from vertex cover on graphs of maximum degree at most B (APX hard for B3). • Given (connected) Graph G with n vertices construct n items and m+n bidders (or bidder types) • For every edge add bidder with value 1 for items and zero otherwise • Additionally, for every item there is a bidder with value 2 for the item. • The optimal pricing gives profit, where is the smallest vertex cover of

  14. APX hardness of (deterministic) item pricing • Given (connected) Graph G with n vertices construct n items and m+n bidders (or bidder types) • For every edge add bidder with value 1 for items and zero otherwise • Additionally, for every item there is a bidder with value 2 for the item. • The optimal pricing gives profit, where is the smallest vertex cover of • Let S be vertex cover, charge 1 for items in S, 2 otherwise • If and price(i)=price(j)=2 then no profit from bidder - reducing price of (say) to one keeps profit unchange.

  15. Pricing Randomized Allocations • Patrick Briest,ShuchiChawla, Robert Kleinberg, S. Matthew Weinberg • Remark: When speaking, say “Bobby Kleinberg”, when writing write “Robert Kleinberg”.

  16. Selling Lotteries • 2 items uniformly value uniformly distributed in [a,b] • Optimal item pricing sets • Offer lottery equal prob item 1 or item 2, at cost • Lose here • Gain here

  17. Different models • Buy-one model: Consumer can only buy one option • Buy-many model: Consumer can buy any number of lotteries and get independent sample from each (will discard multiple copies)

  18. Buy-one model • Polytime algorithm to compute optimal lottery pricing • For item pricing (also known as envy-free unit demand pricing) the optimal item pricing is APX-hard (earlier today) • There is no finite ratio between optimal lottery revenue and optimal item-pricing revenue (for

  19. Lotteries • where , and price of lottery • – collection of lotteries • Bidder type is • Utility of picking lottery is • Utility maximizing lotteries • Max payment for utility maximizing lottery: • Profit of is

  20. Poly time optimal algorithm in case of finite support values of type j bidders - prob of type j bidders – lottery designed for type j - price for lottery Feasible Affordable Type j prefer lottery j

  21. Compare lottery revenue with item pricing revenue • - maximal expected revenue for single bidder when offered optimal revenue maximizing lotteries, bidder values from (joint) distribution • - maximal expected revenue for single bidder when offered optimal revenue maximizing item prices, bidder values from (joint) distribution

  22. Theorem: Assume optimal WLOG Valuation chooses to purchase

  23. Valuation chooses to purchase

  24. Making sense of it (again) • Correlated distributions, unit demand, finite distribution over types: • Deterministic, item pricing • We saw: APX hardness • Lower bound of (m- number of items) or - number of different agent types – not standard complexity assumptions Briest • We saw: Lotteries: optimal solvable Briest et al • Product distributions: • 2 approximation Chawla HatlineMalec Sivan • PTAS – Cai and Daskalakis

  25. 2 Approximation product distribution • bidders, 1 item • 1 bidder, n items

  26. Prophet Inequality • Gambler • games • Each game has a payoff drawn from an independent distribution • After seeing payoff of ith game, can continue to next game or take payoff and leave • Each payoff comes from a distribution • Optimal gambler stategy: backwards induction • Theorem: There is a threshold t such that if the gambler takes the first prize that exceeds t then the gambler gets ½ of the maximal payoff

  27. Proof Choose such that (exists?) Benchmark (highest possible profit for gambler): For

  28. Profit of gambler = “extras” • - event that for all :

  29. Proof of Prophet Inequality Benchmark Gambler profit

  30. More generally Find t such that increases and is continuous decreases and is continuous

  31. Relationship to Auctions • Many bidders, one item, valuation for bidder from • Vickrey 2nd price auction maximzes social welfare ( • Myerson maximize virtual social welfare maximizes revenue • Many items, one bidder, value for item from

  32. Simpler social welfare maximization • Many bidders, 1 item: • There is an anonymous (identical) posted price that gives a approximation to welfare. Directly from Prophet inequality. • Who gets the item? Could be anyone that wants it. There is no restriction on tie breaking. • Many items, one bidder: • There is a single posted price (for all items) that gives a approximation to social welfare.

  33. Simpler revenue maximization • Many bidders, 1 item: • There is a single virtual price that gives a approximation to welfare. Directly from Prophet inequality and that the revenue is the expected virtual welfare. • This means non-anonymous prices • Who gets the item? Could be anyone that wants it. There is no restriction on tie breaking. • Many items, one bidder: • Use same prices for approximation

More Related