Competitive Auctions Review Rattapon Limprasittiporn

Feb 9, 2004Rattapon Limprasittiporn2 Outlines Bibliography Introduction Software seller problem Truthful Competitive Goal & Solution

Feb 9, 2004Rattapon Limprasittiporn3 Bibliography Andrew V. Goldberg Microsoft Research MASSACHUSETTS INSTITUTE OF TECHNOLOGY, Doctor of Philosophy degree in Computer Science, January Digital commerce models and languages. Auctions. Algorithm Design and Analysis. Implementation and Computational Evaluation of Efficient Algorithms. Archival Intermemory.

Feb 9, 2004Rattapon Limprasittiporn4 Bibliography Jason D. Hartline University of Washington Postdoctoral research fellow at Carnegie Mellon University with the ALADDIN Center Economic aspects of algorithms Optimization problems when input is private information of selfish agents Game theoretic

Feb 9, 2004Rattapon Limprasittiporn5 Bibliography Anna R. Karlin Stanford University Theoretical computer Design and analysis of algorithms Probabilistic algorithms Online algorithms

Feb 9, 2004Rattapon Limprasittiporn6 Outlines Bibliography Introduction Software seller problem Truthful Competitive Goal & Solution

Feb 9, 2004Rattapon Limprasittiporn7 Introduction Bidder: person who bid Utility Value Max price that bidder willing to pay Not the price that bidder pays Bidder is happy if he pay less than his utility value Auctioneer: person who sell

Feb 9, 2004Rattapon Limprasittiporn8 Example Bidders: Alice, Bob, and Carrol Bob wins, but... Alice 7 Bob 10 Carrol Nah! 4 6 9

Feb 9, 2004Rattapon Limprasittiporn9 Example Bidder’s Goal Pay minimum price which greater than all other people’s utility values Problem Lots of bidding tactic Single-round sealed-bid auction

Feb 9, 2004Rattapon Limprasittiporn10 Single-Round Sealed-Bid Single-round Each bidder submits bid only once Sealed-bid Bidder blinded from other bidder’s bid Who win? Vickrey auction The highest bid wins Pay 2 nd -highest-bid price

Feb 9, 2004Rattapon Limprasittiporn11 Example Bidders: Alice, Bob, and Carrol Bob wins and pay $7 Alice 7 Bob 10 Carrol

Feb 9, 2004Rattapon Limprasittiporn12 Vickrey Auction At the end The highest bid wins Pay 2 nd -highest-bid price k-item Vickrey auction Have k items to sell Single price auction k highest bidders win All winners pay the k+1 th highest bid

Feb 9, 2004Rattapon Limprasittiporn13 Have 2 items to sell Bidders: Alice, Bob, Carrol, Daniel, and Eve Bob and Denial win Both winner and pay $5 Example Alice 5 Bob 11 Carrol Deniel 12 Eve 2 212

Feb 9, 2004Rattapon Limprasittiporn14 Truthfulness Should bidder bid their utility value? Yes, at least in k-item Vickrey Auction An auction is “truthful” if it encorages bidder to bid their utility K-item Vickrey Auction is a truthful auction

Feb 9, 2004Rattapon Limprasittiporn15 Outlines Bibliography Introduction Software seller problem Truthful Competitive Goal & Solution

Feb 9, 2004Rattapon Limprasittiporn16 Software Seller’s Problem Seller has unlimitted amount of products How can he put them in auction? Choose k that maximizes his revenue Selling 3 items to Alice, Bob, and Deniel is better than sell 2 items to Bob and Deniel Is this a good auction? Alice 5 Bob 11 Carrol Deniel 12 Eve 2 212

Feb 9, 2004Rattapon Limprasittiporn17 Software Seller’s Problem Bidders: Alice, Bob, and Carrol To maximize revenue, seller sells software to Carrol only Alice 10 Bob 30 Carrol

Feb 9, 2004Rattapon Limprasittiporn18 Software Seller’s Problem What if Bob changes his bid from 30 to 11 To maximize revenue, seller sells software to Bob and Carrol at the price of 10 Hey, this is not right! This auction is thus “not truthful” Alice 10 Bob 30 Carrol

Feb 9, 2004Rattapon Limprasittiporn19 Problem Find a good way to “auction” for software seller We are on the seller side What is a “good auction” Truthful Yield “good” revenue Good compared to an “ideal” case “Competitive” (to the ideal case)

Feb 9, 2004Rattapon Limprasittiporn20 Outlines Bibliography Introduction Software seller problem Truthful Competitive Goal & Solution

Feb 9, 2004Rattapon Limprasittiporn21 Truthful (revisit) What is “truthful” auction? Encorage bidders to bid their utility Prevent tactic and strategy How to make an auction “truthful”? Process result of bidder i without looking at his bid “Bid-Independent Auction”

Feb 9, 2004Rattapon Limprasittiporn22 Bid-Independent Auction b = set of all bids that bidders bid For each bidder i Exclude bid from bidder i to get b -i, the set of all bids except the bid from bidder i Compute “auction funtion”, f, on b -i to get threshold t i If bidder i bids more than t i, he wins at price t i, otherwise, he loses

Feb 9, 2004Rattapon Limprasittiporn23 Example b = {5, 11, 4, 12, 2} Let “auction function”, f, = “maximum of” For bidder 1 b -1 = {11, 4, 12, 2} f(b -1 ) = 12 Since 5 < 12, bidder 1 loses the auction For bidder 2 b -2 = {5, 4, 12, 2} f(b -1 ) = 12 Since 11 < 12, bidder 2 loses the auction For bidder 4 b -4 = {5, 11, 4, 2} f(b -4 ) = 11 Since 12 > 11, bidder 4 wins the auction at price 11

Feb 9, 2004Rattapon Limprasittiporn24 Auction Function Auction function, f, is a core of bid- independent auction f is “maximum of” = 1-item Vickrey Auction f is “k th maximum of” = k-item Vickrey Auction Must be “monotone” If b -i > b -j then f(b -i ) > f(b -j ) Every monotone bid-independent auction is truthful

Feb 9, 2004Rattapon Limprasittiporn25 Outlines Bibliography Introduction Software seller problem Truthful Competitive Goal & Solution

Feb 9, 2004Rattapon Limprasittiporn26 Competitive Good revenue compared to an ideal (in seller’s sense) case Ideal case: Optimal single price omniscient auction (F) k highest bids win at price k th highest bid Find k > 1 that yields highest revenue to be the revenue of F Ex. b = {5, 11, 4, 12, 2, 8} “k = 3” yields max revenue of 24 Revenue of F is F(b) = 24 Seller is happy if the revenue is close to F

Feb 9, 2004Rattapon Limprasittiporn27 Competitive Competitive = good revenue Competitive to the ideal case Gives revenue within constant factor far away form F Auction A is competitive if A(b)  F(b) /  for some constant , and for all possible bid input b (worst case analysis)

Feb 9, 2004Rattapon Limprasittiporn28 Example Is 3-item Vickrey Auction competitive? Let b = {20, 20, 20, 1, 1, 1} 3-item Vickrey Auction gives revenue of 3 Optimal single price omniscient auction F gives revenue of 60 3-item Vickrey Auction is not competitive In fact, all deterministic auctions are not competitive!

Feb 9, 2004Rattapon Limprasittiporn29 Outlines Bibliography Introduction Software seller problem Truthful Competitive Goal & Solution

Feb 9, 2004Rattapon Limprasittiporn30 Goal Find a way of auction to make software seller happy Truthful Competitive No deterministic auction is competitive Sol: randomized auction Compute twice might not be the same

Feb 9, 2004Rattapon Limprasittiporn31 Solutions Dual-Price Sampling Optimal Threshold Auction (DSOT) Sampling Cost-Sharing Auction (SCS)

Feb 9, 2004Rattapon Limprasittiporn32 DSOT Partition bids b randomly into two sets b’ and b’’ Use omniscient auction F to compute ideal revenue of b’ and b’’ and get p’ and p’’ p’: price that each winner in b’ pay to get F(b’) p’’: price that each winner in b’’ pay to get F(b’’) Use p’ as a threshold for all bids in b’’ All bids in b’’ less than p’ are rejected All bids in b’’ greater than p’ win at price p’ Use p’’ as a threshold for all bids in b’

Feb 9, 2004Rattapon Limprasittiporn33 Example b = {14, 21, 13, 4, 23, 15, 6, 12, 7} Random partition: b’ = {14, 15, 21, 6, 12, 7} b’’ = {13, 4, 23} Compute threshold F(b’) = 48 which sell 4 items at price 12 = p’ F(b’’) = 26 which sell 2 items at price 13 = p’’ Use p’’ = 13 as a threshold in b’ Bidders who bid 14, 15, 21 win at price 13 Use p’ = 12 as a threshold in b’’ Bidders who bid 13, 23 win at price 12

Feb 9, 2004Rattapon Limprasittiporn34 DSOT Analysis Truthful Bid-Independent Auction Competitive Get some factor of F Multiple price?

Feb 9, 2004Rattapon Limprasittiporn35 Solutions Dual-Price Sampling Optimal Threshold Auction (DSOT) Sampling Cost-Sharing Auction (SCS)

Feb 9, 2004Rattapon Limprasittiporn36 SCS Partition bids b randomly into two sets b’ and b’’ Use omniscient auction F to compute ideal revenue of b’ and b’’ and get F(b’) and F(b’’) The highest k’ bids in b’ that each bid higher than F(b’’) / k’ win at price F(b’’) / k’ The highest k’’ bids in b’’ that each bid higher than F(b’) / k’’ win at price F(b’) / k’’

Feb 9, 2004Rattapon Limprasittiporn37 Example b = {14, 21, 13, 4, 23, 15, 6, 12, 7} Random partition: b’ = {14, 15, 21, 6, 12, 7} b’’ = {13, 4, 23} Compute threshold F(b’) = 48 which sell 4 items F(b’’) = 26 which sell 2 items Two bids in b’, 21 and 15, can share the cost of 26 by paying 13 each Bidders who bid 21 and 15 win at price 13 No bid in b’’ can share the cost of 48 No bid in b’’ wins

Feb 9, 2004Rattapon Limprasittiporn38 SCS Analysis Truthful Bid-Independent Auction Competitive Get some factor of F

Feb 9, 2004Rattapon Limprasittiporn39 Conclusion Two models for software seller DSOT SCS Truthful and competitive Worst case analysis

Feb 9, 2004Rattapon Limprasittiporn40 Thank you Question?