Competitive Auctions Review

Competitive Auctions Review Rattapon Limprasittiporn

Outlines • Bibliography • Introduction • Software seller problem • Truthful • Competitive • Goal & Solution Rattapon Limprasittiporn

Bibliography • Andrew V. Goldberg • Microsoft Research • MASSACHUSETTS INSTITUTE OF TECHNOLOGY, Doctor of Philosophy degree in Computer Science, January 1987. • Digital commerce models and languages. • Auctions. • Algorithm Design and Analysis. • Implementation and Computational Evaluation of Efficient Algorithms. • Archival Intermemory. Rattapon Limprasittiporn

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 Rattapon Limprasittiporn

Bibliography • Anna R. Karlin • Stanford University • Theoretical computer • Design and analysis of algorithms • Probabilistic algorithms • Online algorithms Rattapon Limprasittiporn

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 Rattapon Limprasittiporn

Example • Bidders: Alice, Bob, and Carrol • Bob wins, but ... Alice 7 Bob 10 Carrol 4 3 7 Nah! 6 9 4 Nah! Rattapon Limprasittiporn

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 Rattapon Limprasittiporn

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 2nd-highest-bid price Rattapon Limprasittiporn

7 10 4 Example • Bidders: Alice, Bob, and Carrol • Bob wins and pay $7 Alice 7 Bob 10 Carrol 4 Rattapon Limprasittiporn

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

Example • Have 2 items to sell • Bidders: Alice, Bob, Carrol, Daniel, and Eve • Bob and Denial win • Both winner and pay $5 Alice 5 Bob 11 Carrol 4 Deniel 12 Eve 2 5 11 4 12 2 Rattapon Limprasittiporn

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 Rattapon Limprasittiporn

Alice 5 Bob 11 Carrol 4 Deniel 12 Eve 2 5 11 4 12 2 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? Rattapon Limprasittiporn

Software Seller’s Problem • Bidders: Alice, Bob, and Carrol • To maximize revenue, seller sells software to Carrol only Alice 10 Bob 30 Carrol 40 10 30 40 Rattapon Limprasittiporn

Alice 10 Bob 30 Carrol 40 10 11 40 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” Rattapon Limprasittiporn

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) Rattapon Limprasittiporn

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 bidderiwithout looking at his bid • “Bid-Independent Auction” Rattapon Limprasittiporn

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

Example • b = {5, 11, 4, 12, 2} • Let “auction function”, f, = “maximum of” • For bidder1 • b-1 = {11, 4, 12, 2} • f(b-1) = 12 • Since 5 < 12, bidder1 loses the auction • For bidder2 • b-2 = {5, 4, 12, 2} • f(b-1) = 12 • Since 11 < 12, bidder2 loses the auction • For bidder4 • b-4 = {5, 11, 4, 2} • f(b-4) = 11 • Since 12 > 11 , bidder4 wins the auction at price 11 Rattapon Limprasittiporn

Auction Function • Auction function, f, is a core of bid-independent auction • f is “maximum of” = 1-item Vickrey Auction • f is “kth 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 Rattapon Limprasittiporn

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 kth 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 Rattapon Limprasittiporn

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) ³F(b) / b for some constant b, and for all possible bid input b (worst case analysis) Rattapon Limprasittiporn

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! Rattapon Limprasittiporn

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 Rattapon Limprasittiporn

Solutions • Dual-Price Sampling Optimal Threshold Auction (DSOT) • Sampling Cost-Sharing Auction (SCS) Rattapon Limprasittiporn

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’ Rattapon Limprasittiporn

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 Rattapon Limprasittiporn

DSOT Analysis • Truthful • Bid-Independent Auction • Competitive • Get some factor of F • Multiple price? Rattapon Limprasittiporn

Solutions • Dual-Price Sampling Optimal Threshold Auction (DSOT) • Sampling Cost-Sharing Auction (SCS) Rattapon Limprasittiporn

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’’ Rattapon Limprasittiporn

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 Rattapon Limprasittiporn

SCS Analysis • Truthful • Bid-Independent Auction • Competitive • Get some factor of F Rattapon Limprasittiporn

Conclusion • Two models for software seller • DSOT • SCS • Truthful and competitive • Worst case analysis Rattapon Limprasittiporn

Thank you Question? Rattapon Limprasittiporn

Competitive Auctions Review