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Competitive Auctions

Competitive Auctions. What will we see today?. Were the Auctioneer! Random algorithms Worst case analysis Competitiveness. Our playground. Unlimited number of indivisible goods No value for the auctioneer Truthful auctions Digital goods. Before we begin.

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Competitive Auctions

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  1. Competitive Auctions

  2. What will we see today? • Were the Auctioneer! • Random algorithms • Worst case analysis • Competitiveness

  3. Our playground • Unlimited number of indivisible goods • No value for the auctioneer • Truthful auctions • Digital goods

  4. Before we begin • Normal Auctions (single round sealed bid) • utility vector u • bid vector b • payment vector p • Auction A • Profit is sum of payments

  5. Random Truthfulness • Reminder: Truthful auctions are auctions where each bidder maximizes his profit when bids his utility • Random is probability distribution over deterministic auctions • Random Strong Truthfulness • One natural approach • Our chosen approach • A randomized auction is truthful if it can be described as a probability distribution over deterministic truthful auctions

  6. Bid-independent Auctions

  7. Bid-independent Auctions • Intuition • Masked vector • f a function from masked vectors to prices • Every buyer is offered to pay

  8. Auction • Auction 1: Bid-independent Auction: Af(b)

  9. Examples • Bid vector for buying Lonely-Island new song • 4 bets • What have we got? • 1-item vickery • For k’th largest bid we get • K- item vickery

  10. Bid independent -> truthful • We are offered T(=20) what should we bid? • If U(=15) < T we cant win • If U(=30) >= T any bid >= T will win • Either way U maximizes bidder’s profit max profit T U

  11. Truthful -> Bid-independent • Theorem : A deterministic auction is truthful if and only if it is equivalent to a deterministic bid-independent auction.

  12. Truthful->Bid-independent • For bid vector b and bidder i we fix all bids except bi • Lemma1 For each x where i wins he pays same p • Lemma2 i wins for x>p (possibly for p)

  13. Lemma 1 proof • Lemma1: i pays p • Assume to the contrary • x1,x2 where i pays p1>p2 • Than if Ui = x1 i should lie and tell x2 • =>In contrast to A’s truthfulness p1 p2 u2 u1

  14. Lemma2:Proof • Lemma2: for each x>p (and possibly p) x wins • Assume to the contrary • w exists • w>p • w wins • x exists such that • x>p • x doesn’t win • if U=x i should lie and say w • => In contrast to A’s truthfulness P x w

  15. Truthful->Bid-independent Bid Indepndent is truthful! • Define • Than for any bid b • For bid b if i in A wins and pays p • than also in Af • If loses than • p doesn’t exist • or bi < p

  16. Lets shake things up • Reminder: • Random Auctions • Random Truthful Auctions • A randomized bid-independent auction is a probability distribution over bid-independent auctions • => A randomized auction is truthful iff it is equivalent to a randomized bid-independent auction

  17. Competitiveness DOT

  18. Role models • The competitive notion • Single Price Optimum: • Multi-price Optimum:

  19. DOT • Deterministic Optimal Threshold • single-priced • Define opt(b) as the optimum single price • DOT: • Calculates maximum for rest of the group

  20. Where DOT is optimal • Bids range from [0$,50$] • Bids are i.i.d • DOT optimal for a wide range of problems! • For any bounded support i.i.d(without proof)

  21. Where DOT fails • n bidders(100 bidders) • n/a bid a>>1(1 high paying bidder) • Else bids 1 100

  22. Where DOT fails • For each a bidder : • (n/a-1) a-bidders • profit for p=a is n-a but for p=1 is n-1 • p = 1 • For each 1 bidder • n/a a-bidders • profit for p=1 is n-1 but for p=a is n • p = a • Profit is n/a (number of a bidders) 100

  23. DOT conclusion • Why are we talking worst case? • DOT prevails in Bayesian model • Loses in worst case • When not safe to assume true random source • Competitive outlook is logical

  24. Competitiveness

  25. F-competitive failure • Lemma: For any truthful auction Af and any β≥1, there is a bid vector b such that the expected profit of Af on b is less than F(b)/β

  26. proof • 2 bidders • Define h the smallest value such that • Lets consider the bid {1,H} where H=4βh>1 • Profit is at most • For H bidder : • For 1 bidder : 1

  27. Set our eyes lower • 2-optimal single price bid • The optimal bids that sells at least 2 items • Same as f(b) unless there is one bidder with Hugh utility

  28. Similarly we define the sale of at least m items

  29. β-competitive • Definition: We say that auction A is β-competitive against F-m if for all bid vectors b, the expected profit of A on b satisfies

  30. Determinism sucks • Were going to show that no deterministic auction is βcompetitive • Theorem: Let Af be any symmetric deterministic auction defined by bin-independent function f. Then Af is not competitive. For any m,n there exists a bid vector b of length n such the Af’s profit is at most • Symmetric auction: order of bids doesn’t matter • For example, consider F(2). We can find a bid vector at length 8 such that Af’s profit is at most F(2)/4

  31. Determinism sucks: proof • Lets look at specific m,n at a specific auction Af • Consider bid b where all bids are n or 1 • Let f(j) be the price where j bids are n • n – 1 – j bid 1 • for f(0) > 1 • Consider the bids where all bids are 1

  32. Determinism sucks: proof • k in 0..n-1 the largest integer where f(k) <= 1 • We build a bid with • (k+1) n-bids • (n – k – 1) 1-bids • 1-bidders lose ( f(k+1) > 1) • n-bidders win • Profit : (k+1)f(k) < k + 1

  33. Determinism sucks: proof

  34. Conclusion • Why worst case? • Not truly random source • How competitive? • F is too good • Why random? • Because determinism is not good enough

  35. Random Auctions

  36. Random Auctions • Split the bid vector b in two: b’, b’’ • Use each part to build auction for the other

  37. DSOT

  38. DSOT • Observation: truthful • C competitive to F(2) (without proof) • Unknown C, at least 4

  39. Eccentric millionaires example • Small-time bidders bid small (1) • 2 Eccentric millionaires bid h,h+e b’b’’ 1M 1 1M 1M+1

  40. Eccentric millionaires example • Small-time bidders bid small (1) • 2 Eccentric millionaires bid h,h+e b’b’’ 1M+1 1M 1M 1M+1

  41. Eccentric millionaires example • F(2) profit is 2h(= 2M) • profit is h * Pr[2 high bids are split between auctions] • = h/2(=M/2) • Competitive Ratio of 4

  42. Better bounds: special case • Special case where • b is bounded-range: • Then

  43. Proof • Denote best sale price for at least r items • The price for • Than lets define

  44. So, in special cases it has a very good bound • In worst case, it is C-competitive • C is worse than 4

  45. SCS • Sampling Cost-sharing • CostShare-C: if you have k bidders (highest) which are willing to pay C collectively (bid>C/k). Charge each for C/k • CostShare is truthful • For profit is C, else 0 • I know exactly how much I want to make, regardless of bids

  46. SCS

  47. SCS competitive • if F’=F’’ profit is at least F’F • Auction profit is R = min(F’,F’’) • Suppose F’<F’’ • b’ cannot achieve F’’ • b’’ profit is F’

  48. SCS competitive • Suppose F(2) results is kp • Uniform divison between b’ and b’’: k’ and k’’

  49. Competitive Ratio • Begins as ¼ • Approaches ½ • Tight proof • Consider 2 high bids h,h+e • But we always throw half • Can we improve? • Yes, Costshare(rF’) and Costshare(rF’’) • Competitive ratio is 4/r

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