1. C OMPETITIVE A UCTIONS 1

2. W HAT WILL WE SEE TODAY ? Were the Auctioneer! Random algorithms Worst case analysis Competitiveness 2

3. O UR PLAYGROUND Unlimited number of indivisible goods No value for the auctioneer Truthful auctions Digital goods 3

4. B EFORE WE BEGIN Normal Auctions (single round sealed bid) utility vector u bid vector b payment vector p Auction A Profit is sum of payments 4

5. R ANDOM T RUTHFULNESS 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 5

6. B ID - INDEPENDENT A UCTIONS 6

7. Intuition Masked vector f a function from masked vectors to prices Every buyer is offered to pay 7

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

9. E XAMPLES 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 9

10. B ID 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 10 TU max profit

11. T RUTHFUL -> B ID - INDEPENDENT Theorem : A deterministic auction is truthful if and only if it is equivalent to a deterministic bid- independent auction. 11

12. T RUTHFUL ->B ID - 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) 12

13. L EMMA 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 13 p2 u2 u1 p1

14. L EMMA 2:P ROOF 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 14 Pwx

15. T RUTHFUL ->B ID - INDEPENDENT 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 Bid Indepndent is truthful! 15

16. L ETS 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 16

17. C OMPETITIVENESS DOT 17

18. R OLE MODELS The competitive notion Single Price Optimum: Multi-price Optimum: 18

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

20. W HERE 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) 20

21. W HERE DOT FAILS n bidders(100 bidders) n/a bid a>>1(1 high paying bidder) Else bids 1 21 100

22. W HERE 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) 22 100

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 23

24. C OMPETITIVENESS 24

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 25

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 26

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 27

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

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 29

30. D ETERMINISM SUCKS Were going to show that no deterministic auction is β competitive 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 30

31. D ETERMINISM 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 31

32. D ETERMINISM 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 32

33. D ETERMINISM SUCKS : PROOF 33

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

35. R ANDOM A UCTIONS 35

36. R ANDOM A UCTIONS Split the bid vector b in two: b’, b’’ Use each part to build auction for the other 36

37. DSOT 37

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

39. E CCENTRIC MILLIONAIRES EXAMPLE Small-time bidders bid small (1) 2 Eccentric millionaires bid h,h+e b’b’’ 39 1M 1M+1 1M 1

40. E CCENTRIC MILLIONAIRES EXAMPLE Small-time bidders bid small (1) 2 Eccentric millionaires bid h,h+e b’b’’ 40 1M 1M+1 1M 1M+1

41. E CCENTRIC 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 41

42. B ETTER BOUNDS : SPECIAL CASE Special case where b is bounded-range: Then 42

43. P ROOF Denotebest sale price for at least r items The price for Than lets define 43

44. 44

45. So, in special cases it has a very good bound In worst case, it is C-competitive C is worse than 4 45

46. 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

47. SCS 47

48. 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

49. SCS COMPETITIVE Suppose F(2) results is kp Uniform divison between b’ and b’’: k’ and k’’ 49

50. C OMPETITIVE R ATIO 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 50

51. B OUNDED SUPPLY If we only have k goods Than we use k best bidders and run unlimited supply case Competitive vs 51

52. B OUNDED - SUPPLY TRUTHFULNESS none of the bidders win at a price lower than the highest ignored bid. Use k-vickery to get p-v use auction of unlimited supply on winners get auction price p-A use price max(pv,pA) 52

53. U P TILL N OW Bid independent is truthful Worst case outlook Our benchmarks: F,T Deterministic is just now good enough competitiveness against F(2) Examples of random algorithms DOST: C-competitive SCS : 4-competitive 53

54. C OMPETITIVENESS II is F the best benchmark? 54

55. M ULTI - PRICE F is best single price F(2) comparable to F What about using T? T is only O(log(n)) better Mabye other multi-priced? 55

56. M ONOTONE FUNCTIONS F is better than all monotone auctions Non-monotone example: Hard-coded actions Lets take b* such that half bid 1 and half bid h Lets create function which maximizes profit Acts as omniscient on b* Poorly on other results Lets generalize 56

57. H ARD CODED AUCTIONS Let b* be out bid specific bid will maximize profit on b* bad profit on bids that differ in 1 57

58. M ONOTONE F UNCTIONS Basically, if you bid more you will pay less makes sense, for is higher for the lower bidder DOT,DSOT,SCS, Vickery are monotone 58

59. S UMMARY Bid independent is truthful Worst case outlook competitiveness against F(2) use of random auctions Examples of random algorithms DOST: C-competitive SCS : 4-competitive F is a good benchmark 59

60. Q UESTIONS ? 60