On Random Sampling Auctions for Digital Goods

Transcription

1 On Random Sampling Auctions for Digital Goods Saeed Alaei Azarakhsh Malekian Aravind Srinivasan Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 1

2 Outline Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 2

3 Problem Definition Originally proposed by Goldberg & Hartline. We have a single type of good with unlimited supply There are n bidders with bids v 1 v n. We want a revenue-maximizing incentive compatible auction. We have no prior information on distributions. Benchmark is the optimal uniform price auction: max λ 2 λ v λ Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 3

4 Random Sampling Optimal Price Auction The mechanism: Partition the bids to two groups A and B uniformly at random. Compute the optimal unform price in each group and offer it to the other group. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 4

5 Random Sampling Optimal Price Auction The mechanism: Partition the bids to two groups A and B uniformly at random. Compute the optimal unform price in each group and offer it to the other group. RSOP incentive compatible. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 4

6 Random Sampling Optimal Price Auction The mechanism: Partition the bids to two groups A and B uniformly at random. Compute the optimal unform price in each group and offer it to the other group. RSOP incentive compatible. Conjecture The revenue of RSOP is at least 1 4OPT. i.e. RSOP is 4-competitive. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 4

7 RSOP Example Suppose the bids are {7, 6, 5, 1}. After random partitioning of the bids, A = {6, 1} and B = {7, 5}. We offer 6 to B and 5 to A. we get a revenue of 11 while OPT is 15. Conjecture The worst case performance of RSOP is when bids are {1, 1 2 }. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 5

8 Previous/Present Results Goldberg & Hartline (2001) : OPT RSOP < 7600 Feige et al (2005) : OPT RSOP < 15 Our result (2008) : OPT RSOP < 4.68 Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 6

9 Previous/Present Results Goldberg & Hartline (2001) : OPT RSOP < 7600 Feige et al (2005) : OPT RSOP < 15 Our result (2008) : OPT RSOP < 4.68 Theorem The competitive ratio of RSOP is (λ is the index of the winning bid in OPT ) (e.g. in {7, 6, 5, 1}, λ = 3): < 4.68 λ < 6 < 4 λ > 6 < 3.3 λ (1) Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 6

10 Assumptions We have an infinite number of bids (i.e. n = ), by adding 0 s. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 7

11 Assumptions We have an infinite number of bids (i.e. n = ), by adding 0 s. OPT = 1, by scaling all the bids. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 7

12 Assumptions We have an infinite number of bids (i.e. n = ), by adding 0 s. OPT = 1, by scaling all the bids. v 1 is alwayn B and we only consider the revenue obtained from set B. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 7

13 A lowerbound on RSOP revenue when λ > 10 A dynamic programming method for computing the lower bound given the λ. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 8

14 A lowerbound on RSOP revenue when λ > 10 A dynamic programming method for computing the lower bound given the λ. A second method which independent of λ but assumet is large (i.e. > 5000) and uses Chernoff bound. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 8

15 Random Partition s 8 i 7 Example 6 A = {v 2, v 3, v 4 } 5 B = {v 1, v 5, v 7 } 4 Definition S i = #{v j v j A, j i} i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 9

16 Random Partition s 8 i 7 Example 6 A = {v 2, v 3, v 4 } 5 B = {v 1, v 5, v 7 } 4 = 3/4i = i = i/2 Definition S i = #{v j v j A, j i} i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 9

17 Random Partition Observation S i lim i i or lim Pr i 1 2 [ Si i < 1 2 ɛ ] = i = i/ i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 10

18 Worst Profit Ratio Observations j : S j j < α = i = i/2 α = 3/ i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 11

19 Worst Profit Ratio Observations S j j : < α j Prof (B) Prof (A) 1 α α = i = i/2 α = 3/ i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 11

20 Worst Profit Ratio Observations S j j : < α j Prof (B) Prof (A) 1 α α = i = i/2 α = 3/4 Prof (A) S λ λ i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 11

21 Worst Profit Ratio Observations S j j : < α j Prof (B) Prof (A) 1 α α = i = i/2 α = 3/4 Prof (A) S λ λ i S i Z = min i S i Prof (B) E[Z S λ λ ] Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions i

22 α-event Definition (E α event) = i = i/2 α = 3/4 E α : j : S j j α i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 12

23 α-events E [α,α] = E α E α = i = i/2 α α Z E [α,α] 1 α α i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 13

24 α-events E [α,α] = E α E α Z E [α,α] 1 α α = i = i/2 α 8 α 7 α 6 α 5 α 4 α 3 α 2 α 1 α 0 Z= i Pr[E [α i,α i+1 ]] 1 α i α i Z= i (Pr[Eα i+1 ] Pr[Eα i ]) 1 α i α i 0 i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 13

25 α-events E [α,α] = E α E α Z E [α,α] 1 α α = i = i/2 α 8 α 7 α 6 α 5 α 4 α 3 α 2 α 1 α 0 Z= i Pr[E [α i,α i+1 ]] 1 α i α i Z= i (Pr[Eα i+1 ] Pr[Eα i ]) 1 α i α i 0 i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 13

26 Computing E[Z] Lemma The worst ratio of profit of set B to profit of set B can be computed using the following: E[Z] = i = i Pr[E [αi 1,α i ]] 1 α i α i (Pr[E αi ] Pr[E αi 1 ]) 1 α i α i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 14

27 The Dynamic Program for computing P[E α ] Definition Let P α (k, j) be the probability that for any 1 i k, at most α fraction of the v 1,..., v i are in A and exactly j of v 1,, v k are in A. Let P α (k) = k j=0 P α(k, j), then Pr[E α ] = P α ( ) Dynamic Program for computing P α (k, j) 0 j > αk 1 j = k = 0 P α (k, j) = 1/2P α (k 1, j) j = 0, k > 0 1/2P α (k 1, j) + 1/2P α (k 1, j 1) 0 < j < αk Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 15

28 When λ is large Claim As λ increases, the correlation between S λ /λ and Z decreases so we can separate them. [ ] Sλ Prof (b) E λ Z E[ S λ λ ]E[Z] 1 2 E[Z] We use a variant of Chernoff bound to bound the error caused by separating the two terms. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 16

29 The Dynamic Program for E[ S λ λ Z] Definition Let R α (k, j) the expected value of lowerbound for profit of set A conditioned and multiplied by the probability that for any 1 i k, at most α fraction of the v 1,..., v i are in A and exactly j of v 1,, v k are in A. Dynamic Program for computing R α (k, j) j = 0 or 0 j > αk R α (k, j) = 1/2R α (k 1, j) + 1/2R α (k 1, j 1) 0 < j αk j λ P α(k 1, j) k = λ Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 17

30 The Dynamic Program for E[ S λ λ Z] (Continued) Dynamic Program for computing E [ Sλλ Z E α ] R α (k) = R α ( ) = E j R α (k, j) i=0 E[ S λ λ Z] = i [ Sλ λ E α ] Pr [E α ] (R αi R αi 1 ) 1 α i α i Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 18

31 An upperbound on the revenue of RSOP with large λ Theorem For any given λ, there is a set of bids with λ being the index of the winning price and such that RSOP does not get a revenue of more than 3/8. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 19

32 The equal revenue instances Definition An Equal Revenue Instance with n bids consists of the bids {1, 1 2,, 1 n }. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 20

33 The equal revenue instances Definition An Equal Revenue Instance with n bids consists of the bids {1, 1 2,, 1 n }. Observation In an equal revenue instance, the price offered from each set is the worst price for the other set. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 20

34 The equal revenue instances, RSOP Definition (RSOP ) It is the same as RSOP except that when set A is empty, the price that is offered from A to B is v n instead of 0. The difference between the revenue of RSOP and RSOP is 1/2 n. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 21

35 The equal revenue instances, RSOP Definition (RSOP ) It is the same as RSOP except that when set A is empty, the price that is offered from A to B is v n instead of 0. The difference between the revenue of RSOP and RSOP is 1/2 n. Claim The revenue of RSOP on an equal revenue instance with n + 1 bids less than that with n bids. The proof is by induction. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 21

36 The equal revenue instances, RSOP Definition (RSOP ) It is the same as RSOP except that when set A is empty, the price that is offered from A to B is v n instead of 0. The difference between the revenue of RSOP and RSOP is 1/2 n. Claim The revenue of RSOP on an equal revenue instance with n + 1 bids less than that with n bids. The proof is by induction. Fact Revenue of RSOP for equal revenue instances with n 10 is at most Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 21

37 Revenue RSOP revenue (basic lowerbound) λ E[RSOP] Competitive-Ratio Based on dynamic programming up to n = 5000 and then Chernoff bound. Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 22

38 RSOP revenue (secondary lowerbound) λ E[RSOP] Competitive-Ratio Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 23

39 Questions? Questions? Saeed Alaei, Azarakhsh Malekian, Aravind Srinivasan Random Sampling Auctions... 24