The Competition Complexity of Auctions: Bulow-Klemperer Results for Multidimensional Bidders

Transcription

1 The : Bulow-Klemperer Results for Multidimensional Bidders Oxford, Spring 2017 Alon Eden, Michal Feldman, Ophir Tel-Aviv University Inbal Talgam-Cohen, Marie Curie Hebrew University Matt Princeton *Based on slides by Alon Eden

2 Complexity in AMD One goal of Algorithmic Mechanism Design: Deal with complex allocation of goods settings Goods may not be homogenous Valuations and constraints may be complex 2 Eden et al EC'17 Inbal Talgam-Cohen

3 Complexity in AMD One goal of Algorithmic Mechanism Design: Deal with complex allocation of goods settings Goods may not be homogenous Valuations and constraints may be complex Eg spectrum auctions, cloud computing, ad auctions, 3 Eden et al EC'17 Inbal Talgam-Cohen

4 Revenue maximization Revenue less understood than welfare (even for welfare, some computational issues persist) 4 Eden et al EC'17 Inbal Talgam-Cohen

5 Revenue maximization Revenue less understood than welfare (even for welfare, some computational issues persist) Optimal truthful mechanism known only for handful of complex settings (eg additive buyer with 2 items, 6 uniform iid items [Giannakopolous- Koutsoupias 14, 15]) 5 Eden et al EC'17 Inbal Talgam-Cohen

6 Revenue maximization Revenue less understood than welfare (even for welfare, some computational issues persist) Optimal truthful mechanism known only for handful of complex settings (eg additive buyer with 2 items, 6 uniform iid items [Giannakopolous- Koutsoupias 14, 15]) Common CS solution for complexity: approximation [Hart-Nisan 12, 13, Li-Yao 13, Babioff-et-al 14, Rubinstein- Weinberg 15, Chawla-Miller 16, ] 6 Eden et al EC'17 Inbal Talgam-Cohen

7 Revenue maximization Revenue less understood than welfare (even for welfare, some computational issues persist) Optimal truthful mechanism known only for handful of complex settings (eg additive buyer with 2 items, 6 uniform iid items [Giannakopolous- Koutsoupias 14, 15]) Common CS solution for complexity: approximation [Hart-Nisan 12, 13, Li-Yao 13, Babioff-et-al 14, Rubinstein- Weinberg 15, Chawla-Miller 16, ] Resource augmentation 7 Eden et al EC'17 Inbal Talgam-Cohen

8 Single item welfare maximization Run a 2nd price auction simple, maximizes welfare pointwise (VCG mechanism) v 1 v 2 v n 8 Eden et al EC'17 Inbal Talgam-Cohen

9 Single item welfare maximization Run a 2nd price auction simple, maximizes welfare pointwise (VCG mechanism) v 1 p = v 2 v n 9 Eden et al EC'17 Inbal Talgam-Cohen

10 Single item revenue maximization Single buyer: select price that maximizes p 1 F p ( monopoly price ) v 1 F Price = p 10 Eden et al EC'17 Inbal Talgam-Cohen

11 Single item revenue maximization Single buyer: select price that maximizes p 1 F p ( monopoly price ) Multiple iid buyers: run 2 nd price auction with reserve price p (same p) v 1 F v 2 F Price p (Myerson s auction) v n F 11 Eden et al EC'17 Inbal Talgam-Cohen

12 Single item revenue maximization Single buyer: select price that maximizes p 1 F p ( monopoly price ) Multiple iid buyers: run 2 nd price auction with reserve price p (same p) Assuming regularity v 1 F v 2 F Price p (Myerson s auction) v n F 12 Eden et al EC'17 Inbal Talgam-Cohen

13 Single item revenue maximization Single buyer: select price that maximizes p 1 F p ( monopoly price ) Multiple iid buyers: run 2 nd price auction with reserve price p (same p) v 1 F v 2 F Price p (Myerson s auction) v n F Requires prior knowledge to determine the reserve 13 Eden et al EC'17 Inbal Talgam-Cohen

14 Bulow-Klemperer theorem Thm Expected revenue of the 2 nd price auction with n+1 bidders Expected revenue of the optimal auction with n bidders 14 Eden et al EC'17 Inbal Talgam-Cohen

15 Bulow-Klemperer theorem Thm Expected revenue of the 2 nd price auction with n+1 bidders Expected revenue of the optimal auction with n bidders Robust! No need to learn the distribution No need to change mechanism if the distribution changes The statistics of the data shifts rapidly [Google] Simple! Hardly anything matters more [Milgrom 04] 15 Eden et al EC'17 Inbal Talgam-Cohen

16 Multidimensional settings F 16 Eden et al EC'17 Inbal Talgam-Cohen

17 Multidimensional settings F 1 F 2 F 3 17 Eden et al EC'17 Inbal Talgam-Cohen

18 Multidimensional settings Bidders values are sampled iid from a product distribution over items F 1 F 2 F 3 18 Eden et al EC'17 Inbal Talgam-Cohen

19 Multidimensional settings Bidders values are sampled iid from a product distribution over items F 1 F 2 F 3 Additive: v(, )=v( ) )+v( ) 19 Eden et al EC'17 Inbal Talgam-Cohen

20 Multidimensional settings Revenue maximization is not well understood: Optimal mechanism might necessitate randomization Non-monotone Computationally intractable Only recently, simple approximately optimal mechanisms were devised F 1 F 2 F 3 20 Eden et al EC'17 Inbal Talgam-Cohen

21 Multidimensional settings Either run a randomized, F 1 F 2 F 3 21 Eden et al EC'17 Inbal Talgam-Cohen

22 Multidimensional settings Either run a randomized, hard to compute, F 1 F 2 F 3 22 Eden et al EC'17 Inbal Talgam-Cohen

23 Multidimensional settings Either run a randomized, hard to compute, with infinitely many options mechanism, F 1 F 2 F 3 23 Eden et al EC'17 Inbal Talgam-Cohen

24 Multidimensional settings Either run a randomized, hard to compute, with infinitely many options mechanism, which depends heavily on the distributions F 1 F 2 F 3 24 Eden et al EC'17 Inbal Talgam-Cohen

25 Multidimensional settings Either run a randomized, hard to compute, with infinitely many options mechanism, which depends heavily on the distributions F 1 F 2 F 3 Or add more bidders 25 Eden et al EC'17 Inbal Talgam-Cohen

26 OUR RESULTS 26 Eden et al EC'17 Inbal Talgam-Cohen

27 Multidimensional B-K theorems Competition complexity: Fix an environment with n iid bidders What is x such that the revenue of VCG with n + x bidders is OPT with n bidders 27 Eden et al EC'17 Inbal Talgam-Cohen

28 Multidimensional B-K theorems Competition complexity: Fix an environment with n iid bidders What is x such that the revenue of VCG with n + x bidders is OPT with n bidders Bulow-Klemperer Thm The competition complexity of a single item auction is 1 28 Eden et al EC'17 Inbal Talgam-Cohen

29 Multidimensional B-K theorems Competition complexity: Fix an environment with n iid bidders What is x such that the revenue of VCG with n + x bidders is OPT with n bidders Bulow-Klemperer Thm The competition complexity of a single item auction is 1 Thm [BK] The competition complexity of a single item with m copies is m 29 Eden et al EC'17 Inbal Talgam-Cohen

30 Multidimensional B-K theorems Competition complexity: Fix an environment with n iid bidders What is x such that the revenue of VCG with n + x bidders is OPT with n bidders Thm [EFFTW] The competition complexity of n additive bidders drawn from a product distribution over m items is n + 2(m 1) 30 Eden et al EC'17 Inbal Talgam-Cohen

31 Multidimensional B-K theorems Competition complexity: Fix an environment with n iid bidders What is x such that the revenue of VCG with n + x bidders is OPT with n bidders Thm [EFFTW] Let C be the competition complexity of n additive bidders over m items The competition complexity of n additive bidders with identical downward closed constraints over m items is C + m 1 31 Eden et al EC'17 Inbal Talgam-Cohen

32 Multidimensional B-K theorems Competition complexity: Fix an environment with n iid bidders What is x such that the revenue of VCG with n + x bidders is OPT with n bidders Thm [EFFTW] Let C be the competition complexity of n additive bidders over m items The competition complexity of n additive bidders with randomly drawn downward closed constraints over m items is C + 2(m 1) 32 Eden et al EC'17 Inbal Talgam-Cohen

33 Additive with constraints Constraints = set system over the items Specifies which item sets are feasible Bidder s value for an item set = her value for best feasible subset If all sets are feasible, bidder is additive 33 Eden et al EC'17 Inbal Talgam-Cohen

34 Example of constraints No constraints Total value = $21 $6 $5 $10 34 Eden et al EC'17 Inbal Talgam-Cohen

35 Example of constraints Example of matroid constraints: Only sets of size k = 1 are feasible Total value = $21 $10 Substitutes $6 $5 $10 35 Eden et al EC'17 Inbal Talgam-Cohen

36 Example of constraints Example of downward closed constraints: Sets of size 1 and { } are feasible Total value = $16 $10 Substitutes $6 $5 Complements $10 36 Eden et al EC'17 Inbal Talgam-Cohen

37 Complements in what sense? No complements = gross substitutes: Ԧp Ԧq item prices S in demand( Ԧp) if maximizes utility v i S p(s) S in demand( Ԧp), there is T in demand( Ԧq) with every item in S whose price didn t increase $ $ S $ T $ 37 Eden et al EC'17 Inbal Talgam-Cohen

38 Complements in what sense? No complements = gross substitutes: Ԧp Ԧq item prices S in demand( Ԧp) if maximizes utility v i S p(s) S in demand( Ԧp), there is T in demand( Ԧq) with every item in S whose price didn t increase T 5 6 $ S 10 Ԧp = (5, ε, ε) 38 Eden et al EC'17 Inbal Talgam-Cohen

39 Competition complexity summary Upper bound Valuation Additive Additive st identical downward closed constraints Additive st random downward closed constraints Additive st identical matroid constraints n + 2 m 1 n + 3 m 1 n + 4 m 1 n + 2 m 1 + ρ Lower bounds of Ω n log n + 1 for additive bidders and Ω m for unit demand m bidders are due to ongoing work by [Feldman-Friedler-Rubinstein] and to [Bulow-Klemperer 96]

40 Related work Multidimensional B-K theorems [Roughgarden T Yan 12]: for unit demand bidders, revenue of VCG with m extra bidders revenue of the optimal deterministic DSIC mechanism [Feldman Friedler Rubinstein ongoing]: tradeoffs between enhanced competition and revenue Prior-independent multidimensional mechanisms [Devanur Hartline Karlin Nguyen 11]: unit demand bidders [Roughgarden T Yan 12]: unit demand bidders [Goldner Karlin 16]: additive bidders Sample complexity [Morgenstern Roughgarden 16]: how many samples needed to approximate the optimal mechanism?

41 MULTIDIMENSIONAL B-K THEOREM PROOF SKETCH 41 Eden et al EC'17 Inbal Talgam-Cohen

42 Bulow-Klemperer theorem Thm Revenue of the 2 nd price auction with n+1 bidders Revenue of the optimal auction with n bidders 42 Eden et al EC'17 Inbal Talgam-Cohen

43 Bulow-Klemperer theorem Thm Revenue of the 2 nd price auction with n+1 bidders Revenue of the optimal auction with n bidders Proof (in 3 steps of [Kirkegaard 06]) I Upper-bound the optimal revenue 43 Eden et al EC'17 Inbal Talgam-Cohen

44 Bulow-Klemperer theorem Thm Revenue of the 2 nd price auction with n+1 bidders Revenue of the optimal auction with n bidders Proof (in 3 steps of [Kirkegaard 06]) I Upper-bound the optimal revenue II Find an auction A with more bidders and revenue the upper bound 44 Eden et al EC'17 Inbal Talgam-Cohen

45 Bulow-Klemperer theorem Thm Revenue of the 2 nd price auction with n+1 bidders Revenue of the optimal auction with n bidders Proof (in 3 steps of [Kirkegaard 06]) I Upper-bound the optimal revenue II Find an auction A with more bidders and revenue the upper bound III Show that the 2 nd price auction beats A 45 Eden et al EC'17 Inbal Talgam-Cohen

46 Proof: Step I Upper-bound the optimal revenue Myerson s optimal mechanism v 1 F Price p v 2 F v n F 46

47 Proof: Step II Find an auction A with more bidders and revenue the upper bound v 1 F v 2 F v n F 47 v n+1 F

48 Proof: Step II Find an auction A with more bidders and revenue the upper bound Run Myerson s mechanism on n bidders v 1 F v 2 F v n F 48 v n+1 F

49 Proof: Step II Find an auction A with more bidders and revenue the upper bound Run Myerson s mechanism on n bidders v 1 F v 2 F 49 If Myerson does not allocate, give item to the additional bidder v n F v n+1 F

50 Proof: Step III Show that the 2 nd price auction beats A v 1 F Observation 2 nd price auction is the optimal mechanism out of the mechanisms that always sell 50 v 2 F v n F v n+1 F

51 Competition complexity of a single additive bidder Plan: Follow the 3 steps of the B-K proof I Upper-bound the optimal revenue II Find an auction A with more bidders and revenue the upper bound III Show that VCG beats A 51 Eden et al EC'17 Inbal Talgam-Cohen

52 Competition complexity of a single additive bidder and iid items Plan: Follow the 3 steps of the B-K proof I Upper-bound the optimal revenue II Find an auction A with more bidders and revenue the upper bound III Show that VCG beats A 52 Eden et al EC'17 Inbal Talgam-Cohen

53 I Upper-bound the optimal revenue Single additive bidder and iid items v 1 F v 2 F v m F 53 Eden et al EC'17 Inbal Talgam-Cohen

54 I Upper-bound the optimal revenue Use the duality framework from [Cai Devanur Weinberg 16] OPT E v F m j φ + v j 1 j v j >v j + v j 1 j v j <v j 54 φ v = v 1 F v f(v) is the virtual valuation function

55 I Upper-bound the optimal revenue Use the duality framework from [Cai Devanur Weinberg 16] OPT E v F m j φ + v j 1 j v j >v j + v j 1 j v j <v j Distribution appears in proof only! 55 φ v = v 1 F v f(v) is the virtual valuation function

56 I Upper-bound the optimal revenue Use the duality framework from [Cai Devanur Weinberg 16] OPT E v F m j φ + v j 1 j v j >v j + v j 1 j v j <v j Take item j s virtual value if it s the most attractive item 56 φ v = v 1 F v f(v) is the virtual valuation function

57 I Upper-bound the optimal revenue Use the duality framework from [Cai Devanur Weinberg 16] OPT E v F m j φ + v j 1 j v j >v j + v j 1 j v j <v j Take item j s value if there s a more attractive item 57 φ v = v 1 F v f(v) is the virtual valuation function

58 II Find an auction A with more bidders and revenue upper bound 58

59 II Find an auction A with m bidders and revenue upper bound 59

60 II Find an auction A with m bidders and revenue upper bound VCG for additive bidders 2 nd price auction for each item separately Therefore, we devise a single parameter mechanism that covers item j s contribution to the benchmark E v F m φ + v j 1 j v j >v j + v j 1 j v j <v j 60

61 II Find an auction A j with m bidders and revenue upper bound for item j E v F m φ + v j 1 j v j >v j + v j 1 j v j <v j Run 2 nd price auction with lazy reserve price = φ 1 0 for agent j 0 for agents j j v j F v 1 F v m F Item j 61 Eden et al EC'17 Inbal Talgam-Cohen

62 II Find an auction A j with m bidders and revenue upper bound for item j E v F m φ + v j 1 j v j >v j + v j 1 j v j <v j Case I: v j > v j for all j : j wins if his virtual value is non-negative Expected revenue = Expected virtual value [Myerson 81] v j F v 1 F v m F Item j 62 Eden et al EC'17 Inbal Talgam-Cohen

63 II Find an auction A j with m bidders and revenue upper bound for item j E v F m φ + v j 1 j v j >v j + v j 1 j v j <v j Case II: v j < v j for some j : The second price is at least the value of agent j v j F v 1 F v m F Item j 63 Eden et al EC'17 Inbal Talgam-Cohen

64 III Show that VCG beats A 64 Eden et al EC'17 Inbal Talgam-Cohen

65 III Show that 2 nd price beats A(j) 65 Eden et al EC'17 Inbal Talgam-Cohen

66 III Show that 2 nd price beats A(j) A(j) with m bidders Myerson with m bidders 2nd price with m + 1 bidders 66 Eden et al EC'17 Inbal Talgam-Cohen

67 III Show that 2 nd price beats A(j) A(j) with m bidders Myerson with m bidders 2nd price with m + 1 bidders The competition complexity of a single additive bidder and m iid items is m FF 67 Eden et al EC'17 Inbal Talgam-Cohen

68 Going beyond iid items Single additive bidder and iid items v 1 F 1 v 2 F 2 v m F m 68 Eden et al EC'17 Inbal Talgam-Cohen

69 Going beyond iid items E v 1 F 1 v 2 F 2 v m F m φ j + v j 1 j v j >v j + v j 1 j v j < v j Run 2 nd price auction with lazy reserve price = φ 1 0 for agent j 0 for agents j j v j F j v 1 F j v m F j Item j 69 Eden et al EC'17 Inbal Talgam-Cohen

70 Going beyond iid items E v 1 F 1 v 2 F 2 v m F m φ j + v j 1 j v j >v j + v j 1 j v j < v j Run 2 nd price auction with lazy reserve price = φ 1 0 for agent j 0 for agents j j Cannot couple the event bidder j wins and item j has the highest value v j F j v 1 F j v m F j Item j 70 Eden et al EC'17 Inbal Talgam-Cohen

71 Use a different benchmark E v 1 F 1 v 2 F 2 v m F m φ j + v j 1 j F j (v j )>F j (v j ) + v j 1 j F j (v j )<F j (v j ) v j F j Item j v 1 F j v m F j 71 Eden et al EC'17 Inbal Talgam-Cohen

72 Use a different benchmark E v 1 F 1 v 2 F 2 v m F m φ j + v j 1 j F j (v j )>F j (v j ) + v j 1 j F j (v j )<F j (v j ) v j F j Item j v 1 F j v m F j The competition complexity of a single additive bidder and m items is m

73 Going beyond a single bidder Step I: Benchmark more involved Step II: Devise a more complex single parameter auction A(j) (involves a max) Proving A(j) is greater than item j s contribution to the benchmark is more involved and requires subtle coupling and probabilistic claims BB 73 Eden et al EC'17 Inbal Talgam-Cohen

74 EXTENSION TO CONSTRAINTS 74 Eden et al EC'17 Inbal Talgam-Cohen

75 Recall Example of downward closed constraints: Sets of size 1 and { } are feasible Total value = $16 Substitutes $6 $5 Complements $10 75 Eden et al EC'17 Inbal Talgam-Cohen

76 Extension to downward closed constraints OPT Add Add n VCG n+c Competition complexity C 76 Eden et al EC'17 Inbal Talgam-Cohen

77 Extension to downward closed constraints OPT n DC Larger outcome space OPT Add Add n VCG n+c Competition complexity C 77 Eden et al EC'17 Inbal Talgam-Cohen

78 Extension to downward closed constraints OPT n DC Larger outcome space OPT Add Add n VCG n+c Competition complexity C DC VCG n+c+m 1 78 Eden et al EC'17 Inbal Talgam-Cohen

79 Extension to downward closed constraints OPT n DC Larger outcome space OPT Add Add n VCG n+c Competition complexity C DC VCG n+c+m 1 The competition complexity of n additive bidders with identical downward closed constraints over m items is C + m 1

80 Extension to downward closed constraints Main technical challenge OPT n DC Larger outcome space OPT Add Add n VCG n+c Competition complexity C DC VCG n+c+m 1 The competition complexity of n additive bidders with identical downward closed constraints over m items is C + m 1

81 Claim VCG revenue from selling m items to X = n + C additive bidders whose values are iid draws from F VCG revenue from selling them to X + m 1 bidders with iid values drawn from F, whose valuations are additive st identical downward-closed constraints 81 Eden et al EC'17 Inbal Talgam-Cohen

82 Add DC VCG X VCG X+m 1 VCG for additive bidders 2 nd price auction for each item separately Therefore, the revenue from item j in VCG X Add = 2 nd highest value out of X iid samples from F j 82 Eden et al EC'17 Inbal Talgam-Cohen

83 Add DC VCG X VCG X+m 1 83

84 Add DC VCG X VCG X+m 1 84

85 Add DC VCG X VCG X+m 1 5 F 7 F 2 F 85

86 Add DC VCG X VCG X+m 1 5 F 7 F 2 F

87 Add DC VCG X VCG X+m 1 5 F 7 F 2 F

88 Add DC VCG X VCG X+m 1 5 F 7 F 2 F Claim Revenue for item j in VCGDC X+m value of the highest unallocated bidder for item j 3 2 1

89 Add DC VCG X VCG X+m 1 5 F 7 F 2 F

90 Add DC VCG X VCG X+m 1 5 F 7 F 2 F

91 Add DC VCG X VCG X+m 1 5 F 7 F 2 F

92 Add DC VCG X VCG X+m 1 5 F 7 F 2 F Externality at least

93 Add DC VCG X VCG X+m 1 5 F 7 F 2 F

94 Add DC VCG X VCG X+m 1 5 F 7 F 2 F

95 Add DC VCG X VCG X+m 1 5 F 7 F 2 F

96 Add DC VCG X VCG X+m 1 5 F 7 F 2 F Externality at least

97 Add DC VCG X VCG X+m 1 97 Eden et al EC'17 Inbal Talgam-Cohen

98 Add DC VCG X VCG X+m 1 2 nd highest Highest value VCG X Add (j) = of X samples from F j of unallocated bidder for j DC VCG X+m 1 (j) 98 Eden et al EC'17 Inbal Talgam-Cohen

99 Add DC VCG X VCG X+m 1 2 nd highest Highest value VCG X Add (j) = of X samples from F j of unallocated bidder for j DC VCG X+m 1 (j) 99 Eden et al EC'17 Inbal Talgam-Cohen

100 Add DC VCG X VCG X+m 1 2 nd highest Highest value VCG X Add (j) = of X samples from F j of unallocated bidder for j DC VCG X+m 1 (j) DC Identify X bidders in VCG X+m 1 before sampling their value for item j out of which at most one will be allocated anything 100 Eden et al EC'17 Inbal Talgam-Cohen

101 Add DC VCG X VCG X+m m j 101

102 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) m j 102

103 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 1 Sample valuations for all items but j m j 103

104 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 2 Compute an optimal allocation without item j j 1 m

105 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 2 Compute an optimal allocation without item j Set A of allocated bidders Set ҧ A of unallocated bidders j 1 m

106 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 2 Compute an optimal allocation without item j Set A of allocated bidders Set ҧ A of unallocated bidders j m 7 If j is allocated to bidder in ҧ A in OPT, all other items are allocated as before

107 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Sample values for j for agents in A and compute the optimal allocation where j is allocated to a bidder in A j 1 m

108 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A j 1 m

109 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A Some items might be vacated due to feasibility j 1 m

110 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A Some items might be snatched from other agents j 1 m

111 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A Continue with this process j 1 2 m

112 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A Continue with this process j 1 2 m

113 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A There are A items allocated to agents in A j 2 m

114 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A There are A items allocated to agents in A Map each agent who s item was snatched to the snatched item j 2 m

115 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A There are A items allocated to agents in A j Map each agent who s item was snatched to the snatched item Map each agent who took a vacated item to the item 2 m

116 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A There are A items allocated to agents in A j Map each agent who s item was snatched to the snatched item Map each agent who took a vacated item to the item Every agent who wasn t snatched and didn t take an item has the same allocation 2 m

117 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A There are A items allocated to agents in A j m 2 7 m A allocated

118 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) 3 Compute OPT j A There are A items allocated to agents in A j m Aҧ m A = X + m 1 A m A = X 1 unallocated m A allocated 1 4 5

119 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) j m Aҧ m A = X + m 1 A m A = X 1 unallocated m A allocated 1 4 5

120 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) X bidders whose values for j are iid samples from F j DC At most one is allocated by VCG X+m 1 j 2 m Aҧ m A = X + m 1 A m A = X 1 unallocated m A allocated

121 Add DC VCG X VCG X+m 1 (Assume wlog unique optimal allocation) X bidders whose values for j are iid samples from F j DC At most one is allocated by VCG X+m 1 VCG X Add (j) = 2 nd highest of X samples from F j Highest value of unallocated bidder for j DC VCG X+m 1 (j) 121

122 Extension to downward closed constraints Rev n DC Larger outcome space Rev Add Add n VCG n+c Competition complexity C DC VCG n+c+m 1 The competition complexity of n additive bidders st identical downward closed constraints over m items is 122 C + m 1

123 Extension to downward closed constraints Rev n DC Larger outcome space Rev Add Add n VCG n+c Competition complexity C DC VCG n+c+m 1 Proved! The competition complexity of n additive bidders st identical downward closed constraints over m items is 123 C + m 1

124 A note on tractability VCG is not computationally tractable for general downward closed constraints However: VCG is tractable for matroid constraints Competition complexity is meaningful in its own right Can apply our techniques with maximal-inrange VCG by restricting outcomes to matchings 124 Eden et al EC'17 Inbal Talgam-Cohen

125 Further extensions (preliminary) 1 From competition complexity to approximation In large markets (n m), 2 nd price auction (no extra agents) 1 -approximates OPT 2 2 Non-iid bidders 125 Eden et al EC'17 Inbal Talgam-Cohen

126 Summary Major open problem: Revenue maximization for m items B-K approach: Add competing bidders and maximize welfare Results in: First robust simple mechanisms with provably high revenue for many complex settings Techniques: Bayesian analysis, combinatorial arguments 126 Eden et al EC'17 Inbal Talgam-Cohen

127 Open questions Tighter bounds and tradeoffs Settings with constant competition complexity Partial data on distributions, or large markets Different duality based upper bound? More general settings Beyond downward closed constraints Irregular distributions Affiliation [Bulow-Klemperer 96] Beyond VCG Posted-price mechanisms 127 Eden et al EC'17 Inbal Talgam-Cohen