The Competition Complexity of Auctions: Bulow-Klemperer Results for - PowerPoint PPT Presentation

Multidimensional B-K theorems Competition complexity: Fix an environment with 𝑜 i.i.d. bidders. What is 𝒚 such that the revenue of VCG with 𝒐 + 𝒚 bidders is ≥ OPT with 𝒐 bidders. Thm. [EFFTW] Let 𝑫 be the competition complexity of 𝒐 additive bidders over 𝑛 items. The competition complexity of 𝒐 additive bidders with identical downward closed constraints over 𝑛 items is ≤ 𝑫 + 𝒏 − 𝟐 . Competition Complexity of Auctions 31 Eden et al. EC'17 Inbal Talgam-Cohen

Multidimensional B-K theorems Competition complexity: Fix an environment with 𝑜 i.i.d. bidders. What is 𝒚 such that the revenue of VCG with 𝒐 + 𝒚 bidders is ≥ OPT with 𝒐 bidders. Thm. [EFFTW] Let 𝑫 be the competition complexity of 𝒐 additive bidders over 𝑛 items. The competition complexity of 𝒐 additive bidders with randomly drawn downward closed constraints over 𝑛 items is ≤ 𝑫 + 𝟑(𝒏 − 𝟐) . Competition Complexity of Auctions 32 Eden et al. EC'17 Inbal Talgam-Cohen

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 Competition Complexity of Auctions 33 Eden et al. EC'17 Inbal Talgam-Cohen

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

Example of constraints • Example of “ matroid ” constraints: Only sets of size 𝑙 = 1 are feasible Total value = $21 $10 Substitutes $6 $5 $10 Competition Complexity of Auctions 35 Eden et al. EC'17 Inbal Talgam-Cohen

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

Complements in what sense? • No complements = gross substitutes: – Ԧ 𝑞 ≤ Ԧ 𝑟 item prices – 𝑇 in demand( Ԧ 𝑞) if maximizes utility 𝑤 𝑗 𝑇 − 𝑞(𝑇) – ∀𝑇 in demand( Ԧ 𝑞) , there is 𝑈 in demand(Ԧ 𝑟) with every item in 𝑇 whose price didn ’ t increase 𝑻 $ $ $ $ 𝑼 Competition Complexity of Auctions 37 Eden et al. EC'17 Inbal Talgam-Cohen

Complements in what sense? • No complements = gross substitutes: – Ԧ 𝑞 ≤ Ԧ 𝑟 item prices – 𝑇 in demand( Ԧ 𝑞) if maximizes utility 𝑤 𝑗 𝑇 − 𝑞(𝑇) – ∀𝑇 in demand( Ԧ 𝑞) , there is 𝑈 in demand(Ԧ 𝑟) with every item in 𝑇 whose price didn ’ t increase $ 𝑻 10 𝑞 Ԧ = (5, 𝜗, 𝜗) 5 6 𝑼 Competition Complexity of Auctions 38 Eden et al. EC'17 Inbal Talgam-Cohen

Competition complexity – summary Upper bound Valuation 𝑜 + 2 𝑛 − 1 Additive Additive s.t. identical downward 𝑜 + 3 𝑛 − 1 closed constraints Additive s.t. random downward closed 𝑜 + 4 𝑛 − 1 constraints Additive s.t. identical matroid 𝑜 + 2 𝑛 − 1 + 𝜍 constraints 𝑜 Lower bounds of Ω 𝑜 ⋅ log 𝑛 + 1 for additive bidders and Ω 𝑛 for unit demand bidders are due to ongoing work by [Feldman-Friedler-Rubinstein] and to [Bulow-Klemperer ’ 96]

Related work Multidimensional B-K theorems [Roughgarden T. Yan ‘ 12] : for unit demand bidders, revenue of VCG with 𝒏 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?

MULTIDIMENSIONAL B-K THEOREM PROOF SKETCH Competition Complexity of Auctions 41 Eden et al. EC'17 Inbal Talgam-Cohen

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

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. Competition Complexity of Auctions 43 Eden et al. EC'17 Inbal Talgam-Cohen

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 𝐵 with more bidders and revenue ≥ the upper bound. Competition Complexity of Auctions 44 Eden et al. EC'17 Inbal Talgam-Cohen

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 𝐵 with more bidders and revenue ≥ the upper bound. III. Show that the 2 nd price auction “ beats ” 𝐵 . Competition Complexity of Auctions 45 Eden et al. EC'17 Inbal Talgam-Cohen

Proof: Myerson ’ s optimal mechanism Step I. Upper-bound Price ≥ 𝑞 the optimal revenue. 𝑤 1 ∼ 𝐺 ≥ 𝑤 2 ∼ 𝐺 ≥ . . . . . . ≥ 𝑤 𝑜 ∼ 𝐺 46

Proof: Step II. Find an auction 𝐵 with more bidders and 𝑤 1 ∼ 𝐺 revenue ≥ the upper bound. 𝑤 2 ∼ 𝐺 . . . . . . 𝑤 𝑜 ∼ 𝐺 𝑤 𝑜+1 ∼ 𝐺 47

Proof: Step II. Find an auction 𝐵 with more bidders and 𝑤 1 ∼ 𝐺 revenue ≥ the upper bound. 𝑤 2 ∼ 𝐺 Run Myerson ’ s mechanism on . . . . 𝒐 bidders . . 𝑤 𝑜 ∼ 𝐺 𝑤 𝑜+1 ∼ 𝐺 48

Proof: Step II. Find an auction 𝐵 with more bidders and 𝑤 1 ∼ 𝐺 revenue ≥ the upper bound. 𝑤 2 ∼ 𝐺 Run Myerson ’ s mechanism on . . . . 𝒐 bidders . . 𝑤 𝑜 ∼ 𝐺 If Myerson does not allocate, give item to the additional 𝑤 𝑜+1 ∼ 𝐺 bidder 49

Proof: Step III. Show that the 2 nd price auction “ beats ” 𝐵 . 𝑤 1 ∼ 𝐺 Observation. 2 nd price 𝑤 2 ∼ 𝐺 auction is the optimal . . mechanism out of the . . . . mechanisms that always 𝑤 𝑜 ∼ 𝐺 sell. 𝑤 𝑜+1 ∼ 𝐺 50

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 𝐵 with more bidders and revenue ≥ the upper bound. III. Show that VCG “ beats ” 𝐵 . Competition Complexity of Auctions 51 Eden et al. EC'17 Inbal Talgam-Cohen

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

I. Upper-bound the optimal revenue • Single additive bidder and i.i.d. items 𝑤 1 ∼ 𝐺 𝑤 2 ∼ 𝐺 . . . . . . 𝑤 𝑛 ∼ 𝐺 Competition Complexity of Auctions 53 Eden et al. EC'17 Inbal Talgam-Cohen

I. Upper-bound the optimal revenue Use the duality framework from [Cai Devanur Weinberg ‘ 16] . OPT ≤ 𝜒 + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 <𝑤 𝑘′ E 𝑤∼𝐺 𝑛 ෍ 𝑘 1−𝐺 𝑤 𝜒 𝑤 = 𝑤 − 𝑔(𝑤) is the virtual valuation function. 54

I. Upper-bound the optimal revenue Use the duality framework from [Cai Devanur Weinberg ‘ 16] . OPT ≤ 𝜒 + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 <𝑤 𝑘′ E 𝑤∼𝐺 𝑛 ෍ 𝑘 Distribution appears in proof only! 1−𝐺 𝑤 𝜒 𝑤 = 𝑤 − 𝑔(𝑤) is the virtual valuation function. 55

I. Upper-bound the optimal revenue Use the duality framework from [Cai Devanur Weinberg ‘ 16] . OPT ≤ 𝜒 + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 <𝑤 𝑘′ E 𝑤∼𝐺 𝑛 ෍ 𝑘 Take item 𝑘 ’ s virtual value if it ’ s the most attractive item 1−𝐺 𝑤 𝜒 𝑤 = 𝑤 − 𝑔(𝑤) is the virtual valuation function. 56

I. Upper-bound the optimal revenue Use the duality framework from [Cai Devanur Weinberg ‘ 16] . OPT ≤ 𝜒 + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 <𝑤 𝑘′ E 𝑤∼𝐺 𝑛 ෍ 𝑘 Take item 𝑘 ’ s value if there ’ s a more attractive item 1−𝐺 𝑤 𝜒 𝑤 = 𝑤 − 𝑔(𝑤) is the virtual valuation function. 57

II. Find an auction 𝐵 with more bidders and revenue ≥ upper bound 58

II. Find an auction 𝐵 with 𝑛 bidders and revenue ≥ upper bound 59

II. Find an auction 𝐵 with 𝑛 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 𝒌 ’ s contribution to the benchmark. E 𝑤∼𝐺 𝑛 𝜒 + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘 ′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 <𝑤 𝑘 ′ 60

II. Find an auction 𝐵 𝑘 with 𝑛 bidders and revenue ≥ upper bound for item 𝑘 E 𝑤∼𝐺 𝑛 𝜒 + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 <𝑤 𝑘′ Run 2 nd price auction Item 𝑘 𝑤 𝑘 ∼ 𝐺 with “ lazy ” reserve price = 𝜒 −1 0 for agent 𝑘 𝑤 1 ∼ 𝐺 0 for agents 𝑘 ′ ≠ 𝑘 . . . . . . 𝑤 𝑛 ∼ 𝐺 Competition Complexity of Auctions 61 Eden et al. EC'17 Inbal Talgam-Cohen

II. Find an auction 𝐵 𝑘 with 𝑛 bidders and revenue ≥ upper bound for item 𝑘 E 𝑤∼𝐺 𝑛 𝜒 + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 <𝑤 𝑘′ Case I: 𝑤 𝑘 > 𝑤 𝑘 ′ for all 𝑘 ′ : Item 𝑘 𝑤 𝑘 ∼ 𝐺 𝑘 wins if his virtual value is non-negative. 𝑤 1 ∼ 𝐺 Expected revenue = . . . . . . Expected virtual value 𝑤 𝑛 ∼ 𝐺 [Myerson ’ 81] Competition Complexity of Auctions 62 Eden et al. EC'17 Inbal Talgam-Cohen

II. Find an auction 𝐵 𝑘 with 𝑛 bidders and revenue ≥ upper bound for item 𝑘 E 𝑤∼𝐺 𝑛 𝜒 + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 <𝑤 𝑘′ Case II : 𝑤 𝑘 < 𝑤 𝑘 ′ for some 𝑘 ′ : Item 𝑘 𝑤 𝑘 ∼ 𝐺 The second price is at least 𝑤 1 ∼ 𝐺 the value of agent 𝑘 . . . . . . . 𝑤 𝑛 ∼ 𝐺 Competition Complexity of Auctions 63 Eden et al. EC'17 Inbal Talgam-Cohen

III. Show that VCG “ beats ” 𝐵 Competition Complexity of Auctions 64 Eden et al. EC'17 Inbal Talgam-Cohen

III. Show that 2 nd price “ beats ” 𝐵(𝑘) Competition Complexity of Auctions 65 Eden et al. EC'17 Inbal Talgam-Cohen

III. Show that 2 nd price “ beats ” 𝐵(𝑘) ≤ 2 nd price with 𝑩(𝒌) with 𝒏 bidders ≤ Myerson with 𝒏 bidders 𝒏 + 𝟐 bidders Competition Complexity of Auctions 66 Eden et al. EC'17 Inbal Talgam-Cohen

III. Show that 2 nd price “ beats ” 𝐵(𝑘) ≤ 2 nd price with 𝑩(𝒌) with 𝒏 bidders ≤ Myerson with 𝒏 bidders 𝒏 + 𝟐 bidders  The competition complexity of a single additive bidder and 𝑛 i.i.d. items is ≤ 𝑛 . FF Competition Complexity of Auctions 67 Eden et al. EC'17 Inbal Talgam-Cohen

Going beyond i.i.d items • Single additive bidder and i.i.d. items 𝑤 1 ∼ 𝐺 1 𝑤 2 ∼ 𝐺 2 . . . . . . 𝑤 𝑛 ∼ 𝐺 𝑛 Competition Complexity of Auctions 68 Eden et al. EC'17 Inbal Talgam-Cohen

Going beyond i.i.d items + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 < 𝑤 𝑘′ E 𝑤 1 ∼𝐺 𝜒 𝑘 1 𝑤 2 ∼𝐺 2 … 𝑤 𝑛 ∼𝐺 𝑛 Item 𝑘 𝑤 𝑘 ∼ 𝐺 𝑘 Run 2 nd price auction 𝑤 1 ∼ 𝐺 with “ lazy ” reserve price = 𝑘 𝜒 −1 0 for agent 𝑘 . . . . . . 0 for agents 𝑘 ′ ≠ 𝑘 𝑤 𝑛 ∼ 𝐺 𝑘 Competition Complexity of Auctions 69 Eden et al. EC'17 Inbal Talgam-Cohen

Going beyond i.i.d items + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝑤 𝑘 >𝑤 𝑘′ + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝑤 𝑘 < 𝑤 𝑘′ E 𝑤 1 ∼𝐺 𝜒 𝑘 1 𝑤 2 ∼𝐺 2 … 𝑤 𝑛 ∼𝐺 𝑛 Run 2 nd price auction Item 𝑘 𝑤 𝑘 ∼ 𝐺 𝑘 with “ lazy ” reserve price = 𝜒 −1 0 for agent 𝑘 𝑤 1 ∼ 𝐺 𝑘 0 for agents 𝑘 ′ ≠ 𝑘 . . . . Cannot couple the event . . “ bidder 𝑘 wins ” and 𝑤 𝑛 ∼ 𝐺 𝑘 “ item 𝑘 has the highest value ” Competition Complexity of Auctions 70 Eden et al. EC'17 Inbal Talgam-Cohen

Use a different benchmark + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝐺 𝑘 (𝑤 𝑘 )>𝐺 𝑘′ (𝑤 𝑘′ ) + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝐺 𝑘 (𝑤 𝑘 )<𝐺 𝑘′ (𝑤 𝑘′ ) E 𝑤 1 ∼𝐺 𝜒 𝑘 1 𝑤 2 ∼𝐺 2 … 𝑤 𝑛 ∼𝐺 𝑛 Item 𝑘 𝑤 𝑘 ∼ 𝐺 𝑘 𝑤 1 ∼ 𝐺 𝑘 . . . . . . 𝑤 𝑛 ∼ 𝐺 𝑘 Competition Complexity of Auctions 71 Eden et al. EC'17 Inbal Talgam-Cohen

Use a different benchmark + 𝑤 𝑘 ⋅ 1 ∀𝑘 ′ 𝐺 𝑘 (𝑤 𝑘 )>𝐺 𝑘′ (𝑤 𝑘′ ) + 𝑤 𝑘 ⋅ 1 ∃𝑘 ′ 𝐺 𝑘 (𝑤 𝑘 )<𝐺 𝑘′ (𝑤 𝑘′ ) E 𝑤 1 ∼𝐺 𝜒 𝑘 1 𝑤 2 ∼𝐺 2 … 𝑤 𝑛 ∼𝐺 𝑛 Item 𝑘 𝑤 𝑘 ∼ 𝐺 𝑘 𝑤 1 ∼ 𝐺 𝑘 . . . . . . 𝑤 𝑛 ∼ 𝐺 𝑘  The competition complexity of a single additive bidder and 𝑛 items is ≤ 𝑛 .

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 Competition Complexity of Auctions 73 Eden et al. EC'17 Inbal Talgam-Cohen

EXTENSION TO CONSTRAINTS Competition Complexity of Auctions 74 Eden et al. EC'17 Inbal Talgam-Cohen

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

Extension to downward closed constraints Add ≤ VCG 𝑜+𝐷 Add OPT 𝑜 Competition complexity ≤ 𝐷 Competition Complexity of Auctions 76 Eden et al. EC'17 Inbal Talgam-Cohen

Extension to downward closed constraints DC ≤ Add ≤ VCG 𝑜+𝐷 Add OPT 𝑜 OPT 𝑜 Competition Larger complexity outcome ≤ 𝐷 space Competition Complexity of Auctions 77 Eden et al. EC'17 Inbal Talgam-Cohen

Extension to downward closed constraints DC ≤ Add ≤ VCG 𝑜+𝐷 DC Add OPT 𝑜 ≤ VCG 𝑜+𝐷+𝑛−1 OPT 𝑜 Competition Larger complexity outcome ≤ 𝐷 space Competition Complexity of Auctions 78 Eden et al. EC'17 Inbal Talgam-Cohen

Extension to downward closed constraints DC ≤ Add ≤ VCG 𝑜+𝐷 DC Add OPT 𝑜 ≤ VCG 𝑜+𝐷+𝑛−1 OPT 𝑜 Competition Larger complexity outcome ≤ 𝐷 space The competition complexity of 𝑜 additive bidders with identical downward closed constraints over 𝑛 items is ≤ 𝐷 + 𝑛 − 1 .

Extension to downward closed constraints Main technical challenge DC ≤ Add ≤ VCG 𝑜+𝐷 DC Add OPT 𝑜 ≤ VCG 𝑜+𝐷+𝑛−1 OPT 𝑜 Competition Larger complexity outcome ≤ 𝐷 space The competition complexity of 𝑜 additive bidders with identical downward closed constraints over 𝑛 items is ≤ 𝐷 + 𝑛 − 1 .

Claim. VCG revenue from selling 𝒏 items to 𝒀 = 𝒐 + 𝑫 additive bidders whose values are i.i.d. draws from 𝐺 ≤ VCG revenue from selling them to 𝒀 + 𝒏 − 𝟐 bidders with i.i.d. values drawn from 𝐺 , whose valuations are additive s.t. identical downward-closed constraints. Competition Complexity of Auctions 81 Eden et al. EC'17 Inbal Talgam-Cohen

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 VCG for additive bidders ≡ 2 nd price auction for each item separately. Add = Therefore, the revenue from item 𝒌 in VCG 𝑌 2 nd highest value out of 𝒀 i.i.d. samples from 𝑮 𝒌 . Competition Complexity of Auctions 82 Eden et al. EC'17 Inbal Talgam-Cohen

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 83

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 84

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 85

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 3 2 1 86

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 3 2 1 87

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 Claim. Revenue for item 𝒌 in DC 4 5 1 VCG 𝑌+𝑛−1 ≥ value of the highest unallocated bidder for 3 4 2 item 𝑘 . 3 2 1 88

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 3 2 1 89

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 3 2 1 90

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 3 2 1 91

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 Externality at least 9 3 2 1 92

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 3 2 1 93

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 3 2 1 94

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 3 2 1 95

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 5 ∼ 𝐺 7 ∼ 𝐺 2 ∼ 𝐺 3 6 4 4 5 1 3 4 2 Externality at least 2 3 2 1 96

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 Competition Complexity of Auctions 97 Eden et al. EC'17 Inbal Talgam-Cohen

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 2 nd highest Highest value Add (𝑘) = DC of 𝑌 samples of unallocated ≤ VCG 𝑌 VCG 𝑌+𝑛−1 (𝑘) from 𝐺 bidder for 𝑘 𝑘 Competition Complexity of Auctions 98 Eden et al. EC'17 Inbal Talgam-Cohen

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 2 nd highest Highest value Add (𝑘) = DC of 𝑌 samples ≤ of unallocated ≤ VCG 𝑌 VCG 𝑌+𝑛−1 (𝑘) from 𝐺 bidder for 𝑘 𝑘 Competition Complexity of Auctions 99 Eden et al. EC'17 Inbal Talgam-Cohen

Add ≤ VCG 𝑌+𝑛−1 DC VCG 𝑌 2 nd highest Highest value Add (𝑘) = DC of 𝑌 samples ≤ of unallocated ≤ VCG 𝑌 VCG 𝑌+𝑛−1 (𝑘) from 𝐺 bidder for 𝑘 𝑘 DC Identify 𝑌 bidders in VCG 𝑌+𝑛−1 before sampling their value for item 𝑘 out of which at most one will be allocated anything Competition Complexity of Auctions 100 Eden et al. EC'17 Inbal Talgam-Cohen