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

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

Oxford, Spring 2017

Alon Eden, Michal Feldman, Ophir Friedler @ Tel-Aviv University Inbal Talgam-Cohen, Marie Curie Postdoc @ Hebrew University Matt Weinberg @ Princeton *Based on slides by Alon Eden

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

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 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
• E.g. spectrum auctions, cloud computing, ad

auctions, …

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 3

Revenue maximization

• Revenue less understood than welfare

– (even for welfare, some computational issues persist)

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 4

Revenue maximization

• Revenue less understood than welfare

– (even for welfare, some computational issues persist)

• Optimal truthful mechanism known only for handful
• f complex settings (e.g. additive buyer with 2 items,

6 uniform i.i.d. items... [Giannakopolous- Koutsoupias’14,’15])

Competition Complexity of Auctions 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
• f complex settings (e.g. additive buyer with 2 items,

6 uniform i.i.d. 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, …]

Competition Complexity of Auctions 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
• f complex settings (e.g. additive buyer with 2 items,

6 uniform i.i.d. 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

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 7

Single item welfare maximization

Run a 2nd price auction – simple, maximizes welfare “pointwise”. (VCG mechanism)

𝑤1 𝑤2 𝑤𝑜

≥ ≥ ≥

. . . . . .

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 8

Single item welfare maximization

Run a 2nd price auction – simple, maximizes welfare “pointwise”. (VCG mechanism)

𝑤1 𝑞 = 𝑤2 𝑤𝑜

≥ ≥ ≥

. . . . . .

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 9

Single item revenue maximization

Single buyer: select price that maximizes 𝑞 ⋅ 1 − 𝐺 𝑞 (“monopoly price”).

𝑤1 ∼ 𝐺 Price = 𝑞

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 10

Single item revenue maximization

Single buyer: select price that maximizes 𝑞 ⋅ 1 − 𝐺 𝑞 (“monopoly price”). Multiple i.i.d. buyers: run 2nd price auction with reserve price 𝑞 (same 𝑞). (Myerson’s auction)

𝑤1 ∼ 𝐺 𝑤2 ∼ 𝐺 𝑤𝑜 ∼ 𝐺

≥ ≥ ≥

Price ≥ 𝑞

. . . . . .

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 11

Single item revenue maximization

Single buyer: select price that maximizes 𝑞 ⋅ 1 − 𝐺 𝑞 (“monopoly price”). Multiple i.i.d. buyers: run 2nd price auction with reserve price 𝑞 (same 𝑞). (Myerson’s auction)

𝑤1 ∼ 𝐺 𝑤2 ∼ 𝐺 𝑤𝑜 ∼ 𝐺

≥ ≥ ≥

Price ≥ 𝑞

. . . . . .

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 12

Assuming regularity

Single item revenue maximization

Single buyer: select price that maximizes 𝑞 ⋅ 1 − 𝐺 𝑞 (“monopoly price”). Multiple i.i.d. buyers: run 2nd price auction with reserve price 𝑞 (same 𝑞). (Myerson’s auction)

𝑤1 ∼ 𝐺 𝑤2 ∼ 𝐺 𝑤𝑜 ∼ 𝐺

≥ ≥ ≥

Price ≥ 𝑞

. . .

Requires prior knowledge to determine the reserve

. . .

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 13

Bulow-Klemperer theorem

• Thm. Expected revenue of the 2nd price auction

with n+1 bidders ≥ Expected revenue of the

• ptimal auction with n bidders.

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 14

Bulow-Klemperer theorem

• Thm. Expected revenue of the 2nd price auction

with n+1 bidders ≥ Expected revenue of the

• ptimal 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]

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 15

Multidimensional settings

𝐺

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 16

Multidimensional settings

𝐺

1

𝐺

2

𝐺

3

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

Multidimensional settings

𝐺

1

𝐺

2

𝐺

3

Bidders’ values are sampled i.i.d. from a product distribution

• ver items

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 18

Multidimensional settings

𝐺

1

𝐺

2

𝐺

3

Additive: 𝑤( , )=𝑤( ) )+𝑤( )

Bidders’ values are sampled i.i.d. from a product distribution

• ver items

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 19

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.

𝐺

1

𝐺

2

𝐺

3

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 20

Multidimensional settings

Either run a randomized,

𝐺

1

𝐺

2

𝐺

3

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 21

Multidimensional settings

Either run a randomized, hard to compute,

𝐺

1

𝐺

2

𝐺

3

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 22

Multidimensional settings

Either run a randomized, hard to compute, with infinitely many options mechanism,

𝐺

1

𝐺

2

𝐺

3

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 23

Multidimensional settings

Either run a randomized, hard to compute, with infinitely many options mechanism, which depends heavily on the distributions…

𝐺

1

𝐺

2

𝐺

3

Competition Complexity of Auctions 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… Or add more bidders.

𝐺

1

𝐺

2

𝐺

3

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 25

OUR RESULTS

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 26

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.

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 27

Multidimensional B-K theorems

Bulow-Klemperer Thm. The competition complexity

• f a single item auction is 1.

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.

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 28

Multidimensional B-K theorems

Bulow-Klemperer Thm. The competition complexity

• f a single item auction is 1.

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.

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 29

Multidimensional B-K theorems

• Thm. [BK] The competition complexity of a single

item with 𝒏 copies is 𝒏.

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] The competition complexity of 𝒐

additive bidders drawn from a product distribution

• ver 𝒏 items is ≤ 𝒐 + 𝟑(𝒏 − 𝟐).

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 30

Multidimensional B-K theorems

• 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: 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.

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 31

Multidimensional B-K theorems

• 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: 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.

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 32

Multidimensional B-K theorems

Additive with constraints

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

• 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

Example of constraints

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 34

$6 $10 $21 $5 Total value =

• No constraints

Example of constraints

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 35

$6 $10 $21 $5 $10 Substitutes Total value =

• Example of “matroid” constraints: Only sets of

size 𝑙 = 1 are feasible

$10 $16

Example of constraints

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 36

$6 $10 $5 Substitutes Complements Total value =

• Example of “downward closed” constraints:

Sets of size 1 and { } are feasible

Complements in what sense?

• No complements = gross substitutes:

– Ԧ 𝑞 ≤ Ԧ 𝑟 item prices – 𝑇 in demand( Ԧ 𝑞) if maximizes utility 𝑤𝑗 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

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 37

$ $ $ $

𝑻 𝑼

Complements in what sense?

• No complements = gross substitutes:

– Ԧ 𝑞 ≤ Ԧ 𝑟 item prices – 𝑇 in demand( Ԧ 𝑞) if maximizes utility 𝑤𝑗 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

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 38

$ 𝑻

𝑼 5 6 10 Ԧ 𝑞 = (5, 𝜗, 𝜗)

Competition complexity – summary

Upper bound

Valuation

𝑜 + 2 𝑛 − 1

Additive

𝑜 + 3 𝑛 − 1

Additive s.t. identical downward closed constraints

𝑜 + 4 𝑛 − 1

Additive s.t. random downward closed constraints 𝑜 + 2 𝑛 − 1 + 𝜍 Additive s.t. identical matroid constraints

Lower bounds of Ω 𝑜 ⋅ log

𝑜 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 Eden et al. EC'17 Inbal Talgam-Cohen 41

Bulow-Klemperer theorem

• Thm. Revenue of the 2nd price auction with n+1

bidders ≥ Revenue of the optimal auction with n bidders.

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 42

Bulow-Klemperer theorem

• Thm. Revenue of the 2nd 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 Eden et al. EC'17 Inbal Talgam-Cohen 43

Bulow-Klemperer theorem

• Thm. Revenue of the 2nd 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 Eden et al. EC'17 Inbal Talgam-Cohen 44

Bulow-Klemperer theorem

• Thm. Revenue of the 2nd 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 2nd price auction

“beats” 𝐵.

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 45

Proof:

Step I. Upper-bound the optimal revenue.

𝑤1 ∼ 𝐺 𝑤2 ∼ 𝐺 𝑤𝑜 ∼ 𝐺

≥ ≥ ≥

. . .

Price ≥ 𝑞 Myerson’s optimal mechanism

. . .

46

Proof:

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

𝑤1 ∼ 𝐺 𝑤2 ∼ 𝐺 𝑤𝑜 ∼ 𝐺

. . .

𝑤𝑜+1 ∼ 𝐺

. . .

47

Proof:

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

𝑤1 ∼ 𝐺 𝑤2 ∼ 𝐺 𝑤𝑜 ∼ 𝐺

. . .

𝑤𝑜+1 ∼ 𝐺 Run Myerson’s mechanism on 𝒐 bidders

. . .

48

Proof:

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

𝑤1 ∼ 𝐺 𝑤2 ∼ 𝐺 𝑤𝑜 ∼ 𝐺

. . .

𝑤𝑜+1 ∼ 𝐺 Run Myerson’s mechanism on 𝒐 bidders If Myerson does not allocate, give item to the additional bidder

. . .

49

Proof:

Step III. Show that the 2nd price auction “beats” 𝐵.

• Observation. 2nd price

auction is the optimal mechanism out of the mechanisms that always sell.

𝑤1 ∼ 𝐺 𝑤2 ∼ 𝐺 𝑤𝑜 ∼ 𝐺

. . .

𝑤𝑜+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 Eden et al. EC'17 Inbal Talgam-Cohen 51

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 Eden et al. EC'17 Inbal Talgam-Cohen 52

• I. Upper-bound the optimal revenue
• Single additive bidder and i.i.d. items

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 53

𝑤1 ∼ 𝐺 𝑤2 ∼ 𝐺 𝑤𝑛 ∼ 𝐺

. . . . . .

• I. Upper-bound the optimal revenue

Use the duality framework from [Cai Devanur Weinberg ‘16]. OPT ≤ E𝑤∼𝐺𝑛 ෍

𝑘

𝜒+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘<𝑤𝑘′

𝜒 𝑤 = 𝑤 −

1−𝐺 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 ≤ E𝑤∼𝐺𝑛 ෍

𝑘

𝜒+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘<𝑤𝑘′

𝜒 𝑤 = 𝑤 −

1−𝐺 𝑤 𝑔(𝑤) is the virtual valuation function.

55

Distribution appears in proof only!

• I. Upper-bound the optimal revenue

Use the duality framework from [Cai Devanur Weinberg ‘16]. OPT ≤ E𝑤∼𝐺𝑛 ෍

𝑘

𝜒+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘<𝑤𝑘′

Take item 𝑘’s virtual value if it’s the most attractive item

56

𝜒 𝑤 = 𝑤 −

1−𝐺 𝑤 𝑔(𝑤) is the virtual valuation function.

• I. Upper-bound the optimal revenue

Use the duality framework from [Cai Devanur Weinberg ‘16]. OPT ≤ E𝑤∼𝐺𝑛 ෍

𝑘

𝜒+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘<𝑤𝑘′

Take item 𝑘’s value if there’s a more attractive item

57

𝜒 𝑤 = 𝑤 −

1−𝐺 𝑤 𝑔(𝑤) is the virtual valuation function.

• 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 ≡ 2nd price auction for each item separately. Therefore, we devise a single parameter mechanism that covers item 𝒌’s contribution to the benchmark.

E𝑤∼𝐺𝑛 𝜒+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘<𝑤𝑘′

60

• II. Find an auction 𝐵 𝑘 with 𝑛 bidders

and revenue ≥ upper bound for item 𝑘

E𝑤∼𝐺𝑛 𝜒+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘<𝑤𝑘′

Run 2nd price auction with “lazy” reserve price = 𝜒−1 0 for agent 𝑘 0 for agents 𝑘′ ≠ 𝑘

Item 𝑘

𝑤𝑛 ∼ 𝐺

. . . . . .

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 61

𝑤𝑘 ∼ 𝐺 𝑤1 ∼ 𝐺

E𝑤∼𝐺𝑛 𝜒+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘<𝑤𝑘′

Case I: 𝑤𝑘 > 𝑤𝑘′ for all 𝑘′: 𝑘 wins if his virtual value is non-negative. Expected revenue = Expected virtual value [Myerson’81]

Item 𝑘

𝑤𝑘 ∼ 𝐺 𝑤1 ∼ 𝐺 𝑤𝑛 ∼ 𝐺

. . . . . .

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 62

• II. Find an auction 𝐵 𝑘 with 𝑛 bidders

and revenue ≥ upper bound for item 𝑘

E𝑤∼𝐺𝑛 𝜒+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘<𝑤𝑘′

Case II: 𝑤𝑘 < 𝑤𝑘′ for some 𝑘′: The second price is at least the value of agent 𝑘.

Item 𝑘

𝑤𝑛 ∼ 𝐺

. . . . . .

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 63

• II. Find an auction 𝐵 𝑘 with 𝑛 bidders

and revenue ≥ upper bound for item 𝑘

𝑤𝑘 ∼ 𝐺 𝑤1 ∼ 𝐺

• III. Show that VCG “beats” 𝐵

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 64

• III. Show that 2nd price “beats” 𝐵(𝑘)

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 65

• III. Show that 2nd price “beats” 𝐵(𝑘)

𝑩(𝒌) with 𝒏 bidders ≤ Myerson with 𝒏 bidders

≤ 2nd price with

𝒏 + 𝟐 bidders

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 66

• III. Show that 2nd price “beats” 𝐵(𝑘)

 The competition complexity of a single additive bidder and 𝑛 i.i.d. items is ≤ 𝑛.

FF

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 67

𝑩(𝒌) with 𝒏 bidders ≤ Myerson with 𝒏 bidders

≤ 2nd price with

𝒏 + 𝟐 bidders

Going beyond i.i.d items

• Single additive bidder and i.i.d. items

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 68

𝑤1 ∼ 𝐺

1

𝑤2 ∼ 𝐺

2

𝑤𝑛 ∼ 𝐺

𝑛

. . . . . .

Going beyond i.i.d items

Item 𝑘

𝑤𝑘 ∼ 𝐺

𝑘

𝑤1 ∼ 𝐺

𝑘

𝑤𝑛 ∼ 𝐺

𝑘

. . . . . .

E 𝑤1∼𝐺

1

𝑤2∼𝐺

2

… 𝑤𝑛∼𝐺

𝑛

𝜒𝑘

+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘< 𝑤𝑘′

69

Run 2nd price auction with “lazy” reserve price = 𝜒−1 0 for agent 𝑘 0 for agents 𝑘′ ≠ 𝑘

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

Going beyond i.i.d items

Item 𝑘

. . .

𝑤𝑘 ∼ 𝐺

𝑘

𝑤1 ∼ 𝐺

𝑘

𝑤𝑛 ∼ 𝐺

𝑘

. . .

E 𝑤1∼𝐺

1

𝑤2∼𝐺

2

… 𝑤𝑛∼𝐺

𝑛

𝜒𝑘

+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝑤𝑘>𝑤𝑘′ + 𝑤𝑘 ⋅ 1∃𝑘′ 𝑤𝑘< 𝑤𝑘′

Run 2nd price auction with “lazy” reserve price = 𝜒−1 0 for agent 𝑘 0 for agents 𝑘′ ≠ 𝑘 Cannot couple the event “bidder 𝑘 wins” and “item 𝑘 has the highest value”

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 70

Use a different benchmark

Item 𝑘

. . .

E 𝑤1∼𝐺

1

𝑤2∼𝐺

2

… 𝑤𝑛∼𝐺

𝑛

𝜒𝑘

+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝐺𝑘(𝑤𝑘)>𝐺𝑘′(𝑤𝑘′) + 𝑤𝑘 ⋅ 1∃𝑘′ 𝐺𝑘(𝑤𝑘)<𝐺𝑘′(𝑤𝑘′) 𝑤𝑘 ∼ 𝐺

𝑘

𝑤1 ∼ 𝐺

𝑘

𝑤𝑛 ∼ 𝐺

𝑘

. . .

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 71

Use a different benchmark

Item 𝑘

. . .

E 𝑤1∼𝐺

1

𝑤2∼𝐺

2

… 𝑤𝑛∼𝐺

𝑛

𝜒𝑘

+ 𝑤𝑘 ⋅ 1∀𝑘′ 𝐺𝑘(𝑤𝑘)>𝐺𝑘′(𝑤𝑘′) + 𝑤𝑘 ⋅ 1∃𝑘′ 𝐺𝑘(𝑤𝑘)<𝐺𝑘′(𝑤𝑘′)

 The competition complexity of a single additive bidder and 𝑛 items is ≤ 𝑛.

𝑤𝑘 ∼ 𝐺

𝑘

𝑤1 ∼ 𝐺

𝑘

𝑤𝑛 ∼ 𝐺

𝑘

. . .

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 Eden et al. EC'17 Inbal Talgam-Cohen 73

EXTENSION TO CONSTRAINTS

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

$16

Recall

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 75

$6 $10 $5 Substitutes Complements Total value =

• Example of “downward closed” constraints:

Sets of size 1 and { } are feasible

Extension to downward closed constraints

OPT𝑜

Add≤ VCG𝑜+𝐷 Add

Competition complexity ≤ 𝐷

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 76

Extension to downward closed constraints

OPT𝑜

Add≤ VCG𝑜+𝐷 Add

Competition complexity ≤ 𝐷

OPT𝑜

DC ≤

Larger

• utcome

space

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 77

Extension to downward closed constraints

OPT𝑜

Add≤ VCG𝑜+𝐷 Add

Competition complexity ≤ 𝐷

OPT𝑜

DC ≤

Larger

• utcome

space

≤ VCG𝑜+𝐷+𝑛−1

DC

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 78

Extension to downward closed constraints

OPT𝑜

Add≤ VCG𝑜+𝐷 Add

Competition complexity ≤ 𝐷

OPT𝑜

DC ≤

Larger

• utcome

space

≤ VCG𝑜+𝐷+𝑛−1

DC

The competition complexity of 𝑜 additive bidders with identical downward closed constraints over 𝑛 items is ≤ 𝐷 + 𝑛 − 1.

Extension to downward closed constraints

OPT𝑜

Add≤ VCG𝑜+𝐷 Add

Competition complexity ≤ 𝐷

OPT𝑜

DC ≤

Larger

• utcome

space

≤ VCG𝑜+𝐷+𝑛−1

DC

The competition complexity of 𝑜 additive bidders with identical downward closed constraints over 𝑛 items is ≤ 𝐷 + 𝑛 − 1.

Main technical challenge

• 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 Eden et al. EC'17 Inbal Talgam-Cohen 81

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

VCG for additive bidders ≡ 2nd price auction for each item separately. Therefore, the revenue from item 𝒌 in VCG𝑌

Add =

2nd highest value out of 𝒀 i.i.d. samples from 𝑮𝒌.

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 82

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

83

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

84

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

85

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

86

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

87

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

• Claim. Revenue for item 𝒌 in

VCG𝑌+𝑛−1

DC

≥ value of the highest unallocated bidder for item 𝑘.

88

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

89

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

90

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

91

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

Externality at least 9

92

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

93

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

94

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

95

5 ∼ 𝐺 2 ∼ 𝐺 7 ∼ 𝐺 3 4 6 4 1 5 3 2 4

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

3 1 2

Externality at least 2

96

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 97

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

VCG𝑌

Add(𝑘) =

2nd highest

• f 𝑌 samples

from 𝐺

𝑘

VCG𝑌+𝑛−1

DC

(𝑘) Highest value

• f unallocated

bidder for 𝑘 ≤

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 98

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

VCG𝑌

Add(𝑘) =

2nd highest

• f 𝑌 samples

from 𝐺

𝑘

VCG𝑌+𝑛−1

DC

(𝑘) Highest value

• f unallocated

bidder for 𝑘 ≤

≤

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 99

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

VCG𝑌

Add(𝑘) =

2nd highest

• f 𝑌 samples

from 𝐺

𝑘

VCG𝑌+𝑛−1

DC

(𝑘) Highest value

• f unallocated

bidder for 𝑘 ≤

≤

Identify 𝑌 bidders in VCG𝑌+𝑛−1

DC

before sampling their value for item 𝑘 out of which at most one will be allocated anything

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 100

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

1 2 3 4 5 6 7 j m

…

101

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

1 2 3 4 5 6 7 j m

…

(Assume wlog unique optimal allocation)

102

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 1. Sample valuations for all items but 𝑘.

1 2 3 4 5 6 7 j m

…

(Assume wlog unique optimal allocation)

103

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 2. Compute an optimal allocation without item 𝑘.

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation)

104

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 2. Compute an optimal allocation without item 𝑘.

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation)

Set 𝐵 of allocated bidders

Set ҧ 𝐵 of unallocated bidders

105

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 2. Compute an optimal allocation without item 𝑘.

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation)

Set 𝐵 of allocated bidders

Set ҧ 𝐵 of unallocated bidders

If 𝑘 is allocated to bidder in ҧ 𝐵 in OPT, all other items are allocated as before.

106

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Sample values for 𝑘 for agents in 𝐵 and compute

the optimal allocation where 𝑘 is allocated to a bidder in 𝐵 .

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation)

107

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation)

108

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation) Some items might be vacated due to feasibility

109

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation) Some items might be snatched from other agents

110

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation) Continue with this process

111

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation) Continue with this process

112

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵. There are ≥ 𝐵 items

allocated to agents in 𝐵.

1 2 3 4 5 6 7 j m

(Assume wlog unique optimal allocation)

113

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵. There are ≥ 𝐵 items

allocated to agents in 𝐵.

– Map each agent who’s item was snatched to the snatched item.

2 3 6 7 j m

(Assume wlog unique optimal allocation)

1 4 5

114

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵. There are ≥ 𝐵 items

allocated to agents in 𝐵.

– Map each agent who’s item was snatched to the snatched item. – Map each agent who took a vacated item to the item.

2 3 6 7 j m

(Assume wlog unique optimal allocation)

1 4 5

115

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵. There are ≥ 𝐵 items

allocated to agents in 𝐵.

– 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 3 6 7 j m

(Assume wlog unique optimal allocation)

1 4 5

116

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵. There are ≥ 𝐵 items

allocated to agents in 𝐵.

2 3 6 7 j m

(Assume wlog unique optimal allocation)

1 4 5

≤ 𝑛 − |𝐵| allocated

117

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

• 3. Compute OPT

𝑘∈𝐵. There are ≥ 𝐵 items

allocated to agents in 𝐵.

2 3 6 7 j m

(Assume wlog unique optimal allocation)

1 4 5

≤ 𝑛 − |𝐵| allocated

≥ ҧ 𝐵 − 𝑛 − 𝐵 = 𝑌 + 𝑛 − 1 − 𝐵 − 𝑛 − 𝐵 = 𝑌 − 1 unallocated

118

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

2 3 6 7 j m

(Assume wlog unique optimal allocation)

1 4 5

≤ 𝑛 − |𝐵| allocated

≥ ҧ 𝐵 − 𝑛 − 𝐵 = 𝑌 + 𝑛 − 1 − 𝐵 − 𝑛 − 𝐵 = 𝑌 − 1 unallocated

119

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

2 3 6 7 j m

(Assume wlog unique optimal allocation)

1 4 5

≤ 𝑛 − |𝐵| allocated

≥ ҧ 𝐵 − 𝑛 − 𝐵 = 𝑌 + 𝑛 − 1 − 𝐵 − 𝑛 − 𝐵 = 𝑌 − 1 unallocated

𝑌 bidders whose values for 𝑘 are i.i.d. samples from 𝐺

𝑘.

At most one is allocated by VCG𝑌+𝑛−1

DC

.

120

VCG𝑌

Add ≤ VCG𝑌+𝑛−1 DC

…

(Assume wlog unique optimal allocation) 𝑌 bidders whose values for 𝑘 are i.i.d. samples from 𝐺

𝑘.

At most one is allocated by VCG𝑌+𝑛−1

DC

.

VCG𝑌

Add(𝑘) =

2nd highest

• f 𝑌 samples

from 𝐺

𝑘

VCG𝑌+𝑛−1

DC

(𝑘) Highest value

• f unallocated

bidder for 𝑘 ≤

≤

121

Extension to downward closed constraints

Rev𝑜

Add ≤ VCG𝑜+𝐷 Add

Competition complexity ≤ 𝐷

Rev𝑜

DC ≤

Larger

• utcome

space

≤ VCG𝑜+𝐷+𝑛−1

DC

The competition complexity of 𝑜 additive bidders s.t. identical downward closed constraints over 𝑛 items is ≤ 𝐷 + 𝑛 − 1.

122

Extension to downward closed constraints

Rev𝑜

Add ≤ VCG𝑜+𝐷 Add

Competition complexity ≤ 𝐷

Rev𝑜

DC ≤

Larger

• utcome

space

≤ VCG𝑜+𝐷+𝑛−1

DC

The competition complexity of 𝑜 additive bidders s.t. identical downward closed constraints over 𝑛 items is ≤ 𝐷 + 𝑛 − 1.

Proved!

123

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-in-

range VCG” by restricting outcomes to matchings

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 124

Further extensions (preliminary)

• 1. From competition complexity to

approximation

– In large markets (𝑜 ≫ 𝑛), 2nd price auction (no extra agents)

1 2-approximates OPT

• 2. Non-i.i.d. bidders

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 125

Summary

• Major open problem: Revenue maximization

for 𝑛 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

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 126

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

Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen 127