the competition complexity of auctions: bulow-klemperer ... complexity.pdfthe competition complexity...
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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
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Complexity in AMD
One goal of Algorithmic Mechanism Design:
Deal with complex allocation of goods settings
โข Goods may not be homogenous
โข Valuations and constraints may be complex
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Complexity in AMD
One goal of Algorithmic Mechanism Design:
Deal with complex allocation of goods settings
โข Goods may not be homogenous
โข Valuations and constraints may be complex
โข E.g. spectrum auctions, cloud computing, ad auctions, โฆ
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Revenue maximization
โข Revenue less understood than welfare
โ (even for welfare, some computational issues persist)
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Revenue maximization
โข Revenue less understood than welfare
โ (even for welfare, some computational issues persist)
โข Optimal truthful mechanism known only for handful of complex settings (e.g. additive buyer with 2 items, 6 uniform i.i.d. items... [Giannakopolous-Koutsoupiasโ14,โ15])
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Revenue maximization
โข Revenue less understood than welfare
โ (even for welfare, some computational issues persist)
โข Optimal truthful mechanism known only for handful of complex settings (e.g. additive buyer with 2 items, 6 uniform i.i.d. items... [Giannakopolous-Koutsoupiasโ14,โ15])
โข Common CS solution for complexity: approximation
โ [Hart-Nisanโ12,โ13, Li-Yaoโ13, Babioff-et-al.โ14, Rubinstein-Weinbergโ15, Chawla-Millerโ16, โฆ]
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Revenue maximization
โข Revenue less understood than welfare
โ (even for welfare, some computational issues persist)
โข Optimal truthful mechanism known only for handful of complex settings (e.g. additive buyer with 2 items, 6 uniform i.i.d. items... [Giannakopolous-Koutsoupiasโ14,โ15])
โข Common CS solution for complexity: approximation
โ [Hart-Nisanโ12,โ13, Li-Yaoโ13, Babioff-et-al.โ14, Rubinstein-Weinbergโ15, Chawla-Millerโ16, โฆ]
โข Resource augmentationCompetition Complexity of Auctions
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Single item welfare maximization
Run a 2nd price auction โsimple, maximizes welfare โpointwiseโ.
(VCG mechanism)
๐ฃ1
๐ฃ2
๐ฃ๐
โฅ
โฅ
โฅ
.
.
.
.
.
.
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Single item welfare maximization
Run a 2nd price auction โsimple, maximizes welfare โpointwiseโ.
(VCG mechanism)
๐ฃ1
๐ = ๐ฃ2
๐ฃ๐
โฅ
โฅ
โฅ
.
.
.
.
.
.
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Single item revenue maximization
Single buyer: select price that
maximizes ๐ โ 1 โ ๐น ๐
(โmonopoly priceโ).๐ฃ1 โผ ๐น
Price = ๐
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Single item revenue maximization
Single buyer: select price that
maximizes ๐ โ 1 โ ๐น ๐
(โmonopoly priceโ).
Multiple i.i.d. buyers: run 2nd price auction with reserve price ๐ (same ๐).
(Myersonโs auction)
๐ฃ1 โผ ๐น
๐ฃ2 โผ ๐น
๐ฃ๐ โผ ๐น
โฅ
โฅ
โฅ
Price โฅ ๐
.
.
.
.
.
.
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Single item revenue maximization
Single buyer: select price that
maximizes ๐ โ 1 โ ๐น ๐
(โmonopoly priceโ).
Multiple i.i.d. buyers: run 2nd price auction with reserve price ๐ (same ๐).
(Myersonโs auction)
๐ฃ1 โผ ๐น
๐ฃ2 โผ ๐น
๐ฃ๐ โผ ๐น
โฅ
โฅ
โฅ
Price โฅ ๐
.
.
.
.
.
.
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Assuming regularity
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Single item revenue maximization
Single buyer: select price that
maximizes ๐ โ 1 โ ๐น ๐
(โmonopoly priceโ).
Multiple i.i.d. buyers: run 2nd price auction with reserve price ๐ (same ๐).
(Myersonโs auction)
๐ฃ1 โผ ๐น
๐ฃ2 โผ ๐น
๐ฃ๐ โผ ๐น
โฅ
โฅ
โฅ
Price โฅ ๐
.
.
.
Requires prior knowledge to determine the reserve
.
.
.
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Bulow-Klemperer theorem
Thm. Expected revenue of the 2nd price auction with n+1 bidders โฅ Expected revenue of the optimal auction with n bidders.
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Bulow-Klemperer theorem
Thm. Expected revenue of the 2nd price auction with n+1 bidders โฅ 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. โThe statistics of the data shifts rapidlyโ [Google]
Simple! โHardly anything matters moreโ [Milgromโ04]
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Multidimensional settings
๐น
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Multidimensional settings
๐น1
๐น2
๐น3
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Multidimensional settings
๐น1
๐น2
๐น3
Biddersโ values are sampled i.i.d. from a product distribution over items
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Multidimensional settings
๐น1
๐น2
๐น3
Additive: ๐ฃ( , )=๐ฃ( ) )+๐ฃ( )
Biddersโ values are sampled i.i.d. from a product distribution over items
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Multidimensional settings
โข Revenue maximization is not well understood:
โข Optimal mechanism mightnecessitate randomization.
โข Non-monotone.
โข Computationally intractable.
โข Only recently, simple approximately optimal mechanisms were devised.
๐น1
๐น2
๐น3
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Multidimensional settings
Either run a randomized,
๐น1
๐น2
๐น3
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Multidimensional settings
Either run a randomized,
hard to compute,๐น1
๐น2
๐น3
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Multidimensional settings
Either run a randomized,
hard to compute,
with infinitely many options
mechanism,
๐น1
๐น2
๐น3
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Multidimensional settings
Either run a randomized,
hard to compute,
with infinitely many options
mechanism, which depends
heavily on the distributionsโฆ
๐น1
๐น2
๐น3
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Multidimensional settings
Either run a randomized,
hard to compute,
with infinitely many options
mechanism, which depends
heavily on the distributionsโฆ
Or add more bidders.
๐น1
๐น2
๐น3
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OUR RESULTS
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Competition complexity: Fix an environment with ๐i.i.d. bidders. What is ๐ such that the revenue of VCGwith ๐ + ๐ bidders is โฅ OPT with ๐ bidders.
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Multidimensional B-K theorems
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Bulow-Klemperer Thm. The competition complexity of a single item auction is 1.
Competition complexity: Fix an environment with ๐i.i.d. bidders. What is ๐ such that the revenue of VCGwith ๐ + ๐ bidders is โฅ OPT with ๐ bidders.
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Multidimensional B-K theorems
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Bulow-Klemperer Thm. The competition complexity of a single item auction is 1.
Competition complexity: Fix an environment with ๐i.i.d. bidders. What is ๐ such that the revenue of VCGwith ๐ + ๐ bidders is โฅ OPT with ๐ bidders.
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Multidimensional B-K theorems
Thm. [BK] The competition complexity of a single item with ๐ copies is ๐.
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Competition complexity: Fix an environment with ๐i.i.d. bidders. What is ๐ such that the revenue of VCGwith ๐ + ๐ bidders is โฅ OPT with ๐ bidders.
Thm. [EFFTW] The competition complexity of ๐additive bidders drawn from a product distribution over ๐ items is โค ๐ + ๐(๐โ ๐).
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Multidimensional B-K theorems
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Thm. [EFFTW] Let ๐ช be the competition complexity of ๐additive bidders over ๐ items. The competition complexity of ๐ additive bidders with identical downward closed constraints over ๐ items is โค ๐ช +๐โ ๐.
Competition complexity: Fix an environment with ๐i.i.d. bidders. What is ๐ such that the revenue of VCGwith ๐ + ๐ bidders is โฅ OPT with ๐ bidders.
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Multidimensional B-K theorems
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Thm. [EFFTW] Let ๐ช be the competition complexity of ๐additive bidders over ๐ items. The competition complexity of ๐ additive bidders with randomly drawn downward closed constraints over ๐ items is โค ๐ช+ ๐(๐ โ ๐).
Competition complexity: Fix an environment with ๐i.i.d. bidders. What is ๐ such that the revenue of VCGwith ๐ + ๐ bidders is โฅ OPT with ๐ bidders.
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Multidimensional B-K theorems
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Additive with constraints
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โข Constraints = set system over the items
โ Specifies which item sets are feasible
โข Bidderโs value for an item set = her value for best feasible subset
โข If all sets are feasible, bidder is additive
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Example of constraints
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$6
$10
$21
$5
Total value =
โข No constraints
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Example of constraints
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$6
$10
$21
$5
$10
Substitutes
Total value =
โข Example of โmatroidโ constraints: Only sets of size ๐ = 1 are feasible
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$10$16
Example of constraints
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$6
$10
$5
Substitutes
Complements
Total value =
โข Example of โdownward closedโ constraints: Sets of size 1 and { } are feasible
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Complements in what sense?
โข No complements = gross substitutes:
โ ิฆ๐ โค ิฆ๐ item prices
โ ๐ in demand( ิฆ๐) if maximizes utility ๐ฃ๐ ๐ โ ๐(๐)
โ โ๐ in demand( ิฆ๐), there is ๐ in demand( ิฆ๐) with every item in ๐ whose price didnโt increase
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$
$$
$
๐บ
๐ป
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Complements in what sense?
โข No complements = gross substitutes:
โ ิฆ๐ โค ิฆ๐ item prices
โ ๐ in demand( ิฆ๐) if maximizes utility ๐ฃ๐ ๐ โ ๐(๐)
โ โ๐ in demand( ิฆ๐), there is ๐ in demand( ิฆ๐) with every item in ๐ whose price didnโt increase
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
38
$๐บ
๐ป 5 6
10 ิฆ๐ = (5, ๐, ๐)
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Competition complexity โ summary Upper boundValuation
๐ + 2 ๐ โ 1Additive
๐ + 3 ๐ โ 1Additive s.t. identical downward closed constraints
๐ + 4 ๐ โ 1Additive s.t. random downward closed constraints
๐ + 2 ๐ โ 1 + ๐Additive s.t. identical matroidconstraints
Lower bounds of ฮฉ ๐ โ log๐
๐+ 1 for additive bidders and ฮฉ ๐ for unit demand
bidders are due to ongoing work by [Feldman-Friedler-Rubinstein] and to [Bulow-Klempererโ96]
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Related workMultidimensional B-K theorems
[Roughgarden T. Yan โ12]: for unit demand bidders, revenue of VCG with ๐ extra bidders โฅ revenue of the optimal deterministic DSIC mechanism.
[Feldman Friedler Rubinstein โ ongoing]: tradeoffs between enhanced competition and revenue.
Prior-independent multidimensional mechanisms
[Devanur Hartline Karlin Nguyen โ11]: unit demand bidders.
[Roughgarden T. Yan โ12]: unit demand bidders.
[Goldner Karlin โ16]: additive bidders.
Sample complexity
[Morgenstern Roughgarden โ16]: how many samples needed to approximate the optimal mechanism?
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MULTIDIMENSIONAL B-K THEOREMPROOF SKETCH
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Bulow-Klemperer theorem
Thm. Revenue of the 2nd price auction with n+1 bidders โฅ Revenue of the optimal auction with n bidders.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Bulow-Klemperer theorem
Thm. Revenue of the 2nd price auction with n+1 bidders โฅ Revenue of the optimal auction with n bidders.
Proof. (in 3 steps of [Kirkegaardโ06])
I. Upper-bound the optimal revenue.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Bulow-Klemperer theorem
Thm. Revenue of the 2nd price auction with n+1 bidders โฅ Revenue of the optimal auction with n bidders.
Proof. (in 3 steps of [Kirkegaardโ06])
I. Upper-bound the optimal revenue.
II. Find an auction ๐ด with more bidders and revenue โฅ the upper bound.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Bulow-Klemperer theorem
Thm. Revenue of the 2nd price auction with n+1 bidders โฅ Revenue of the optimal auction with n bidders.
Proof. (in 3 steps of [Kirkegaardโ06])
I. Upper-bound the optimal revenue.
II. Find an auction ๐ด with more bidders and revenue โฅ the upper bound.
III. Show that the 2nd price auction โbeatsโ ๐ด.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Proof:
Step I. Upper-bound the optimal revenue.
๐ฃ1 โผ ๐น
๐ฃ2 โผ ๐น
๐ฃ๐ โผ ๐น
โฅ
โฅ
โฅ
.
.
.
Price โฅ ๐
Myersonโs optimal mechanism
.
.
.
46
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Proof:
Step II. Find an auction ๐ดwith more bidders and revenue โฅ the upper bound.
๐ฃ1 โผ ๐น
๐ฃ2 โผ ๐น
๐ฃ๐ โผ ๐น
.
.
.
๐ฃ๐+1 โผ ๐น
.
.
.
47
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Proof:
Step II. Find an auction ๐ดwith more bidders and revenue โฅ the upper bound.
๐ฃ1 โผ ๐น
๐ฃ2 โผ ๐น
๐ฃ๐ โผ ๐น
.
.
.
๐ฃ๐+1 โผ ๐น
Run Myersonโsmechanism on๐ bidders
.
.
.
48
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Proof:
Step II. Find an auction ๐ดwith more bidders and revenue โฅ the upper bound.
๐ฃ1 โผ ๐น
๐ฃ2 โผ ๐น
๐ฃ๐ โผ ๐น
.
.
.
๐ฃ๐+1 โผ ๐น
Run Myersonโsmechanism on๐ bidders
If Myerson does not allocate, give item to the additionalbidder
.
.
.
49
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Proof:
Step III. Show that the 2nd
price auction โbeatsโ ๐ด.
Observation. 2nd price
auction is the optimal mechanism out of the mechanisms that always sell.
๐ฃ1 โผ ๐น
๐ฃ2 โผ ๐น
๐ฃ๐ โผ ๐น
.
.
.
๐ฃ๐+1 โผ ๐น
.
.
.
50
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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 ๐ด with more bidders and revenue โฅ the upper bound.
III. Show that VCG โbeatsโ ๐ด.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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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 ๐ด with more bidders and revenue โฅ the upper bound.
III. Show that VCG โbeatsโ ๐ด.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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I. Upper-bound the optimal revenue
โข Single additive bidder and i.i.d. items
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
53
๐ฃ1 โผ ๐น
๐ฃ2 โผ ๐น
๐ฃ๐ โผ ๐น
.
.
.
.
.
.
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I. Upper-bound the optimal revenue
Use the duality framework from [Cai DevanurWeinberg โ16].
OPT โค
E๐ฃโผ๐น๐
๐
๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐<๐ฃ๐โฒ
๐ ๐ฃ = ๐ฃ โ1โ๐น ๐ฃ
๐(๐ฃ)is the virtual valuation function.54
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I. Upper-bound the optimal revenue
Use the duality framework from [Cai DevanurWeinberg โ16].
OPT โค
E๐ฃโผ๐น๐
๐
๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐<๐ฃ๐โฒ
๐ ๐ฃ = ๐ฃ โ1โ๐น ๐ฃ
๐(๐ฃ)is the virtual valuation function.55
Distribution appears in proof only!
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I. Upper-bound the optimal revenue
Use the duality framework from [Cai DevanurWeinberg โ16].
OPT โค
E๐ฃโผ๐น๐
๐
๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐<๐ฃ๐โฒ
Take item ๐โs virtual value if itโs the mostattractive item
56 ๐ ๐ฃ = ๐ฃ โ1โ๐น ๐ฃ
๐(๐ฃ)is the virtual valuation function.
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I. Upper-bound the optimal revenue
Use the duality framework from [Cai DevanurWeinberg โ16].
OPT โค
E๐ฃโผ๐น๐
๐
๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐<๐ฃ๐โฒ
Take item ๐โs value if thereโs a more attractive item
57 ๐ ๐ฃ = ๐ฃ โ1โ๐น ๐ฃ
๐(๐ฃ)is the virtual valuation function.
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II. Find an auction ๐ด with more bidders and revenue โฅ upper bound
58
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II. Find an auction ๐ด with ๐ bidders and revenue โฅ upper bound
59
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II. Find an auction ๐ด with ๐ bidders and revenue โฅ upper bound
VCG for additive bidders โก 2nd price auction for each item separately.
Therefore, we devise a single parameter mechanism that covers item ๐โs contribution to the benchmark.
E๐ฃโผ๐น๐ ๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐<๐ฃ๐โฒ
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II. Find an auction ๐ด ๐ with ๐ bidders and revenue โฅ upper bound for item ๐
E๐ฃโผ๐น๐ ๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐<๐ฃ๐โฒ
Run 2nd price auctionwith โlazyโ reserve price =
๐โ1 0 for agent ๐
0 for agents ๐โฒ โ ๐
Item ๐
๐ฃ๐ โผ ๐น
.
.
.
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
61
๐ฃ๐ โผ ๐น
๐ฃ1 โผ ๐น
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E๐ฃโผ๐น๐ ๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐<๐ฃ๐โฒ
Case I: ๐ฃ๐ > ๐ฃ๐โฒ for all ๐โฒ:
๐ wins if his virtual value is
non-negative.
Expected revenue =
Expected virtual value
[Myersonโ81]
Item ๐๐ฃ๐ โผ ๐น
๐ฃ1 โผ ๐น
๐ฃ๐ โผ ๐น
.
.
.
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
62
II. Find an auction ๐ด ๐ with ๐ bidders and revenue โฅ upper bound for item ๐
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E๐ฃโผ๐น๐ ๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐<๐ฃ๐โฒ
Case II: ๐ฃ๐ < ๐ฃ๐โฒ for some ๐โฒ:
The second price is at least
the value of agent ๐.
Item ๐
๐ฃ๐ โผ ๐น
.
.
.
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
63
II. Find an auction ๐ด ๐ with ๐ bidders and revenue โฅ upper bound for item ๐
๐ฃ๐ โผ ๐น
๐ฃ1 โผ ๐น
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III. Show that VCG โbeatsโ ๐ด
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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III. Show that 2nd price โbeatsโ ๐ด(๐)
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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III. Show that 2nd price โbeatsโ ๐ด(๐)
๐จ(๐) with๐ bidders
โคMyerson with๐ bidders
โค2nd price with๐+ ๐ bidders
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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III. Show that 2nd price โbeatsโ ๐ด(๐)
The competition complexity of a single additive bidder and ๐ i.i.d. items is โค ๐.
FFCompetition Complexity of Auctions
Eden et al. EC'17 Inbal Talgam-Cohen67
๐จ(๐) with๐ bidders
โคMyerson with๐ bidders
โค2nd price with๐+ ๐ bidders
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Going beyond i.i.d items
โข Single additive bidder and i.i.d. items
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
68
๐ฃ1 โผ ๐น1
๐ฃ2 โผ ๐น2
๐ฃ๐ โผ ๐น๐
.
.
.
.
.
.
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Going beyond i.i.d items
Item ๐๐ฃ๐ โผ ๐น๐
๐ฃ1 โผ ๐น๐
๐ฃ๐ โผ ๐น๐
.
.
.
.
.
.
E ๐ฃ1โผ๐น1๐ฃ2โผ๐น2โฆ๐ฃ๐โผ๐น๐
๐๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ
+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐< ๐ฃ๐โฒ
69
Run 2nd price auctionwith โlazyโ reserve price =
๐โ1 0 for agent ๐
0 for agents ๐โฒ โ ๐
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Going beyond i.i.d items
Item ๐
.
.
.
๐ฃ๐ โผ ๐น๐
๐ฃ1 โผ ๐น๐
๐ฃ๐ โผ ๐น๐
.
.
.
E ๐ฃ1โผ๐น1๐ฃ2โผ๐น2โฆ๐ฃ๐โผ๐น๐
๐๐+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐>๐ฃ๐โฒ
+ ๐ฃ๐ โ 1โ๐โฒ ๐ฃ๐< ๐ฃ๐โฒ
Run 2nd price auctionwith โlazyโ reserve price = ๐โ1 0 for agent ๐0 for agents ๐โฒ โ ๐Cannot couple the event โbidder ๐ winsโ and โitem ๐ has the highest valueโ
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen70
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Use a different benchmark
Item ๐
.
.
.
E ๐ฃ1โผ๐น1๐ฃ2โผ๐น2โฆ๐ฃ๐โผ๐น๐
๐๐+ ๐ฃ๐ โ 1โ๐โฒ ๐น๐(๐ฃ๐)>๐น๐โฒ(๐ฃ๐โฒ)
+ ๐ฃ๐ โ 1โ๐โฒ ๐น๐(๐ฃ๐)<๐น๐โฒ(๐ฃ๐โฒ)
๐ฃ๐ โผ ๐น๐
๐ฃ1 โผ ๐น๐
๐ฃ๐ โผ ๐น๐
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen71
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Use a different benchmark
Item ๐
.
.
.
E ๐ฃ1โผ๐น1๐ฃ2โผ๐น2โฆ๐ฃ๐โผ๐น๐
๐๐+ ๐ฃ๐ โ 1โ๐โฒ ๐น๐(๐ฃ๐)>๐น๐โฒ(๐ฃ๐โฒ)
+ ๐ฃ๐ โ 1โ๐โฒ ๐น๐(๐ฃ๐)<๐น๐โฒ(๐ฃ๐โฒ)
The competition complexity of a single additive bidder and ๐ items is โค ๐.
๐ฃ๐ โผ ๐น๐
๐ฃ1 โผ ๐น๐
๐ฃ๐ โผ ๐น๐
.
.
.
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Going beyond a single bidder
โข Step I:
โ Benchmark more involved
โข Step II:
โ Devise a more complex single parameter auction A(j) (involves a max)
โ Proving A(j) is greater than item jโs contribution to the benchmark is more involved and requires subtle coupling and probabilistic claims
BBCompetition Complexity of Auctions
Eden et al. EC'17 Inbal Talgam-Cohen73
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EXTENSION TO CONSTRAINTS
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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$16
Recall
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
75
$6
$10
$5
Substitutes
Complements
Total value =
โข Example of โdownward closedโ constraints: Sets of size 1 and { } are feasible
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Extension to downward closed constraints
OPT๐Addโค VCG๐+๐ถ
Add
Competitioncomplexity โค ๐ถ
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Extension to downward closed constraints
OPT๐Addโค VCG๐+๐ถ
Add
Competitioncomplexity โค ๐ถ
OPT๐DC โค
Larger outcomespace
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
77
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Extension to downward closed constraints
OPT๐Addโค VCG๐+๐ถ
Add
Competitioncomplexity โค ๐ถ
OPT๐DC โค
Larger outcomespace
โค VCG๐+๐ถ+๐โ1DC
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
78
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Extension to downward closed constraints
OPT๐Addโค VCG๐+๐ถ
Add
Competitioncomplexity โค ๐ถ
OPT๐DC โค
Larger outcomespace
โค VCG๐+๐ถ+๐โ1DC
The competition complexity of ๐ additive bidders with identical downward closed constraints over ๐ items is โค ๐ถ +๐ โ 1.
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Extension to downward closed constraints
OPT๐Addโค VCG๐+๐ถ
Add
Competitioncomplexity โค ๐ถ
OPT๐DC โค
Larger outcomespace
โค VCG๐+๐ถ+๐โ1DC
The competition complexity of ๐ additive bidders with identical downward closed constraints over ๐ items is โค ๐ถ +๐ โ 1.
Main technical challenge
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Claim. VCG revenue from selling ๐ items to ๐ฟ = ๐ + ๐ชadditive bidders whose values are i.i.d. draws from ๐น
โคVCG revenue from selling them to ๐ฟ +๐โ ๐ bidders with i.i.d. values drawn from ๐น, whose valuations are additive s.t. identical downward-closed constraints.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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VCG๐Add โค VCG๐+๐โ1
DC
VCG for additive bidders โก 2nd price auction for each item separately.
Therefore, the revenue from item ๐ in VCG๐Add =
2nd highest value out of ๐ฟ i.i.d. samples from ๐ญ๐.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
82
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VCG๐Add โค VCG๐+๐โ1
DC
83
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VCG๐Add โค VCG๐+๐โ1
DC
84
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
VCG๐Add โค VCG๐+๐โ1
DC
85
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 1286
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 1287
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 12
Claim. Revenue for item ๐ in
VCG๐+๐โ1DC โฅ value of the
highest unallocated bidder for item ๐.
88
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 1289
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 1290
![Page 91: The Competition Complexity of Auctions: Bulow-Klemperer ... Complexity.pdfThe Competition Complexity of Auctions: Bulow-Klemperer Results for Multidimensional Bidders Oxford, Spring](https://reader034.vdocument.in/reader034/viewer/2022042808/5f8a6c8f5257844adf1166b7/html5/thumbnails/91.jpg)
5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 1291
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 12
Externality at least 9
92
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 1293
![Page 94: The Competition Complexity of Auctions: Bulow-Klemperer ... Complexity.pdfThe Competition Complexity of Auctions: Bulow-Klemperer Results for Multidimensional Bidders Oxford, Spring](https://reader034.vdocument.in/reader034/viewer/2022042808/5f8a6c8f5257844adf1166b7/html5/thumbnails/94.jpg)
5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 1294
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 1295
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5 โผ ๐น 2 โผ ๐น7 โผ ๐น
3 46
4 15
3 24
VCG๐Add โค VCG๐+๐โ1
DC
3 12
Externality at least 2
96
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VCG๐Add โค VCG๐+๐โ1
DC
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
97
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VCG๐Add โค VCG๐+๐โ1
DC
VCG๐Add(๐) =
2nd highest
of ๐ samplesfrom ๐น๐
VCG๐+๐โ1DC (๐)
Highest value
of unallocated
bidder for ๐
โค
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
98
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VCG๐Add โค VCG๐+๐โ1
DC
VCG๐Add(๐) =
2nd highest
of ๐ samplesfrom ๐น๐
VCG๐+๐โ1DC (๐)
Highest value
of unallocated
bidder for ๐
โคโค
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
99
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VCG๐Add โค VCG๐+๐โ1
DC
VCG๐Add(๐) =
2nd highest
of ๐ samplesfrom ๐น๐
VCG๐+๐โ1DC (๐)
Highest value
of unallocated
bidder for ๐
โคโค
Identify ๐ bidders in VCG๐+๐โ1DC
before sampling their value for item ๐ out of which at most one will be allocated anything
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
100
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
1 2 3 4 5 6 7 jmโฆ
101
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
1 2 3 4 5 6 7 jmโฆ
(Assume wlog unique optimal allocation)
102
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
1. Sample valuations for all items but ๐.
1 2 3 4 5 6 7 jmโฆ
(Assume wlog unique optimal allocation)
103
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
2. Compute an optimal allocation without item ๐.
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
104
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
2. Compute an optimal allocation without item ๐.
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Set ๐ด of allocatedbidders
Set าง๐ด of unallocatedbidders
105
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
2. Compute an optimal allocation without item ๐.
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Set ๐ด of allocatedbidders
Set าง๐ด of unallocatedbidders
If ๐ is allocated to bidder in าง๐ด in OPT,
all other items are allocated as before.
106
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Sample values for ๐ for agents in ๐ด and compute the optimal allocation where ๐ is allocated to a bidder in ๐ด .
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
107
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
108
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Some items might be vacated due to feasibility
109
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Some items might be snatched from other agents
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Continue with this process
111
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Continue with this process
112
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด. There are โฅ ๐ด items
allocated to agents in ๐ด.
1
2
3 4 56
7
j
m
(Assume wlog unique optimal allocation)
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด. There are โฅ ๐ด items
allocated to agents in ๐ด.โ Map each agent whoโs item was snatched to the snatched item.
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด. There are โฅ ๐ด items
allocated to agents in ๐ด.โ Map each agent whoโs item was snatched to the snatched item.
โ 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
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด. There are โฅ ๐ด items
allocated to agents in ๐ด.โ Map each agent whoโs item was snatched to the snatched item.
โ Map each agent who took a vacated item to the item.
โ Every agent who wasnโt snatched and didnโt take an itemhas the same allocation.
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด. There are โฅ ๐ด items
allocated to agents in ๐ด.
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
โค ๐ โ |๐ด| allocated
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
3. Compute OPT๐โ๐ด. There are โฅ ๐ด items
allocated to agents in ๐ด.
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
โค ๐ โ |๐ด| allocatedโฅ าง๐ด โ ๐ โ ๐ด= ๐ +๐ โ 1 โ ๐ด โ
๐ โ ๐ด= ๐ โ 1 unallocated
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
โค ๐ โ |๐ด| allocatedโฅ าง๐ด โ ๐ โ ๐ด= ๐ +๐ โ 1 โ ๐ด โ
๐ โ ๐ด= ๐ โ 1 unallocated
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
โค ๐ โ |๐ด| allocatedโฅ าง๐ด โ ๐ โ ๐ด= ๐ +๐ โ 1 โ ๐ด โ
๐ โ ๐ด= ๐ โ 1 unallocated
๐ bidders whose values for ๐ are i.i.d. samples from ๐น๐ .
At most one is allocated by VCG๐+๐โ1DC .
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VCG๐Add โค VCG๐+๐โ1
DC
โฆ
(Assume wlog unique optimal allocation)
๐ bidders whose values for ๐ are i.i.d. samples from ๐น๐ .
At most one is allocated by VCG๐+๐โ1DC .
VCG๐Add(๐) =
2nd highest
of ๐ samplesfrom ๐น๐
VCG๐+๐โ1DC (๐)
Highest value
of unallocated
bidder for ๐
โคโค
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Extension to downward closed constraints
Rev๐Addโค VCG๐+๐ถ
Add
Competitioncomplexity โค ๐ถ
Rev๐DC โค
Larger outcomespace
โค VCG๐+๐ถ+๐โ1DC
The competition complexity of ๐ additive bidders s.t.identical downward closed constraints over ๐ items is โค ๐ถ +๐ โ 1. 122
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Extension to downward closed constraints
Rev๐Addโค VCG๐+๐ถ
Add
Competitioncomplexity โค ๐ถ
Rev๐DC โค
Larger outcomespace
โค VCG๐+๐ถ+๐โ1DC
The competition complexity of ๐ additive bidders s.t.identical downward closed constraints over ๐ items is โค ๐ถ +๐ โ 1.
Proved!
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A note on tractability
VCG is not computationally tractable for general downward closed constraints. However:
โข VCG is tractable for matroid constraints
โข Competition complexity is meaningful in its own right
โข Can apply our techniques with โmaximal-in-range VCGโ by restricting outcomes to matchings
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Further extensions (preliminary)
1. From competition complexity to approximation
โ In large markets (๐ โซ ๐), 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
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Summary
โข Major open problem: Revenue maximization for ๐ items
โข B-K approach: Add competing bidders and maximize welfare
โข Results in: First robust simple mechanisms with provably high revenue for many complex settings
โข Techniques: Bayesian analysis, combinatorial arguments
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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Open questions
โข Tighter bounds and tradeoffs โ Settings with constant competition complexityโ Partial data on distributions, or large marketsโ Different duality based upper bound?
โข More general settingsโ Beyond downward closed constraintsโ Irregular distributionsโ Affiliation [Bulow-Klempererโ96]
โข Beyond VCG โ Posted-price mechanisms
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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