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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
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, âŚ
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
3
Revenue maximization
⢠Revenue less understood than welfare
â (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
â (even for welfare, some computational issues persist)
⢠Optimal truthful mechanism known only for handful of complex settings (e.g. additive buyer with 2 items, 6 uniform i.i.d. items... [Giannakopolous-Koutsoupiasâ14,â15])
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
5
Revenue maximization
⢠Revenue less understood than welfare
â (even for welfare, some computational issues persist)
⢠Optimal truthful mechanism known only for handful of complex settings (e.g. additive buyer with 2 items, 6 uniform i.i.d. items... [Giannakopolous-Koutsoupiasâ14,â15])
⢠Common CS solution for complexity: approximation
â [Hart-Nisanâ12,â13, Li-Yaoâ13, Babioff-et-al.â14, Rubinstein-Weinbergâ15, Chawla-Millerâ16, âŚ]
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
6
Revenue maximization
⢠Revenue less understood than welfare
â (even for welfare, some computational issues persist)
⢠Optimal truthful mechanism known only for handful of complex settings (e.g. additive buyer with 2 items, 6 uniform i.i.d. items... [Giannakopolous-Koutsoupiasâ14,â15])
⢠Common CS solution for complexity: approximation
â [Hart-Nisanâ12,â13, Li-Yaoâ13, Babioff-et-al.â14, Rubinstein-Weinbergâ15, Chawla-Millerâ16, âŚ]
⢠Resource augmentationCompetition Complexity of Auctions
Eden et al. EC'17 Inbal Talgam-Cohen7
Single item welfare maximization
Run a 2nd price auction âsimple, maximizes welfare âpointwiseâ.
(VCG mechanism)
đŁ1
đŁ2
đŁđ
âĽ
âĽ
âĽ
.
.
.
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
8
Single item welfare maximization
Run a 2nd price auction âsimple, maximizes welfare âpointwiseâ.
(VCG mechanism)
đŁ1
đ = đŁ2
đŁđ
âĽ
âĽ
âĽ
.
.
.
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
9
Single item revenue maximization
Single buyer: select price that
maximizes đ â 1 â đš đ
(âmonopoly priceâ).đŁ1 âź đš
Price = đ
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
10
Single item revenue maximization
Single buyer: select price that
maximizes đ â 1 â đš đ
(âmonopoly priceâ).
Multiple i.i.d. buyers: run 2nd price auction with reserve price đ (same đ).
(Myersonâs auction)
đŁ1 âź đš
đŁ2 âź đš
đŁđ âź đš
âĽ
âĽ
âĽ
Price ⼠đ
.
.
.
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
11
Single item revenue maximization
Single buyer: select price that
maximizes đ â 1 â đš đ
(âmonopoly priceâ).
Multiple i.i.d. buyers: run 2nd price auction with reserve price đ (same đ).
(Myersonâs auction)
đŁ1 âź đš
đŁ2 âź đš
đŁđ âź đš
âĽ
âĽ
âĽ
Price ⼠đ
.
.
.
.
.
.
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 đ â 1 â đš đ
(âmonopoly priceâ).
Multiple i.i.d. buyers: run 2nd price auction with reserve price đ (same đ).
(Myersonâs auction)
đŁ1 âź đš
đŁ2 âź đš
đŁđ âź đš
âĽ
âĽ
âĽ
Price ⼠đ
.
.
.
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 ⼠Expected revenue of the optimal 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 ⼠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. âThe statistics of the data shifts rapidlyâ [Google]
Simple! âHardly anything matters moreâ [Milgromâ04]
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
15
Multidimensional settings
đš
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
16
Multidimensional settings
đš1
đš2
đš3
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
17
Multidimensional settings
đš1
đš2
đš3
Biddersâ values are sampled i.i.d. from a product distribution over items
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
18
Multidimensional settings
đš1
đš2
đš3
Additive: đŁ( , )=đŁ( ) )+đŁ( )
Biddersâ values are sampled i.i.d. from a product distribution over items
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
19
Multidimensional settings
⢠Revenue maximization is not well understood:
⢠Optimal mechanism mightnecessitate randomization.
⢠Non-monotone.
⢠Computationally intractable.
⢠Only recently, simple approximately optimal mechanisms were devised.
đš1
đš2
đš3
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
20
Multidimensional settings
Either run a randomized,
đš1
đš2
đš3
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
21
Multidimensional settings
Either run a randomized,
hard to compute,đš1
đš2
đš3
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,
đš1
đš2
đš3
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 distributionsâŚ
đš1
đš2
đš3
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 distributionsâŚ
Or add more bidders.
đš1
đš2
đš3
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 đi.i.d. bidders. What is đ such that the revenue of VCGwith đ + đ bidders is ⼠OPT with đ bidders.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
27
Multidimensional B-K theorems
Bulow-Klemperer Thm. The competition complexity of a single item auction is 1.
Competition complexity: Fix an environment with đi.i.d. bidders. What is đ such that the revenue of VCGwith đ + đ bidders is ⼠OPT with đ bidders.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
28
Multidimensional B-K theorems
Bulow-Klemperer Thm. The competition complexity of a single item auction is 1.
Competition complexity: Fix an environment with đi.i.d. bidders. What is đ such that the revenue of VCGwith đ + đ bidders is ⼠OPT with đ 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 đ copies is đ.
Competition complexity: Fix an environment with đi.i.d. bidders. What is đ such that the revenue of VCGwith đ + đ bidders is ⼠OPT with đ bidders.
Thm. [EFFTW] The competition complexity of đadditive bidders drawn from a product distribution over đ items is ⤠đ + đ(đâ đ).
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
30
Multidimensional B-K theorems
Thm. [EFFTW] Let đŞ be the competition complexity of đadditive bidders over đ items. The competition complexity of đ additive bidders with identical downward closed constraints over đ items is ⤠đŞ +đâ đ.
Competition complexity: Fix an environment with đi.i.d. bidders. What is đ such that the revenue of VCGwith đ + đ bidders is ⼠OPT with đ bidders.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
31
Multidimensional B-K theorems
Thm. [EFFTW] Let đŞ be the competition complexity of đadditive bidders over đ items. The competition complexity of đ additive bidders with randomly drawn downward closed constraints over đ items is ⤠đŞ+ đ(đ â đ).
Competition complexity: Fix an environment with đi.i.d. bidders. What is đ such that the revenue of VCGwith đ + đ bidders is ⼠OPT with đ 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
â Specifies which item sets are feasible
⢠Bidderâs 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 âmatroidâ constraints: Only sets of size đ = 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 âdownward closedâ constraints: Sets of size 1 and { } are feasible
Complements in what sense?
⢠No complements = gross substitutes:
â ÔŚđ ⤠Ԍđ item prices
â đ in demand( ÔŚđ) if maximizes utility đŁđ đ â đ(đ)
â âđ in demand( ÔŚđ), there is đ in demand( ÔŚđ) with every item in đ whose price didnât increase
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
37
$
$$
$
đş
đť
Complements in what sense?
⢠No complements = gross substitutes:
â ÔŚđ ⤠Ԍđ item prices
â đ in demand( ÔŚđ) if maximizes utility đŁđ đ â đ(đ)
â âđ in demand( ÔŚđ), there is đ in demand( ÔŚđ) with every item in đ whose price didnât increase
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
38
$đş
đť 5 6
10 ÔŚđ = (5, đ, đ)
Competition complexity â summary Upper boundValuation
đ + 2 đ â 1Additive
đ + 3 đ â 1Additive s.t. identical downward closed constraints
đ + 4 đ â 1Additive s.t. random downward closed constraints
đ + 2 đ â 1 + đAdditive s.t. identical matroidconstraints
Lower bounds of Ί đ â logđ
đ+ 1 for additive bidders and Ί đ for unit demand
bidders are due to ongoing work by [Feldman-Friedler-Rubinstein] and to [Bulow-Klempererâ96]
Related workMultidimensional B-K theorems
[Roughgarden T. Yan â12]: for unit demand bidders, revenue of VCG with đ extra bidders ⼠revenue of the optimal deterministic DSIC mechanism.
[Feldman Friedler Rubinstein â ongoing]: tradeoffs between enhanced competition and revenue.
Prior-independent multidimensional mechanisms
[Devanur Hartline Karlin Nguyen â11]: unit demand bidders.
[Roughgarden T. Yan â12]: unit demand bidders.
[Goldner Karlin â16]: additive bidders.
Sample complexity
[Morgenstern Roughgarden â16]: how many samples needed to approximate the optimal mechanism?
MULTIDIMENSIONAL B-K THEOREMPROOF 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 ⼠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 ⼠Revenue of the optimal auction with n bidders.
Proof. (in 3 steps of [Kirkegaardâ06])
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 ⼠Revenue of the optimal auction with n bidders.
Proof. (in 3 steps of [Kirkegaardâ06])
I. Upper-bound the optimal revenue.
II. Find an auction đ´ with more bidders and revenue ⼠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 ⼠Revenue of the optimal auction with n bidders.
Proof. (in 3 steps of [Kirkegaardâ06])
I. Upper-bound the optimal revenue.
II. Find an auction đ´ with more bidders and revenue ⼠the upper bound.
III. Show that the 2nd price auction âbeatsâ đ´.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
45
Proof:
Step I. Upper-bound the optimal revenue.
đŁ1 âź đš
đŁ2 âź đš
đŁđ âź đš
âĽ
âĽ
âĽ
.
.
.
Price ⼠đ
Myersonâs optimal mechanism
.
.
.
46
Proof:
Step II. Find an auction đ´with more bidders and revenue ⼠the upper bound.
đŁ1 âź đš
đŁ2 âź đš
đŁđ âź đš
.
.
.
đŁđ+1 âź đš
.
.
.
47
Proof:
Step II. Find an auction đ´with more bidders and revenue ⼠the upper bound.
đŁ1 âź đš
đŁ2 âź đš
đŁđ âź đš
.
.
.
đŁđ+1 âź đš
Run Myersonâsmechanism onđ bidders
.
.
.
48
Proof:
Step II. Find an auction đ´with more bidders and revenue ⼠the upper bound.
đŁ1 âź đš
đŁ2 âź đš
đŁđ âź đš
.
.
.
đŁđ+1 âź đš
Run Myersonâsmechanism onđ bidders
If Myerson does not allocate, give item to the additionalbidder
.
.
.
49
Proof:
Step III. Show that the 2nd
price auction âbeatsâ đ´.
Observation. 2nd price
auction is the optimal mechanism out of the mechanisms that always sell.
đŁ1 âź đš
đŁ2 âź đš
đŁđ âź đš
.
.
.
đŁđ+1 âź đš
.
.
.
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 đ´ with more bidders and revenue ⼠the upper bound.
III. Show that VCG âbeatsâ đ´.
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 đ´ with more bidders and revenue ⼠the upper bound.
III. Show that VCG âbeatsâ đ´.
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
đŁ1 âź đš
đŁ2 âź đš
đŁđ âź đš
.
.
.
.
.
.
I. Upper-bound the optimal revenue
Use the duality framework from [Cai DevanurWeinberg â16].
OPT â¤
EđŁâźđšđ
đ
đ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛+ đŁđ â 1âđⲠđŁđ<đŁđâ˛
đ đŁ = đŁ â1âđš đŁ
đ(đŁ)is the virtual valuation function.54
I. Upper-bound the optimal revenue
Use the duality framework from [Cai DevanurWeinberg â16].
OPT â¤
EđŁâźđšđ
đ
đ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛+ đŁđ â 1âđⲠđŁđ<đŁđâ˛
đ đŁ = đŁ â1âđš đŁ
đ(đŁ)is the virtual valuation function.55
Distribution appears in proof only!
I. Upper-bound the optimal revenue
Use the duality framework from [Cai DevanurWeinberg â16].
OPT â¤
EđŁâźđšđ
đ
đ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛+ đŁđ â 1âđⲠđŁđ<đŁđâ˛
Take item đâs virtual value if itâs the mostattractive item
56 đ đŁ = đŁ â1âđš đŁ
đ(đŁ)is the virtual valuation function.
I. Upper-bound the optimal revenue
Use the duality framework from [Cai DevanurWeinberg â16].
OPT â¤
EđŁâźđšđ
đ
đ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛+ đŁđ â 1âđⲠđŁđ<đŁđâ˛
Take item đâs value if thereâs a more attractive item
57 đ đŁ = đŁ â1âđš đŁ
đ(đŁ)is the virtual valuation function.
II. Find an auction đ´ with more bidders and revenue ⼠upper bound
58
II. Find an auction đ´ with đ bidders and revenue ⼠upper bound
59
II. Find an auction đ´ with đ bidders and revenue ⼠upper bound
VCG for additive bidders ⥠2nd price auction for each item separately.
Therefore, we devise a single parameter mechanism that covers item đâs contribution to the benchmark.
EđŁâźđšđ đ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛+ đŁđ â 1âđⲠđŁđ<đŁđâ˛
60
II. Find an auction đ´ đ with đ bidders and revenue ⼠upper bound for item đ
EđŁâźđšđ đ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛+ đŁđ â 1âđⲠđŁđ<đŁđâ˛
Run 2nd price auctionwith âlazyâ reserve price =
đâ1 0 for agent đ
0 for agents đⲠâ đ
Item đ
đŁđ âź đš
.
.
.
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
61
đŁđ âź đš
đŁ1 âź đš
EđŁâźđšđ đ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛+ đŁđ â 1âđⲠđŁđ<đŁđâ˛
Case I: đŁđ > đŁđⲠfor all đâ˛:
đ wins if his virtual value is
non-negative.
Expected revenue =
Expected virtual value
[Myersonâ81]
Item đđŁđ âź đš
đŁ1 âź đš
đŁđ âź đš
.
.
.
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
62
II. Find an auction đ´ đ with đ bidders and revenue ⼠upper bound for item đ
EđŁâźđšđ đ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛+ đŁđ â 1âđⲠđŁđ<đŁđâ˛
Case II: đŁđ < đŁđⲠfor some đâ˛:
The second price is at least
the value of agent đ.
Item đ
đŁđ âź đš
.
.
.
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
63
II. Find an auction đ´ đ with đ bidders and revenue ⼠upper bound for item đ
đŁđ âź đš
đŁ1 âź đš
III. Show that VCG âbeatsâ đ´
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
64
III. Show that 2nd price âbeatsâ đ´(đ)
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
65
III. Show that 2nd price âbeatsâ đ´(đ)
đ¨(đ) withđ bidders
â¤Myerson withđ bidders
â¤2nd price withđ+ đ bidders
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
66
III. Show that 2nd price âbeatsâ đ´(đ)
The competition complexity of a single additive bidder and đ i.i.d. items is ⤠đ.
FFCompetition Complexity of Auctions
Eden et al. EC'17 Inbal Talgam-Cohen67
đ¨(đ) withđ bidders
â¤Myerson withđ bidders
â¤2nd price withđ+ đ 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
đŁ1 âź đš1
đŁ2 âź đš2
đŁđ âź đšđ
.
.
.
.
.
.
Going beyond i.i.d items
Item đđŁđ âź đšđ
đŁ1 âź đšđ
đŁđ âź đšđ
.
.
.
.
.
.
E đŁ1âźđš1đŁ2âźđš2âŚđŁđâźđšđ
đđ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛
+ đŁđ â 1âđⲠđŁđ< đŁđâ˛
69
Run 2nd price auctionwith âlazyâ reserve price =
đâ1 0 for agent đ
0 for agents đⲠâ đ
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
Going beyond i.i.d items
Item đ
.
.
.
đŁđ âź đšđ
đŁ1 âź đšđ
đŁđ âź đšđ
.
.
.
E đŁ1âźđš1đŁ2âźđš2âŚđŁđâźđšđ
đđ+ đŁđ â 1âđⲠđŁđ>đŁđâ˛
+ đŁđ â 1âđⲠđŁđ< đŁđâ˛
Run 2nd price auctionwith âlazyâ reserve price = đâ1 0 for agent đ0 for agents đⲠâ đCannot couple the event âbidder đ winsâ and âitem đ has the highest valueâ
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen70
Use a different benchmark
Item đ
.
.
.
E đŁ1âźđš1đŁ2âźđš2âŚđŁđâźđšđ
đđ+ đŁđ â 1âđⲠđšđ(đŁđ)>đšđâ˛(đŁđâ˛)
+ đŁđ â 1âđⲠđšđ(đŁđ)<đšđâ˛(đŁđâ˛)
đŁđ âź đšđ
đŁ1 âź đšđ
đŁđ âź đšđ
.
.
.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen71
Use a different benchmark
Item đ
.
.
.
E đŁ1âźđš1đŁ2âźđš2âŚđŁđâźđšđ
đđ+ đŁđ â 1âđⲠđšđ(đŁđ)>đšđâ˛(đŁđâ˛)
+ đŁđ â 1âđⲠđšđ(đŁđ)<đšđâ˛(đŁđâ˛)
The competition complexity of a single additive bidder and đ items is ⤠đ.
đŁđ âź đšđ
đŁ1 âź đšđ
đŁđ âź đšđ
.
.
.
Going beyond a single bidder
⢠Step I:
â Benchmark more involved
⢠Step II:
â Devise a more complex single parameter auction A(j) (involves a max)
â Proving A(j) is greater than item jâs 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
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 âdownward closedâ constraints: Sets of size 1 and { } are feasible
Extension to downward closed constraints
OPTđAdd⤠VCGđ+đś
Add
Competitioncomplexity ⤠đś
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
76
Extension to downward closed constraints
OPTđAdd⤠VCGđ+đś
Add
Competitioncomplexity ⤠đś
OPTđDC â¤
Larger outcomespace
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
77
Extension to downward closed constraints
OPTđAdd⤠VCGđ+đś
Add
Competitioncomplexity ⤠đś
OPTđDC â¤
Larger outcomespace
⤠VCGđ+đś+đâ1DC
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
78
Extension to downward closed constraints
OPTđAdd⤠VCGđ+đś
Add
Competitioncomplexity ⤠đś
OPTđDC â¤
Larger outcomespace
⤠VCGđ+đś+đâ1DC
The competition complexity of đ additive bidders with identical downward closed constraints over đ items is ⤠đś +đ â 1.
Extension to downward closed constraints
OPTđAdd⤠VCGđ+đś
Add
Competitioncomplexity ⤠đś
OPTđDC â¤
Larger outcomespace
⤠VCGđ+đś+đâ1DC
The competition complexity of đ additive bidders with identical downward closed constraints over đ items is ⤠đś +đ â 1.
Main technical challenge
Claim. VCG revenue from selling đ items to đż = đ + đŞadditive bidders whose values are i.i.d. draws from đš
â¤VCG revenue from selling them to đż +đâ đ bidders with i.i.d. values drawn from đš, whose valuations are additive s.t. identical downward-closed constraints.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
81
VCGđAdd ⤠VCGđ+đâ1
DC
VCG for additive bidders ⥠2nd price auction for each item separately.
Therefore, the revenue from item đ in VCGđAdd =
2nd highest value out of đż i.i.d. samples from đđ.
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
82
VCGđAdd ⤠VCGđ+đâ1
DC
83
VCGđAdd ⤠VCGđ+đâ1
DC
84
5 âź đš 2 âź đš7 âź đš
VCGđAdd ⤠VCGđ+đâ1
DC
85
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 1286
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 1287
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 12
Claim. Revenue for item đ in
VCGđ+đâ1DC ⼠value of the
highest unallocated bidder for item đ.
88
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 1289
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 1290
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 1291
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 12
Externality at least 9
92
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 1293
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 1294
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 1295
5 âź đš 2 âź đš7 âź đš
3 46
4 15
3 24
VCGđAdd ⤠VCGđ+đâ1
DC
3 12
Externality at least 2
96
VCGđAdd ⤠VCGđ+đâ1
DC
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
97
VCGđAdd ⤠VCGđ+đâ1
DC
VCGđAdd(đ) =
2nd highest
of đ samplesfrom đšđ
VCGđ+đâ1DC (đ)
Highest value
of unallocated
bidder for đ
â¤
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
98
VCGđAdd ⤠VCGđ+đâ1
DC
VCGđAdd(đ) =
2nd highest
of đ samplesfrom đšđ
VCGđ+đâ1DC (đ)
Highest value
of unallocated
bidder for đ
â¤â¤
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
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VCGđAdd ⤠VCGđ+đâ1
DC
VCGđAdd(đ) =
2nd highest
of đ samplesfrom đšđ
VCGđ+đâ1DC (đ)
Highest value
of unallocated
bidder for đ
â¤â¤
Identify đ bidders in VCGđ+đâ1DC
before sampling their value for item đ out of which at most one will be allocated anything
Competition Complexity of Auctions Eden et al. EC'17 Inbal Talgam-Cohen
100
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
1 2 3 4 5 6 7 jmâŚ
101
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
1 2 3 4 5 6 7 jmâŚ
(Assume wlog unique optimal allocation)
102
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
1. Sample valuations for all items but đ.
1 2 3 4 5 6 7 jmâŚ
(Assume wlog unique optimal allocation)
103
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
2. Compute an optimal allocation without item đ.
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
104
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
2. Compute an optimal allocation without item đ.
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Set đ´ of allocatedbidders
Set ҧđ´ of unallocatedbidders
105
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
2. Compute an optimal allocation without item đ.
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Set đ´ of allocatedbidders
Set ҧđ´ of unallocatedbidders
If đ is allocated to bidder in ҧđ´ in OPT,
all other items are allocated as before.
106
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Sample values for đ for agents in đ´ and compute the optimal allocation where đ is allocated to a bidder in đ´ .
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
107
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
108
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Some items might be vacated due to feasibility
109
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Some items might be snatched from other agents
110
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Continue with this process
111
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´
1
2
3
4
5 6
7
j
m
(Assume wlog unique optimal allocation)
Continue with this process
112
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´. There are ⼠đ´ items
allocated to agents in đ´.
1
2
3 4 56
7
j
m
(Assume wlog unique optimal allocation)
113
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´. There are ⼠đ´ items
allocated to agents in đ´.â Map each agent whoâs item was snatched to the snatched item.
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
114
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´. There are ⼠đ´ items
allocated to agents in đ´.â Map each agent whoâs item was snatched to the snatched item.
â 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
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´. There are ⼠đ´ items
allocated to agents in đ´.â Map each agent whoâs item was snatched to the snatched item.
â Map each agent who took a vacated item to the item.
â Every agent who wasnât snatched and didnât take an itemhas the same allocation.
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
116
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´. There are ⼠đ´ items
allocated to agents in đ´.
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
⤠đ â |đ´| allocated
117
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
3. Compute OPTđâđ´. There are ⼠đ´ items
allocated to agents in đ´.
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
⤠đ â |đ´| allocated⼠ҧđ´ â đ â đ´= đ +đ â 1 â đ´ â
đ â đ´= đ â 1 unallocated
118
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
⤠đ â |đ´| allocated⼠ҧđ´ â đ â đ´= đ +đ â 1 â đ´ â
đ â đ´= đ â 1 unallocated
119
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
2
3 6
7
j
m
(Assume wlog unique optimal allocation)
1 4 5
⤠đ â |đ´| allocated⼠ҧđ´ â đ â đ´= đ +đ â 1 â đ´ â
đ â đ´= đ â 1 unallocated
đ bidders whose values for đ are i.i.d. samples from đšđ .
At most one is allocated by VCGđ+đâ1DC .
120
VCGđAdd ⤠VCGđ+đâ1
DC
âŚ
(Assume wlog unique optimal allocation)
đ bidders whose values for đ are i.i.d. samples from đšđ .
At most one is allocated by VCGđ+đâ1DC .
VCGđAdd(đ) =
2nd highest
of đ samplesfrom đšđ
VCGđ+đâ1DC (đ)
Highest value
of unallocated
bidder for đ
â¤â¤
121
Extension to downward closed constraints
RevđAdd⤠VCGđ+đś
Add
Competitioncomplexity ⤠đś
RevđDC â¤
Larger outcomespace
⤠VCGđ+đś+đâ1DC
The competition complexity of đ additive bidders s.t.identical downward closed constraints over đ items is ⤠đś +đ â 1. 122
Extension to downward closed constraints
RevđAdd⤠VCGđ+đś
Add
Competitioncomplexity ⤠đś
RevđDC â¤
Larger outcomespace
⤠VCGđ+đś+đâ1DC
The competition complexity of đ additive bidders s.t.identical downward closed constraints over đ items is ⤠đś +đ â 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 âmaximal-in-range VCGâ 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
â In large markets (đ ⍠đ), 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
⢠Major open problem: Revenue maximization for đ 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
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Open questions
⢠Tighter bounds and tradeoffs â Settings with constant competition complexityâ Partial data on distributions, or large marketsâ Different duality based upper bound?
⢠More general settingsâ Beyond downward closed constraintsâ Irregular distributionsâ Affiliation [Bulow-Klempererâ96]
⢠Beyond VCG â Posted-price mechanisms
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