Composable and Efficient mechanisms

The Price of Anarchy in AuctionsPart II: The Smoothness FrameworkJason HartlineNorthwestern UniversityVasilis SyrgkanisCornell UniversityDecember 11, 2013Part II: High-level goalsPoA in auctions (as games of incomplete information):

Single-Item First Price, All-Pay, Second Price Auctions

Simultaneous Single Item Auctions

Position Auctions: GSP, GFP

Combinatorial auctions

2General ApproachReduce analysis of complex setting to simple setting.

Conclusion for simple setting X, proved under restriction P, extends to complex setting Y

X: complete information PNE to Y: incomplete information BNE

X: single auction to Y: composition of auctions

3Best-Response Analysis4First Extension TheoremComplete info PNE to BNE with correlated values

5First Extension TheoremTarget setting. First Price Bayes-Nash Equilibrium with asymmetric correlated values

Simple setting. Complete information Pure Nash Equilibrium

Thm. If proof of PNE PoA based on own-value based deviation argument then PoA of BNE also good

Complete info PNE to BNE with correlated values6References:Roughgarden STOC09Lucier, Paes Leme EC11Roughgarden EC12Syrgkanis 12Syrgkanis, Tardos STOC13

First-Price Auction Refresher

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First-Price Auction Target: BNE with correlated values

8First-Price Auction Target: BNE with correlated values

9First-Price Auction Simpler: PNE and complete Information

10First-Price Auction Simpler: PNE and complete Information11

First-Price Auction Simpler: PNE and complete Information12By PNE condition

First-Price Auction Simpler: PNE and complete Information13

Direct extensions14Problem in previous PNE proof

15First-Price Auction Simpler: PNE and complete Information16

Recall PoA=1 ProofOwn-value deviations(price and other values oblivious)17

OROwn-value deviations(price and other values oblivious)18

Own-value deviations(price and other values oblivious)19

Own-value deviations(price and other values oblivious)20

Own-value deviations(price and other values oblivious)21

Key Deviation PropertySmoothness Property2223Note. Smoothness is property of auction not equilibrium24Applies to any auction: Not First-Price Auction specific25Finally2627]Just redo PNE proof in expectation over values.

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31First Extension TheoremComplete info PNE to BNE with correlated values32Second Extension Theorem Single auction to simultaneous auctions PNE complete information33Second Extension TheoremSingle auction to simultaneous auctions PNE complete information34References:Roughgarden STOC09Roughgarden EC12Syrgkanis 12Syrgkanis, Tardos STOC13

Simultaneous First-Price AuctionsUnit-demand bidders

Simultaneous First-Price AuctionsUnit-demand biddersSimultaneous First-Price Auctions

Simultaneous First-Price Auctions39BNE PoA?40But Half-way there41What we showed:

Second Extension TheoremSingle auction to simultaneous auctionsPNE complete information

42Third Extension Theorem Complete info PNE to BNE with independent values43Third Extension TheoremComplete info PNE to BNE with independent values44References:Christodoulou et al. ICALP08Roughgarden EC12Syrgkanis 12Syrgkanis, Tardos STOC13

Does this extend to BNE PoA?45Relax First Extension Theorem to allow for dependence on opponents values

To counterbalance: assume independent values

46BNE (independent valuations)

Need to construct feasible BNE deviations

Each player random samples the others values and deviates as if that was the true values of his opponents

Above works out, due to independence of value distributions47

BNE (independent valuations)

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BNE (independent valuations)49

BNE (independent valuations)Sum of deviating utilities50Sum of complete information setting deviating utilities

BNE (independent valuations)51By smoothness on the left

BNE (independent valuations)52Rest is easy53Third Extension TheoremComplete info PNE to BNE with independent values54Third Extension TheoremComplete info PNE to BNE with independent values55Direct approach: Arguing about distributions56Direct approachFocusing on complete info PNE, might be restrictive in some settings

Working with the distributions directly can potentially yield better bounds

Arguing about distributions57References:Feldman et al. STOC13

Single-item auction BNE

58Single-item auction BNE

59What does it buy usCorrelated deviating strategies across multiple auctionsDecomposition of deviation analysis to separate deviations imposes independent randomness

60What does it buy usCorrelation can achieve higher deviating utility

61What does it buy usCorrelation can achieve higher deviating utility

62Direct approachDrawing deviation from price distribution!

Buys correlation across auctions

Better bounds beyond submodular

Arguing about distributions63Second Price Payment Rules64

Second price$2$5$7$3$4Pays $5$99$0$0$0$0Pays $0

65Second Price and Overbidding66Smoothness of Vickrey Auction67Smoothness of Vickrey Auction68Sneak Peek of Examples69Generalized First-Price Auction

AdvertisersSlotsCTRs70Matching Markets Greedy Mechanism

Unit-Demand BiddersItems

71Single-Minded Combinatorial Auction

Single-Minded BiddersItems

7273ExamplesExamples74Generalized First-Price Auction

AdvertisersSlotsCTRs75Smoothness of GFP

AdvertisersSlotsCTRs76Smoothness of GFP

AdvertisersSlotsCTRs77Smoothness of GFP

AdvertisersSlotsCTRs78Matching Markets Greedy Mechanism

Unit-Demand BiddersItems

79Matching Markets Greedy Mechanism

Unit-Demand BiddersItems

80Matching Markets Greedy Mechanism

Unit-Demand BiddersItems

81Incentives improve algorithmic approximationGreedy on true values: 2-approx.

At equilibrium:Player 2 never goes for first itemToo expensiveSo allocation is efficient

Unit-Demand BiddersItems

82Single-Minded Combinatorial Auction

Single-Minded BiddersItems

83Optimal Algorithm

Single-Minded BiddersItems

84Linear inefficiency!

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Single-Minded BiddersItems

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Single-Minded BiddersItems

87Bad Example Corrected

Large players cannot block all small players88Smoothness of Approximation Algorithm

Single-Minded BiddersItems

89Smoothness of Approximation Algorithm90Approximation improves efficiency91Some ReferencesSmoothness Roughgarden STOC09, Lucier, Paes Leme EC11, Roughgarden EC12, Syrgkanis 12, Syrgkanis, Tardos STOC13 Simultaneous First-Second Price Single-Item AuctionsBikhchandani GEB96, Christodoulou, Kovacs, Schapira ICALP08, Bhawalkar, Roughgarden SODA11, Hassidim, Kaplan, Mansour, Nisan EC11, Feldman, Fu, Gravin, Lucier STOC13Auctions based on Greedy Allocation AlgorithmsLucier, Borodin SODA10AdAuctions (GSP, GFP)Paes-Leme Tardos FOCS10, Lucier, Paes-Leme + CKKK EC11Sequential First/Second Price AuctionsPaes Leme, Syrgkanis, Tardos SODA12, Syrgkanis, Tardos EC12Multi-Unit Auctions Bart de Keijzer et al. ESA13All above can be thought as smoothness proofs and some are compositions of auctionsThis conferencePrice of Anarchy in Auctions and MechanismsDutting, Henzinger, Stanberger. Valuation Compressions in VCG-Based Combinatorial AuctionsJose R. Correa, Andreas S. Schulz and Nicolas E. Stier-Moses. The Price of Anarchy of the Proportional Allocation Mechanism RevisitedJason Hartline, Darrell Hoy and Sam Taggart.Interim Smoothness for Auction Welfare and Revenue. (poster)Michal Feldman, Vasilis Syrgkanis and Brendan Lucier. Limits of Efficiency in Sequential AuctionsBrendan Lucier, Yaron Singer, Vasilis Syrgkanis and Eva Tardos. Equilibrium in Combinatorial Public ProjectsPrice of Anarchy in GamesXinran He and David Kempe. Price of Anarchy for the N-player Competitive Cascade Game with Submodular Activation FunctionsMona Rahn and Guido Schfer. Bounding the Inefficiency of Altruism Through Social Contribution GamesYoram Bachrach, Vasilis Syrgkanis and Milan Vojnovic. Incentives and Efficiency in Uncertain Collaborative Environments