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Intrinsic Robustness ofthe Price of Anarchy

Tim RoughgardenJuly 3,2013

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Basic Knowledge

• PNE

• Optimal Solution Improvement upon given dictatorial control over everyone’s actions

• Price of Anarchy

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Introduction

• Why need more robust bounds? • Hard to coordinate on one of multiple Equilibrium • PNE is computationally intractable • PNE does not exist

Need a more robust bounds to some wider range of outcome

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Basic Knowledge

• MNE Ex: “Rock-Paper-Scissors”

[Always exist/hard to compute]

• CorEq [Easy to compute/hard to learn]

• No Regret [CCE] [Easy to compute /learn]

PNE

MNE

CorEq

No Regret [CCE]

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Smooth Game • Definition [on Cost-minimization game]

[

C(s)=

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Smooth Game

• Definition of Robust POA [on Cost-minimization game] ()

• Relaxation of Smoothness

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Smooth Game • Definition [on Payoff-Maximization Game]

• Definition of Robust POA

[

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Example & Non-Example

Example• Congestion Game With Affine Cost Function• Valid Utility Game• Simultaneous Second-Price Auctions

Non-Example• Network Formation Game• Symmetric Congestion Games with Singleton Strategies

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Example

Congestion Game With Affine Cost Function

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Example

Congestion Game With Affine Cost Function

We claim that Congestion Game With Affine Cost Function

(

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Example

Valid Utility Game

Definition of “Valid”:

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Example

Valid Utility Game

We claim that Congestion Game With Affine Cost Function

( 𝑓𝑟𝑜𝑚 h𝑡 𝑒𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑎𝑟𝑖𝑡𝑦 𝑜𝑓 𝑉 )

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Example

Simultaneous Second-price Auctions

Each good is allocated independent, at a price equal to the second highest price

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Example

Simultaneous Second-price Auctions

This game satisfies the following relaxation of (1,1)-smoothness

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Example

Simultaneous Second-price Auctions [the relaxed smoothness condition]

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Tight Class of Game

Definition

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Extension Theorem

Static Version

,

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Extension Theorem

Repeat Play and No-Regret Sequences

𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑎𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑠1 ,𝑠2 ,… ,𝑠𝑇𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠𝑜𝑓 𝑎(𝜆 , µ)−𝑠𝑚𝑜𝑜𝑡h𝑔𝑎𝑚𝑒

(h 𝑦𝑝𝑜𝑡 h𝑒𝑡𝑖𝑐𝑎𝑙𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡𝑜𝑓 𝑝𝑙𝑎𝑦𝑒𝑟 𝑖 𝑖𝑛𝑡𝑖𝑚𝑒𝑡)

(𝑏𝑦 𝑡 h𝑒𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛𝑜𝑓 𝑠𝑚𝑜𝑜𝑡 h𝑛𝑒𝑠𝑠)

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Extension Theorem

Repeat Play and No-Regret Sequences

𝑖 𝑓 𝑤𝑒 𝑐𝑜𝑛𝑐𝑒𝑟𝑛𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑖𝑛𝑒𝑣𝑒𝑟𝑦 𝑝𝑙𝑎𝑦𝑒𝑟𝑠𝑒𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒𝑠 h      𝑣𝑎𝑛𝑖𝑠 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑒𝑔𝑟𝑒𝑡 :

(

(𝑛𝑜𝑟𝑒𝑔𝑟𝑒𝑡𝑚𝑎𝑘𝑒𝑠 h𝑡 𝑒 𝑙𝑎𝑠𝑡 𝑡𝑒𝑟𝑚𝑔𝑜𝑒𝑠 𝑡𝑜0𝑎𝑠𝑇→ 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦 )

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Extension Theorem

Repeat Version

(𝑎𝑠𝑇→ 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦 ) (𝑝𝑟𝑖𝑐𝑒𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 h𝑎𝑛𝑎𝑟𝑐 𝑦 )

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Extension Theorem

Repeat Version (Mixed-Strategy )

(𝑎𝑠𝑇→ 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦 )

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Approximate Equilibria

• Approximate Equilibria (cost-minimization games )

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Bicriteria Bound

• Smooth Closed Sets of Cost-Minimization Games 𝒢 :𝑎𝑠𝑒𝑡 𝑜𝑓 𝑐𝑜𝑠𝑡𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛𝑔𝑎𝑚𝑒𝑠 h𝑡 𝑎𝑡 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑𝑢𝑛𝑑𝑒𝑟 𝑝𝑙𝑎𝑦𝑒𝑟 𝑑𝑒𝑙𝑒𝑡𝑖𝑜𝑛𝑠𝑎𝑛𝑑𝑑𝑢𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑠𝐺∈𝒢 :𝐺 h h𝑤 𝑖𝑐 𝑖𝑠 ( 𝜆 ,𝜇 )− h𝑠𝑚𝑜𝑜𝑡�̂� :𝐺𝑎𝑚𝑒 h𝑡 𝑎𝑡 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚𝐺𝑏𝑦𝑑𝑢𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑛𝑔 h𝑒𝑎𝑐 𝑝𝑙𝑎𝑦𝑒𝑟 𝑖𝑛𝑖 𝑡𝑖𝑚𝑒𝑠

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Other Topic in this paper

• Congestion games Are Tight [To General Case]• Shortest Best-Response Sequencing (Best-Response Dynamics)

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Subsequent Work

• Guarantees with Irrational Players• Relaxing the Smoothness Condition• The POA in Games of Incomplete Information• Limits of Smoothness

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Thanks for your attention!