Download - The Price of Anarchy on Boston road 13 th Statphy workshop. Aug 11, 2005 NECSI summer school 2005 HyeJin Youn ( KAIST ) Fabian Roth ( ETH, Switzerland.

The Price of Anarchy on Boston road

13th Statphy workshop. Aug 11, 2005

NECSI summer school 2005

HyeJin Youn (KAIST)

Fabian Roth (ETH, Switzerland)

Matthew Silver (MIT)

Marie-Helen Cloutier (Canada)

Peter Ittzes (Collegium Budapest)

Hawoong Jeong(KAIST)

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http://www.kaist.ac.kr/index.html

A basic traffic problem

• agents from S to T at minimum cost

S T

C(x) = Ax+B

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Latency function C(X) = AX + B


Two Optimization Strategies• Two types of mindsets

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Decentralised control: Each agent minimizes

personal cost

There always exists a user-equilibrium/Nash equilibrium (Beckmann 1956)

Global Optimisation

User optimizations

Centralised controlMinimising Global Cost


The “Price of Anarchy”

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Decentralised control: Each agent minimizes

personal cost

There always exists a user-equilibrium/Nash equilibrium (Beckmann 1956)

Global Optimum

User Optimum

Centralised controlMinimising Global Cost

Price of Anarchy

Koutsoupias & Papadimitriou, 1999

Price of Anarchy <= 4/3 (Roughgarden & Tardos, 2000)

• Examples: Road Traffic, Network Routing, Prisoners Dilemma

Price of Anarchy: Simple Example

S E

C=10

C(X) = X

Global Optimum = ?

10 Agents from S EC = latency function (cost)

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Global Optimum


S E

C=10

C(X) = X

Global Optimum = 5x10 + 5x5 = 75

X = 5

X = 5

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Global Optimum


S E

C=10

C(X) = X

User Equilibrium = ?

X = 5

X = 5

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User Optimum



S E

C=10

C(X) = X

X = 5 + 1

X = 5 - 1

+1

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User Optimum

user cost = 5 + 1 < 10



S E

C=10

C(X) = X

X = 6 + 1

X = 4 - 1

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User Optimum

again+1

user cost = 6 + 1 < 10



S E

C=10

C(X) = X

X = 8

X = 2

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User Optimum

user cost = 7 + 1 < 10


again+1


S E

C=10

C(X) = X

X = 9

X = 1

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User Optimum

user cost = 8 + 1 < 10


again+1


S E

C=10

C(X) = X

He is indifferent: C = 9 + 1 = 10

X = 10

X = 0

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User Optimum



S E

C=10

C(X) = X

User Equilibrium = 10 x10 = 100

X = 10

X = 0

Global Optimum = 5x10 + 5x5 = 75

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User Optimum


4/3= upper bound of Price of Anarchy


Braess’s Paradox

S T

x

x1

1

0

Send 1 Unit of Flow

User Equilibrium without middle arc = 1.5

User Equilibrium with middle arc = 2

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Increasing user optimum at extra cost

Price of Anarchy = 2/1.5 = 4/34/3


Simulation Questions

• Price of Anarchy on a real world– the Boston Road Network

• Control factors– # of Agents– Topology

• Reducing the Price of Anarchy without raising Global Optimum– Semi-centralised control (Akella et al, ~2004)

– Network Redesign: Destroy Arcs (Braess’s paradox)

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Boston Road Map

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Boston Road Network

Start

End

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(node 59, edges 108, regular-like ) Latency function = ax + b

Width1, 2, 3 length


User Equilibrium Global Optimum

Number of Agents: 1

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Number of Agents: 2

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Number of Agents: 3

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Number of Agents: 4

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Number of Agents: 10

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Number of Agents: 5

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Number of Agents: 6

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Number of Agents: 7

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Number of Agents: 8

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Number of Agents: 9

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Variation of POA with Agent #

# of Agents

POA

Reminder: POA = UE/GO

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Affect of Arc Removal on UE

Arc

Total Agent Cost

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Affect of an Arc Removal on UE

Severe increase

Increase

Mild to no increase

Decrease

Start

End

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Conclusions• Price of Anarchy on a real world

– the Boston Road Network• Control factors

– # of Agents• Reducing the Price of Anarchy without raising Global Optimum

– Network Redesign: Destroy Arcs (Braess’s paradox)

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Flow from to Central Square to Copley Square could be improved by removing some streets

• Importance of Dynamics of fitness landscape ( how topology matters? )• Removal of a node flattening rugged fitness landscape

–Enlarging search spaces –how to map on prisoner’s dilemma–prisoner’s dilemma get agents better when they look further.

but traffic doesn’t have such a benefit to cooperators ( tax? )