The Price of Anarchy on Complex Networks

The Price of Anarchy on Complex Networks

KIAS Conference July, 2006

HyeJin Youn, Hawoong Jeong

Complex Systems & Statistical Physics Lab.

(Dept. of Physics, KAIST, Korea)

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

Marriage map between 100 richest people in Korea

Importance of networks & dynamics

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Protein Interaction Network

H.Jeong et al (2001)

Importance of networks & dynamics

Internet routing

∝ 1/width

Network Dynamics• “States” of both the nodes and the edges can change

• Dynamics of the networks : The topology of the network itself often evolves in time

• Dynamics on the networks : Agents are moving on the networks (E.g. Zero-range process, Contact process, Cascading failure, Shortest paths & OPTIMAL PATH)

∝ # of travelers

∝ length

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Latency function (like time or cost per person)

lengthwidth

Latency travelersof #1


Network flow with congestion

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Based on the model of Roughgarden & Tardos, 2000

S

Cost function on path i

Latency function

T

width of path i

length of path i

# of agent on path i

Given network with many agents going from S (source) to T (target), what will be the optimized distribution of agents for best performance??


Optimizations in physics

• Euler-Lagrange differential equation• minimal free energy in thermodynamic physics• Fitting experimental DATA with formula• Low temperature behavior of disordered magnets• …

Centralized controlMinimizing Global Cost

Centralized controlMinimizing Global Cost

Decentralized controlEach agent minimizes its own personal cost

Decentralized controlEach agent minimizes its own personal cost

User Optimization(Nash equilibrium)User Optimization(Nash equilibrium)

Global OptimizationGlobal Optimization

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• There are two types of optimizations!!!


The “Price of Anarchy”

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Koutsoupias & Papadimitriou, 1999

Price of Anarchy (Roughgarden & Tardos, 2000)

1 ≤

total cost of

Centralized controlMinimizing Global

Cost

Decentralized control

Each agent minimizes

its own personal cost

total cost ofGlobal Optimum

User OptimumPrice of AnarchyPrice of Anarchy

“Price we have to pay not being coordinated by central agency”“Price of being selfish”


Price of Anarchy: Contrived Example

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Pigous’s example: Congestion sensitive network

S T

What will be the min total cost, i.e. Global Optimum = ?

10 agents want toGo from S to T.

If xa=x, then xb=10-x, ∴ total cost=10ᆞ x +

(10-x) ᆞ (10-x) = x2-10x+100=(x-5)2+75∴ xa=xb=5 with total cost 75



Global Optimum = 5x10 + 5x5 = 75

75/10 = 7.5min driving in average CSSPL

xa = xb =5

S T

Envy

BUTBUT

The upper agents get envious of people with lower costs!


What will be the User Optimum?

(Nash Equilibrium: everyone happy)CSSPL


xa = 5

xb = 5S T


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user cost = 5 + 1 < 10


Move toLower path

+1 S T

xa = 5-1

xb = 5+1


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again+1

S T

xa = 4-1

xb = 6+1

user cost = 6 + 1 < 10


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again+1

S T

xa = 3-1

xb = 7+1

user cost = 7 + 1 < 10


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again+1

S T

xa = 2-1

xb = 8+1

user cost = 8 + 1 < 10


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User Optimum = 10 x10 = 100

avg 10min travel time > avg 7.5-min travel time

again+1

S T

xa = 1-1

xb = 9+1


User Optimum = 10 x10 = 100Global Optimum = 5x10 + 5x5 = 75

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4/3 Price of Price of Anarchy!Anarchy!

S T

xa = 5 vs 0

xb = 5 vs 10

There is a gap between global optimum & user optimum!


More realistic/complex example

• Assumption: traffic reaches at equilibrium

• Price of Anarchy on a real world– the Boston Road Network– (with real geometrical information like w

idth, length, one-way etc)– Traffic from central square (S) to copley

square (T)

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Boston Road Map

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Boston Road Network

Start

End

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(nodes 59, edges 108, regular-like)

Latency function = ax + b

lengthWidth


More realistic/complex example• Assumption: traffic reaches at equilibrium

• Price of Anarchy on a real world– the Boston Road Network– (with real geometrical information)

• Global optimum : mapping to Min-cost Max-flow problem

• User optimum ~ approximate optimum: Metropolis Algorithm and Annealing method to find out the optimum configurations

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Number of traveler =1

User Optimum Global Optimum


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Number of Agents: 20

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Congestion distribution on the edges


Reminder: POA = UE/GO

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Variation of POA with Agent #

number of agents

Pri

ce o

f A

narc

hy


Why Price of Anarchy decreases?

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• Fitness landscape for a simple case:

S T

l(xa)= 55

l(xb)= xb

Fitness for User Optimum

5

Strategy a

Strategy b

l(xb)= xb

Fitness for Global Optimum

cb (xb)= xb2

cb (xb)= 5xb

2.5

l(xa)= 5

l(xa)= 5

l(xb)= 2xb

2.5


POA too small??

• More general edge latency function– n > 1

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- Roughgarden-Tardos

• Linear latency function:

S T

C=1

C(X) = X^3

When n=3UO = 1GO = 0.37*1 + (0.63)^4 = 0.528

POA = UA/GO = 1.894 Bigger than 4/3 (n=1)

1

0.63

0.37

Nash Equilibrium 4/3 x (Global Optimum)Nash Equilibrium 4/3 x (Global Optimum)


Where to use??

• To write a paper …• Network design for better traffic?

Making network more efficient without central government??

• Lower PoA ~ better(?) system(∵even w/o central control, user optimum

is closer to global optimum, better!)

• Let’s make better network with lower PoA– Simple thought (by stupid government): constr

uct more roads with tax money! Braess paradox

(counter-intuitive consequences)

Braess’s Paradox

T

x

x10

10

0: cost-free express road

User Optimum without middle arc = 150 = Global Optimum

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Price of Anarchy = 200/150 = 4/34/3

increase

User Optimum with middle arc = 200

SS

Again 10 travelers want to move from S to T.


Boston Road Network

Start

End

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Affect of an Arc Removal on User Optimum

Start

End

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negative


0 20 40 60 80 1000

500

1000

1500

2000

2500

0 10 20 30 40 50 60-4

-2

0

2

4

6

8

10

12congestionnegative NEpositive NE

NE

Edge index

53 out of 108 edges are identified as deteriorating inefficiency! (ΔPoA<0)19 out of 53 edges are found having made the decentralised system cost more! (ΔNE<0)

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Affect of Arc Removal on User Optimum

edge index

Cost

in

crem

en

t

PoA=UO/GO


More systematic approaches

• Model network analysis– Regular Lattice– Erdos-Renyi Network– Small-world Network– Scale-free Network

• Multiple Sources & Targets

• Any correlation between PoA and other topological quantities?

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PoAC network representation

SW(N=100, r=3, p=0,0.1)

Regular Lattice (N=100)

Number of Agents = 60

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PoAC network representation

ER(N=100, k=6) BA(N=100, m3)

Thick and black edge: x+10 (wide and long)Thin and grey: 10x+1 size of node: PoAC method of spreading: using Kamada-Kawai(free) in Pajek except SW

Number of Agents = 60

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PoA (s-t pair)

<PoA>s-t pairs

Num

ber

densi

ty Number of agents = 60

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PoAC distribution

Number of agents = 60

PoA centrality

Num

ber

densi

ty

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<PoA>all S-T pairs, network

Number of Agents

<PoA

>

SW worst in POA!

SF good!RL best!

ER bad!

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BC(s)*BC(t)

<PoA

>bc(

s)bc

(t)


PoA (s-t pair) BC correlation

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k(S)*k(T)

<PoA

>k(S

)k(T

)


PoA (S-T pair) degree correlation

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Summary & Conclusion• Price of Anarchy on a network : price that a decentralized system s

hould pay for not being coordinated, • can be understood as a measure of inefficiency of the system.

• Price of Anarchy on a real world (Boston Road Network) - It is small, but it does exist!

• Reducing the Price of Anarchy

- network modification (Braess’s paradox)- Structural guidance of selfish users to the global optimiStructural guidance of selfish users to the global optimi

zed zed • Efficiency in traffic dynamics: RL>BA>ER>SW??• Correlation with topological properties? Degree?• More works are ongoing…

Flow from to Central Square to Copley Square could be improved by removing some streets (NOT adding new streets!)

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Job opening at KAIST

• Funding: 2nd phase Brain Korea 21 Project• Several PostDoc & Research Professor positi

ons are available in many fields.• For more information, please contact

H. Jeong ([email protected])

The Price of Anarchy on Complex Networks

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