Post on 31-Mar-2015
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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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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Price of Anarchy: Simple Example
10 Agents from S EC = latency function (cost)
Global Optimum
S E
C=10
C(X) = X
User Equilibrium = ?
X = 5
X = 5
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10 Agents from S EC = latency function (cost)
User Optimum
Price of Anarchy: Simple Example
S E
C=10
C(X) = X
X = 5 + 1
X = 5 - 1
+1
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10 Agents from S EC = latency function (cost)
User Optimum
user cost = 5 + 1 < 10
Price of Anarchy: Simple Example
S E
C=10
C(X) = X
X = 6 + 1
X = 4 - 1
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10 Agents from S EC = latency function (cost)
User Optimum
again+1
user cost = 6 + 1 < 10
Price of Anarchy: Simple Example
S E
C=10
C(X) = X
X = 8
X = 2
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10 Agents from S EC = latency function (cost)
User Optimum
user cost = 7 + 1 < 10
Price of Anarchy: Simple Example
again+1
S E
C=10
C(X) = X
X = 9
X = 1
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10 Agents from S EC = latency function (cost)
User Optimum
user cost = 8 + 1 < 10
Price of Anarchy: Simple Example
again+1
S E
C=10
C(X) = X
He is indifferent: C = 9 + 1 = 10
X = 10
X = 0
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10 Agents from S EC = latency function (cost)
User Optimum
Price of Anarchy: Simple Example
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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10 Agents from S EC = latency function (cost)
User Optimum
Price of Anarchy: Simple Example
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 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: 10
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User Equilibrium Global Optimum
Number of Agents: 6
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User Equilibrium Global Optimum
Number of Agents: 7
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User Equilibrium Global Optimum
Number of Agents: 8
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User Equilibrium Global Optimum
Number of Agents: 9
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User Equilibrium Global Optimum
Number of Agents: 15
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User Equilibrium Global Optimum
Number of Agents: 20
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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? )