The Price of Anarchy on Boston road 13 th Statphy workshop. Aug 11, 2005 NECSI summer school 2005...
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The Price of Anarchy on Boston road 13 th Statphy workshop. Aug 11, 2005 NECSI summer school 2005 HyeJin Youn (KAIST) Fabian Roth (ETH, Switzerlan d) Matthew Silver (MIT) Marie-Helen Cloutier (Canad a) Peter Ittzes (Collegium Buda pest) Hawoong Jeong(KAIST) CSSPL
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Transcript of The Price of Anarchy on Boston road 13 th Statphy workshop. Aug 11, 2005 NECSI summer school 2005...
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)
CSSPL
A basic traffic problem
• agents from S to T at minimum cost
S T
C(x) = Ax+B
CSSPL
Latency function C(X) = AX + B
Two Optimization Strategies• Two types of mindsets
CSSPL
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”
CSSPL
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)
CSSPL
Global Optimum
S E
C=10
C(X) = X
Global Optimum = 5x10 + 5x5 = 75
X = 5
X = 5
CSSPL
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
CSSPL
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
CSSPL
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
CSSPL
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
CSSPL
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
CSSPL
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
CSSPL
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
CSSPL
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
CSSPL
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)
CSSPL
Boston Road Network
Start
End
CSSPL
(node 59, edges 108, regular-like ) Latency function = ax + b
Width1, 2, 3 length
User Equilibrium Global Optimum
Number of Agents: 10
CSSPL
User Equilibrium Global Optimum
Number of Agents: 6
CSSPL
User Equilibrium Global Optimum
Number of Agents: 7
CSSPL
User Equilibrium Global Optimum
Number of Agents: 8
CSSPL
User Equilibrium Global Optimum
Number of Agents: 9
CSSPL
User Equilibrium Global Optimum
Number of Agents: 15
CSSPL
User Equilibrium Global Optimum
Number of Agents: 20
CSSPL
Variation of POA with Agent #
# of Agents
POA
Reminder: POA = UE/GO
CSSPL
Affect of Arc Removal on UE
Arc
Total Agent Cost
CSSPL
Affect of an Arc Removal on UE
Severe increase
Increase
Mild to no increase
Decrease
Start
End
CSSPL
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)
CSSPL
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? )
