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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, Switzerland ) Matthew Silver ( MIT ) Marie-Helen Cloutier ( Canada ) Peter Ittzes ( Collegium Budapest ) Hawoong Jeong ( KAIST ) CSSPL

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A basic traffic problem agents from S to T at minimum cost ST C(x) = Ax+B CSSPL Latency function C(X) = AX + B

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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 control Minimising Global Cost

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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 control Minimising Global Cost Price of Anarchy Koutsoupias & Papadimitriou, 1999 Price of Anarchy <= 4/3 (Roughgarden & Tardos, 2000) Examples: Road Traffic, Network Routing, Prisoners Dilemma

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Price of Anarchy: Simple Example SE C=10 C(X) = X Global Optimum = ? 10 Agents from S E C = latency function (cost) CSSPL Global Optimum

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SE C=10 C(X) = X Global Optimum = 5x10 + 5x5 = 75 X = 5 CSSPL Price of Anarchy: Simple Example 10 Agents from S E C = latency function (cost) Global Optimum

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SE C=10 C(X) = X User Equilibrium = ? X = 5 CSSPL 10 Agents from S E C = latency function (cost) User Optimum Price of Anarchy: Simple Example

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SE C=10 C(X) = X X = 5 + 1 X = 5 - 1 +1 CSSPL 10 Agents from S E C = latency function (cost) User Optimum user cost = 5 + 1 < 10 Price of Anarchy: Simple Example

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SE C=10 C(X) = X X = 6 + 1 X = 4 - 1 CSSPL 10 Agents from S E C = latency function (cost) User Optimum again +1 user cost = 6 + 1 < 10 Price of Anarchy: Simple Example

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SE C=10 C(X) = X X = 8 X = 2 CSSPL 10 Agents from S E C = latency function (cost) User Optimum user cost = 7 + 1 < 10 Price of Anarchy: Simple Example again +1

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SE C=10 C(X) = X X = 9 X = 1 CSSPL 10 Agents from S E C = latency function (cost) User Optimum user cost = 8 + 1 < 10 Price of Anarchy: Simple Example again +1

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SE C=10 C(X) = X He is indifferent: C = 9 + 1 = 10 X = 10 X = 0 CSSPL 10 Agents from S E C = latency function (cost) User Optimum Price of Anarchy: Simple Example

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SE C=10 C(X) = X User Equilibrium = 10 x10 = 100 X = 10 X = 0 Global Optimum = 5x10 + 5x5 = 75 CSSPL 10 Agents from S E C = latency function (cost) User Optimum Price of Anarchy: Simple Example 4/3= upper bound of Price of Anarchy

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Braesss Paradox S T x x 1 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 4/3 Price of Anarchy = 2/1.5 = 4/3

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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 (Braesss paradox) CSSPL

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Boston Road Map CSSPL

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Boston Road Network Start End CSSPL (node 59, edges 108, regular-like ) Latency function = ax + b Width 1, 2, 3 length

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User Equilibrium Global Optimum Number of Agents: 1 CSSPL

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User Equilibrium Global Optimum Number of Agents: 2 CSSPL

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User Equilibrium Global Optimum Number of Agents: 3 CSSPL

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User Equilibrium Global Optimum Number of Agents: 4 CSSPL

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User Equilibrium Global Optimum Number of Agents: 10 CSSPL

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User Equilibrium Global Optimum Number of Agents: 5 CSSPL

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User Equilibrium Global Optimum Number of Agents: 6 CSSPL

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User Equilibrium Global Optimum Number of Agents: 7 CSSPL

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User Equilibrium Global Optimum Number of Agents: 8 CSSPL

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User Equilibrium Global Optimum Number of Agents: 9 CSSPL

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User Equilibrium Global Optimum Number of Agents: 15 CSSPL

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User Equilibrium Global Optimum Number of Agents: 20 CSSPL

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Variation of POA with Agent # # of Agents POA Reminder: POA = UE/GO CSSPL

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Affect of Arc Removal on UE Arc Total Agent Cost CSSPL

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Affect of an Arc Removal on UE Severe increase Increase Mild to no increase Decrease Start End CSSPL

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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 (Braesss 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 prisoners dilemma –prisoners dilemma get agents better when they look further. but traffic doesnt have such a benefit to cooperators ( tax? )

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