# 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.

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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

A basic traffic problem agents from S to T at minimum cost ST 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 control Minimising 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 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

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

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

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

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

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

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

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

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

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

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

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

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

User Equilibrium Global Optimum Number of Agents: 1 CSSPL

User Equilibrium Global Optimum Number of Agents: 2 CSSPL

User Equilibrium Global Optimum Number of Agents: 3 CSSPL

User Equilibrium Global Optimum Number of Agents: 4 CSSPL

User Equilibrium Global Optimum Number of Agents: 10 CSSPL

User Equilibrium Global Optimum Number of Agents: 5 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 (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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