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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.

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Presentation on theme: "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."— Presentation transcript:

1 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

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

3 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

4 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

5 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

6 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

7 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

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

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

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

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

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

13 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

14 Braesss Paradox S T x x 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

15 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

16 Boston Road Map CSSPL

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

18 User Equilibrium Global Optimum Number of Agents: 1 CSSPL

19 User Equilibrium Global Optimum Number of Agents: 2 CSSPL

20 User Equilibrium Global Optimum Number of Agents: 3 CSSPL

21 User Equilibrium Global Optimum Number of Agents: 4 CSSPL

22 User Equilibrium Global Optimum Number of Agents: 10 CSSPL

23 User Equilibrium Global Optimum Number of Agents: 5 CSSPL

24 User Equilibrium Global Optimum Number of Agents: 6 CSSPL

25 User Equilibrium Global Optimum Number of Agents: 7 CSSPL

26 User Equilibrium Global Optimum Number of Agents: 8 CSSPL

27 User Equilibrium Global Optimum Number of Agents: 9 CSSPL

28 User Equilibrium Global Optimum Number of Agents: 15 CSSPL

29 User Equilibrium Global Optimum Number of Agents: 20 CSSPL

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

31 Affect of Arc Removal on UE Arc Total Agent Cost CSSPL

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

33 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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