The Price of Anarchy on Complex Networks KIAS Conference July, 2006 HyeJin Youn, Hawoong Jeong Complex Systems & Statistical Physics Lab. (Dept. of Physics, KAIST, Korea) CSSPL

Marriage map between 100 richest people in Korea Importance of networks & dynamics CSSPL

Protein Interaction Network H.Jeong et al (2001) Importance of networks & dynamics Internet routing

∝ 1/width Network Dynamics “States” of both the nodes and the edges can change Dynamics of the networks : The topology of the network itself often evolves in time Dynamics on the networks : Agents are moving on the networks (E.g. Zero-range process, Contact process, Cascading failure, Shortest paths & OPTIMAL PATH) ∝ # of travelers ∝ length CSSPL Latency function (like time or cost per person)

Network flow with congestion CSSPL Based on the model of Roughgarden & Tardos, 2000 S Cost function on path i Latency function T width of path i length of path i # of agent on path i Given network with many agents going from S (source) to T (target), what will be the optimized distribution of agents for best performance??

Optimizations in physics Euler-Lagrange differential equation minimal free energy in thermodynamic physics Fitting experimental DATA with formula Low temperature behavior of disordered magnets … Centralized control Minimizing Global Cost Centralized control Minimizing Global Cost Decentralized control Each agent minimizes its own personal cost Decentralized control Each agent minimizes its own personal cost User Optimization (Nash equilibrium) Global Optimization CSSPL There are two types of optimizations!!!

The “Price of Anarchy” CSSPL Koutsoupias & Papadimitriou, 1999 Price of Anarchy (Roughgarden & Tardos, 2000) 1 ≤ total cost of Centralized control Minimizing Global Cost Decentralized control Each agent minimizes its own personal cost total cost of Global Optimum User Optimum Price of Anarchy “Price we have to pay not being coordinated by central agency” “Price of being selfish”

Price of Anarchy: Contrived Example CSSPL Pigous’s example: Congestion sensitive network ST What will be the min total cost, i.e. Global Optimum = ? 10 agents want to Go from S to T. If x a =x, then x b =10-x, ∴ total cost=10 ᆞ x + (10-x) ᆞ (10-x) = x 2 -10x+100=(x-5) ∴ x a =x b =5 with total cost 75

Price of Anarchy: Contrived Example Global Optimum = 5x10 + 5x5 = 75  75/10 = 7.5min driving in average CSSPL x a = x b = 5 ST Envy BUT The upper agents get envious of people with lower costs!

What will be the User Optimum? (Nash Equilibrium: everyone happy) CSSPL Price of Anarchy: Contrived Example x a = 5 x b = 5 ST

CSSPL user cost = < 10 Price of Anarchy: Contrived Example Move to Lower path +1 ST x a = 5-1 x b = 5+1 

CSSPL Price of Anarchy: Contrived Example again +1 ST x a = 4-1 x b = 6+1 user cost = < 10 

CSSPL Price of Anarchy: Contrived Example again +1 ST x a = 3-1 x b = 7+1 user cost = < 10 

CSSPL Price of Anarchy: Contrived Example again +1 ST x a = 2-1 x b = 8+1 user cost = < 10 

CSSPL Price of Anarchy: Contrived Example User Optimum = 10 x10 = 100  avg 10min travel time > avg 7.5-min travel time again +1 ST x a = 1-1 x b = 9+1

User Optimum = 10 x10 = 100 Global Optimum = 5x10 + 5x5 = 75 CSSPL Price of Anarchy: Contrived Example 4/3 Price of Anarchy! ST x a = 5 vs 0 x b = 5 vs 10 There is a gap between global optimum & user optimum!

More realistic/complex example Assumption: traffic reaches at equilibrium Price of Anarchy on a real world –the Boston Road Network –(with real geometrical information like width, length, one-way etc) –Traffic from central square (S) to copley square (T) CSSPL

Boston Road Map CSSPL

Boston Road Network Start End CSSPL (nodes 59, edges 108, regular-like) Latency function = ax + b lengthWidth

More realistic/complex example Assumption: traffic reaches at equilibrium Price of Anarchy on a real world –the Boston Road Network –(with real geometrical information) Global optimum : mapping to Min-cost Max-flow problem User optimum ~ approximate optimum: Metropolis Algorithm and Annealing method to find out the optimum configurations CSSPL

Number of traveler =1 User Optimum Global Optimum

CSSPL Number of traveler =2 User Optimum Global Optimum

CSSPL Number of traveler =3 User Optimum Global Optimum

CSSPL Number of traveler =4 User Optimum Global Optimum

CSSPL Number of traveler =5 User Optimum Global Optimum

CSSPL Number of traveler =6 User Optimum Global Optimum

CSSPL Number of traveler =7 User Optimum Global Optimum

CSSPL Number of traveler =8 User Optimum Global Optimum

CSSPL Number of traveler =9 User Optimum Global Optimum

CSSPL Number of traveler =10 User Optimum Global Optimum

CSSPL Number of traveler =11 User Optimum Global Optimum

CSSPL Number of traveler =12 User Optimum Global Optimum

CSSPL Number of traveler =13 User Optimum Global Optimum

CSSPL Number of traveler =14 User Optimum Global Optimum

CSSPL Number of traveler =15 User Optimum Global Optimum

CSSPL Number of traveler =16 User Optimum Global Optimum

CSSPL Number of traveler =17 User Optimum Global Optimum

CSSPL Number of traveler =18 User Optimum Global Optimum

CSSPL Number of traveler =19 User Optimum Global Optimum

Number of Agents: 20 CSSPL Congestion distribution on the edges

Reminder: POA = UE/GO CSSPL Variation of POA with Agent # number of agents Price of Anarchy

Why Price of Anarchy decreases? CSSPL Fitness landscape for a simple case: ST l(x a )= 5 5 l(x b )= x b Fitness for User Optimum 5 Strategy a Strategy b l(x b )= x b Fitness for Global Optimum c b (x b )= x b 2 c b (x b )= 5x b 2.5 l(x a )= 5 l(x b )= 2x b 2.5

POA too small?? More general edge latency function – n > 1 CSSPL - Roughgarden-Tardos Linear latency function: ST C=1 C(X) = X^3 When n=3 UO = 1 GO = 0.37*1 + (0.63)^4 = POA = UA/GO = Bigger than 4/3 (n=1) Nash Equilibrium 4/3 x (Global Optimum)

Upper bound of POA CSSPL n POA n GO More general edge latency function Possibility of getting higher order of POA if using latency function with higher exponents

Where to use?? To write a paper … Network design for better traffic?

Making network more efficient without central government?? Lower PoA ~ better(?) system ( ∵ even w/o central control, user optimum is closer to global optimum, better!) Let’s make better network with lower PoA –Simple thought (by stupid government): construct more roads with tax money!  Braess paradox (counter-intuitive consequences)

Braess’s Paradox T x x 10 0 : cost-free express road User Optimum without middle arc = 150 = Global Optimum CSSPL Price of Anarchy 4/3 = 200/150 = 4/3 increase User Optimum with middle arc = 200 S S Again 10 travelers want to move from S to T.

Boston Road Network Start End CSSPL

Affect of an Arc Removal on User Optimum Start End CSSPL negative

53 out of 108 edges are identified as deteriorating inefficiency! (ΔPoA<0) 19 out of 53 edges are found having made the decentralised system cost more! (ΔNE<0) CSSPL Affect of Arc Removal on User Optimum edge index Cost increment PoA=UO/GO

More systematic approaches Model network analysis –Regular Lattice –Erdos-Renyi Network –Small-world Network –Scale-free Network Multiple Sources & Targets Any correlation between PoA and other topological quantities? CSSPL

PoAC network representation SW(N=100, r=3, p=0,0.1) Regular Lattice (N=100) Number of Agents = 60 CSSPL

PoAC network representation ER(N=100, k=6)BA(N=100, m3) Thick and black edge: x+10 (wide and long) Thin and grey: 10x+1 size of node: PoAC method of spreading: using Kamada-Kawai(free) in Pajek except SW Number of Agents = 60 CSSPL

PoA (s-t pair) s-t pairs Number density Number of agents = 60 CSSPL

PoAC distribution Number of agents = 60 PoA centrality Number density CSSPL

all S-T pairs, network Number of Agents SW worst in POA! SF good! RL best! ER bad! CSSPL

BC(s)*BC(t) bc(s)bc(t) Number of agents = 60 PoA (s-t pair) BC correlation CSSPL

k(S)*k(T) k(S)k(T) Number of agents = 60 PoA (S-T pair) degree correlation CSSPL

Summary & Conclusion Price of Anarchy on a network : price that a decentralized system should pay for not being coordinated, can be understood as a measure of inefficiency of the system. Price of Anarchy on a real world (Boston Road Network) - It is small, but it does exist! Reducing the Price of Anarchy - network modification (Braess’s paradox) - Structural guidance of selfish users to the global optimized Efficiency in traffic dynamics: RL>BA>ER>SW?? Correlation with topological properties? Degree? More works are ongoing… Flow from to Central Square to Copley Square could be improved by removing some streets (NOT adding new streets!) CSSPL

Job opening at KAIST Funding: 2 nd phase Brain Korea 21 Project Several PostDoc & Research Professor positions are available in many fields. For more information, please contact H. Jeong