# Selfish Routing Games Kook Jin Ahn Daniel Wagner.

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Selfish Routing Games Kook Jin Ahn Daniel Wagner

Nonatomic Model Network G=(V,E) Source-sink pairs or commodities r={(s 1,t 1 ),…,(s k,t k )} –Send f p flow from s i to t i through path p –P i is a set of all s i -t i paths –∑ p ∈ Pi f p = 1 Each commodity pays c e (f e ) per unit flow on edge e. –f e : total amount of flow that goes through e. –Continuous, non-decreasing –Total Cost : ∑ i ∑ p ∈ Pi ∑ e ∈ p f p c e (f e )=∑ e ∈ E f e c e (f e )

Atomic Model Network G=(V,E) Source-sink pairs or commodities r={(s 1,t 1 ),…,(s k,t k )} –Send a unit flow from s i to t i through a single s i -t i path p i Each commodity pays c e (f e ) per unit flow onp edge e. –f e : total amount of flow that goes through e. –Non-decreasing –Total Cost : ∑ i ∑ e ∈ pi c e (f e )=∑ e ∈ E f e c e (f e )

Nonatomic Equilibrium f is an equilibrium if for every commodity i and every pair of paths p,p’ ∈ P i such that f p >0, C p (f)≤C p’ (f).

Pigou’s Example c e1 (x)=1 c e2 (x)=x Optimal flow f e1 = 1/2 f e2 = 1/2 Equilibrium f e1 = 0 f e2 = 1

Existence, Uniqueness, and Potential Function Existence There exists an equilibrium flow f for every nonatomic selfish routing game (G,r,c). Potential function Φ(f)=∑ e ∫ fe c e (x)dx Uniqueness If f 1 and f 2 are equilibrium flows for a nonatomic selfish routing game (G,r,c), then c e (f 1 e )=c e (f 2 e ) for all e.

Equilibrium and Optimal Flows An optimal flow of a nonatomic selfish routing game (G,r,c) is an equilibrium flow of a nonatomic selfish routing game (G,r,c’) where c’ e (x)=xc e (x).

Price of Anarchy If xc e (x)≤r ∫ x c e (x)dx for all e, the price of anarchy is at most r. If (G,r,c) is a nonatomic instance such that c e are polynomials with degree at most p for all e, the price of anarchy is at most p+1.

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