Selfish Routing 第３回 3.1-3.3

3.1 Overview

Description Typical Representative Price of Anarchy Linear Quadratic Cubic Polynomials of degree M / M / 1 Delay Functions

3.2 The Price of Anarchy with Linear Cost Functions

3.2.1 Preliminaries

Lemma 3.2.1 (G , r, c) : an instance with linear cost functions. ce(x) =aex+be : linear cost function (a ) a feasible flow f is at Nash equilibrium for (G, r, c) iff for each commodity i and si-ti paths P1, P2 ∈ Ｐ i with fP1 > 0, (b) a feasible flow f * is optimal for (G, r, c) iff for each commodity i and si-ti paths P1, P2 ∈ Ｐ i with f *P1 > 0,

Corollary 3.2.2 If (G, r, c) is an instance in which each edge cost function ce is of the form ce(x) = aex, then a flow feasible for (G, r, c) is optimal iff it is at Nash equilibrium

Lemma 3.2.3 Suppose (G, r, c) has linear cost functions and f is a flow at Nash equilibrium then, c*e( fe / 2 )=ce( fe ) for each edge e the flow f / 2 is optimal for (G, r/2, c)

3.2.3 Proof of Upper Bound

Lemma 3.2.4 f : a flow for an instance (G , r, c) with linear cost functions. (式 2.3) p.19 ce : the cost function of edge e

Proof ce(x) =aex+be for each edge e

Lemma 3.2.5 (G, r, c) : an instance with linear cost functions f * : an optimal flow For every δ > 0, a feasible flow for the instance (G, (1+δ)r, c) has cost at least

Proof Fix an instance (G, r, c) with linear cost functions, an optimal flow f *, and a value for the parameter δ > 0. Suppose f is a feasible flow for (G, (1+δ)r, c). cost functions ce(x) : linear ⇒　xce(x) : convex functions

Proof f * : an optimal flow for (G, r, c) f : a feasible flow for (G, (1+δ)r, c). linear cost functions : semiconvex ( Def. 2.4.2(d) ) Applying Proposition 2.4.4(d)

Proof f * : an optimal flow for (G, r, c) f : a feasible flow for (G, (1+δ)r, c) Proposition 2.4.4(d): For every feasible flow f, f : a feasible flow for (G, r, c)

Lemma 3.2.5 (G, r, c) : an instance with linear cost functions f * : an optimal flow For every δ > 0, a feasible flow for the instance (G, (1+δ)r, c) has cost at least

Theorem 3.2.6 If (G, r, c) has linear cost functions, then

Proof f : a flow at Nash equilibrium for (G, r, c). (Lemma 3.2.3) (Lemma 3.2.4, 3.2.3)

3.3 A General Upper Bound on the Price of Anarchy

3.3.1 The Anarchy Value

: constant function s t f * : optimal flow f : Nash flow

Definition 3.3.1 Let c be a cost function. The anarchy value α(c) of c is with the understanding that 0/0 = 1

If c is a continuously differentiable and semiconvex Proposition 3.3.2 If c is a continuously differentiable and semiconvex cost function, then where λ ∈ [0,1] solves c*(λr) = c(r), μ= c(λr) / c(r) ∈ [0,1], and 0/0 is defined to be 1.

c : a continuously differentiable , semiconvex Proof c : a continuously differentiable , semiconvex s t

Proof c : a continuously differentiable , semiconvex ( nondecreasing ) ⇒ c* : continuous c*(x) = c(x) + x・c(x) ≧ c(x) for all x There is a value λ∈ [0,1] solving c*(λr) = c(r) (By the Intermediate Value Theorem)

Intermediate Value Theorem f (x) : continuous

Intermediate Value Theorem c* (x) : continuous, nondecreasing

Nash s t Optimal s t

Proof

Definition 3.3.3 The anarchy value α(C) of a set C of cost functions is

3.3.2 Proof of the Upper Bound

Lemma 3.3.6 From Definition 3.3.1 and 3.3.3 Let C be a set of cost functions with anarchy valueα(C). For c ∈ C and x, r 0, From Definition 3.3.1 and 3.3.3

Theorem 3.3.7 Let C be a set of cost functions with anarchy valueα(C), and ( G, r, c ) an instance with cost functions in C. Then

Proof (Corollary 2.6.6) f *: optimal flows f : Nash flows for (G, r, c) with cost function in the set C (Lemma 3.3.6 x = f * , r = f ) (Corollary 2.6.6)