1. Selfish Routing
第３回

2. 3.1 Overview

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

4. 3.2 The Price of Anarchy with Linear Cost Functions

5. 3.2.1 Preliminaries

6. 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,

7. 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

8. 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)

9. 3.2.3 Proof of Upper Bound

10. 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

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

12. 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

13. 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

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

17. 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)

19. 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

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

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

22. 3.3 A General Upper Bound on the Price of Anarchy

23. 3.3.1 The Anarchy Value

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

25. 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

26. 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.

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

28. 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)

29. Intermediate Value Theorem
f (x) : continuous

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

31. Nash s t
Optimal s t

32. Proof

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

34. 3.3.2 Proof of the Upper Bound

35. 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 ,
From Definition and 3.3.3

36. 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

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