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How Bad is Selfish Routing

Tim Roughgarden Eva Tardos

presented by Yajun Wang (yalding@cs.ust.hk)

for COMP670O Spring 2006, HKUST

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Problem Formulation: Traffic Model

• Given the rate of traffic between each pair ofnodes in a network, find an assignment of trafficto minimize the total latency.

• On each edge, the latency is load dependent

v

s t

w

l(x) = x

l(x) = xl(x) = 1

l(x) = 1

v

s t

w

l(x) = x

l(x) = xl(x) = 1

l(x) = 1

l(x) = 0

Braess’s Paradox

• Each player controls a negligible fraction of theoverall traffic.

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Formal Model

• Graph G = (V, E) and k source-destination pairs {si, ti}

• P = ∪iPi

• Pi denotes the set of (simple) si − ti paths, and

• A flow is a function:

f : P → R+

• A flow is feasible if :∑P∈Pi

fP = ri

• Each edge has a nonnegative, differentiable, nondecreasinglatency function le(·)

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Cost for Flows

• Let (G, r, l) be an instance , and f is a flow.

• Latency of a path P

• Cost of a flow f:

• Players are small flows behave ”greedily” and ”selfishly”

lP (f) =∑

e∈P le(fe)

C(f) =∑

P∈P lP (f)fP =∑

e∈E le(fe)fe

There are infinite number of players, each carry a negligibleamout of flow.

fe =∑

P :e∈P fP

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Flows at Nash Equilibrium

• Definition (Nash Equilibrium):

A flow f is feasible for instance (G, r, l) is at NashEquilibrium if for all i ∈ {1, . . . , k}, P1, P2 ∈ Pi,and δ ∈ [0, fP1 ], we have lP1(f) ≤ lP2(f̃), where

f̃P =

fP − δ if P = P1

fP + δ if P = P2

fP if P /∈ {P1, P2}

• Lemma: A flow f feasible for instance (G, r, l) is at NashEquilibrium if and only if for all i ∈ {1, . . . , k}, P1, P2 ∈Pi with fP1 > 0, lP1(f) ≤ lP2(f).

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Optimal Flows via Convex Programming

• NonLinear Programming Formulation

∑P∈Pi

fP = ri ∀i ∈ {1, . . . , k}

fe =∑

P∈P:e∈P

fP ∀e ∈ E

fP ≥ 0 ∀P ∈ P

Min∑e∈E

ce(fe)

subject to:

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Characteristic of Optimal Flows

• Lemma: A flow f is optimal for a convex program of theprevious form if and only if for every i ∈ {1, . . . , k} andP1, P2 ∈ Pi with fP1 > 0, c′P1(f) ≤ c′P2(f).

c′P (f) =∑

e∈P c′e(fe)

Let c′e be the derivative ddxce(x)

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Characteristic of Optimal Flows

• Lemma: A flow f is optimal for a convex program of theprevious form if and only if for every i ∈ {1, . . . , k} andP1, P2 ∈ Pi with fP1 > 0, c′P1(f) ≤ c′P2(f).

c′P (f) =∑

e∈P c′e(fe)

Let c′e be the derivative ddxce(x)

• Lemma: A flow f feasible for instance (G, r, l) is at NashEquilibrium if and only if for all i ∈ {1, . . . , k}, P1, P2 ∈Pi with fP1 > 0, lP1(f) ≤ lP2(f).

C(f) =k∑

i=1

Li(f)ri

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Nash Equilibrium and Optimal Flow

• Corollary: Let (G, r, l) be an instance in which x · le(x)is a convex function for each edge e, with marginal costfunctions l∗e . Then a flow f feasible for (G, r, l) is optimalif and only if it is at Nash equilibrium for the instance(G, r, l∗)

l∗e(fe) = (le(fe)fe)′ = le(fe) + l′e(fe)fe

Marginal cost function:

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Nash Equilibrium and Optimal Flow (cont’)

• Lemma: An instance (G, r, l) with continuous, nonde-creasing latency functions admits a feasible flow at Nashequilibrium. Moreover, if f, f̃ are flows at Nash equilib-rium, then C(f) = C(f̃).

Proof:

∑P∈Pi

fP = ri ∀i ∈ {1, . . . , k}

fe =∑

P∈P:e∈P

fP ∀e ∈ E

fP ≥ 0 ∀P ∈ P

Min∑e∈E

he(fe)

Set he(x) =∫ x

0le(t)dt

Note, h′e(x) = le(x)

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”Unique” Nash Equilibrium

• Lemma: An instance (G, r, l) with continuous, nonde-creasing latency functions admits a feasible flow at Nashequilibrium. Moreover, if f, f̃ are flows at Nash equilib-rium, then C(f) = C(f̃).

Proof (cont’) :

Min∑e∈E

he(fe)

Set he(x) =∫ x

0le(t)dt

If fe 6= f̃e, the functionhe(x) must be linear and leis a constant function

This implies le(fe) = le(f̃e).

C(f) =∑k

i=1 Li(f)ri = C(f̃).

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Nontrivial Upper Bound for Price of Anarchy

ρ = ρ(G, r, l) = C(f)C(f∗)

ρ(G, r, l) ≤ α

For instance (G, r, l), let f∗ be an optimal flow andf be a flow at Nash equilibrium.

Corollary: Suppose the instance (G, r, l) and the constant α ≥ 1satisfy:

x · le(x) ≤ α ·∫ x

0le(t)dt

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Nontrivial Upper Bound for Price of Anarchy (cont’)

ρ(G, r, l) ≤ α

Corollary: Suppose the instance (G, r, l) and the constant α ≥ 1satisfy:

x · le(x) ≤ α ·∫ x

0le(t)dt

Proof: C(f) =∑e∈E

le(fe)fe

≤ α∑e∈E

∫ fe

0

le(t)dt

≤ α∑e∈E

∫ f∗e

0

le(t)dt

≤ α∑e∈E

le(f∗e )f∗e

= α · C(f∗)

N.E optimizes this ob-jective function.

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Upper Bound for Polynomial Latency Function

ρ(G, r, l) ≤ p + 1

Corollary: Suppose the instance (G, r, l) has the latency func-tions:

le(x) =∑p

i=0 ae,ixi

Remarks: It is not tight.

ae,i ≥ 0

le(x) = aex + be for ae, be ≥ 0 ρ ≤ 2

For higher degree polynomial latency functions:

ρ = O( pln p )

Tight Bound: ρ ≤ 4/3

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A Bicriteria Result for General Latency Functions

Negative Result: :

If l(x) = xp:

1− p(p + 1)−(p+1)/p → 0

s t

l(x) = 1

l(x) = x

ρ = 4/3

Optimal flows assgins (p+1)−1/p on thelower link, which has a total latency:

ρ →∞

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Augment Analysis for General Latency Function

Let

• Theorem: If f is a flow at Nash equilibrium for (G, r, l)and f∗ is feasible for (G, 2r, l), then C(f) ≤ C(f∗)

l̄e(x) ={

le(fe) if x ≤ fe

le(x) if x ≥ fe

∑e

l̄e(f∗e )f∗e − C(f∗) =∑e∈E

f∗e (l̄e(f∗e )− le(f∗e ))

≤∑e∈E

le(fe)fe

= C(f)

∑e

l̄P (f∗)f∗P ≥∑

i

∑P∈Pi

Li(f)f∗P

=∑

i

2Li(f)ri

= 2C(f)

l̄P (f∗) ≥ l̄P (f0) ≥ Li(f)

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Worst-Case Ratio with Linear Latency Fuctions

• Lemma: If (G, r, l) be an instance with edge latency func-tions le(x) = aex + be for each edge e ∈ E. Then

le = aex + be with ae, be ≥ 0

l∗e = 2aex + be

(a) a flow f is at Nash equilibrium in G if and onlyif for P, P ′ ∈ Pi with fP > 0,∑

e∈P aefe + be ≤∑

e∈P ′ aefe + be

(b) a flow f∗ is (globally) Optimal in G if and onlyif for P, P ′ ∈ Pi with f∗P > 0,∑

e∈P 2aef∗e + be ≤

∑e∈P ′ 2aef

∗e + be

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Worst-Case Ratio with Linear Latency Fuctions (cont’)

• Lemma: Suppose (G, r, l) has linear latency functions andf is a flow at Nash equilibrium. Then

(a) The flow f/2 is optimal for (G, r/2, l)

(b) the marginal cost of increasing the flow on a path P forf/2 equals the latency of P for f

Creating optimal flow in two steps: (f is at Nash equilibrium)

(1) Send a flow optimal for instance (G, r/2, l). C(f)/4

(2) Augment to one optimal for instance (G, r, l). C(f)/2

l∗P (f/2) = lP (f)

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Augment Cost for Linear Latency Functions

• Lemma: (G, r, l) has linear latency functions and f∗ isan optimal flow. Let L∗

i (f∗) be the minimum marginal

cost for si− ti paths. For any δ > 0, a feasible flow f for(G, (1 + δ)r, l):

x · le(x) = aex2 + be is convex.

C(f) ≥ C(f∗) + δ∑k

i=1 L∗i (f

∗)ri

le(fe)fe ≥ le(f∗e )f∗ + (fe − f∗)l∗e(f∗e )

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Augment Cost for Linear Latency Functions

• Proof:

C(f) =∑e∈E

le(fe)fe

≥∑e∈E

le(f∗e )f∗e +∑e∈E

(fe − f∗e )l∗e(f∗e )

= C(f∗) +k∑

i=1

∑P∈Pi

l∗P (f∗)(fP − f∗P )

≥ C(f∗) +k∑

i=1

L∗i (f

∗)∑

P∈Pi

(fP − f∗P )

= C(f∗) + δk∑

i=1

L∗i (f

∗)ri

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Worst-Case Ratio with Linear Latency Fuctions (cont’)

• Lemma: If (G, r, l) has linear latency functions, thenρ(G, r, l) ≤ 4/3

Let f be a flow at N.E. f/2 is optimal for(G, r/2, l). Moreover, L∗

i (f/2) = Li(f).Proof:

C(f∗) ≥ C(f/2) +k∑

i=1

L∗i (f/2)

ri

2

= C(f/2) +12

k∑i=1

Li(f)ri

= C(f/2) +12C(f)

≥ 34C(f)

C(f/2) =14aef

2e +

12befe

≥ 14

∑e

(aef2e + befe)

=14C(f)

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Extensions:

• Approximate Nash Equilibrium:

If f is at ε N.E, and f∗ is feasible for (G, 2r, l), thenC(f) ≤ 1+ε

1−εC(f∗).

• Finite Agents: Splittable Flow

C(f) ≤ C(f∗).

• Finite Agents: Unsplittable Flow

If for some α < 2, le(x+ri) ≤ α · le(x), x ∈ [0,∑

j 6=i rj ]

C(f) ≤ α2−αC(f∗).