Course: Price of AnarchyProfessor: Michal Feldman

Student: Iddan Golomb26/02/2014

Non-Atomic Selfish Routing

Talk OutlineIntroduction

What are non-atomic selfish routing gamesPoA interpretation

Main result – Reduction to Pigou-like networksPigou-like networksProof of the main resultAnalysis of consequences

How to improve the situationCapacity augmentationMarginal cost pricing

Summing-up

Motivation

Non-Atomic Selfish Routing (1)

Directed graph (network): G(V,E)Source-target vertex pairs: (s1,t1),…, (sk,tk)

Paths: Pi from si to ti Flow: Non-negative vector over paths. Rate: Total flow. f is feasible for r if: Latency: Function over E:

Non-negativeNon-decreasingContinuous (differentiable)

Instance: (G,r,l)

:i

P iP P

i f r

:el R R

Non-Atomic Selfish Routing (2)

Utilitarian cost:

Edges:

Paths: Non-atomic: Many players, negligible

influence each Examples – Driving on roads, packet

routing over the internet, etc.

e e ee E

C f l f f

( ) ( )p P

P

C f l f f

Price of Anarchy Interpretation

PoA: Pure N.E. (non-atomic)In our case, we will show:

N.E. exists All N.E. flows have same total cost

Examples when PoA is interesting:Limited influence on starting point (“in the

wild”)Limited traffic regulationOptimal flow is instable

PoA ≥ 1The smaller, the betterIf grows with #players bad sign…

( . . )

( )

Cost N E flow

Cost optimal flow

Pigou’s Example

N.E: C(f)=1Optimal:

PoA=4/3Questions:

General graphs?General latency functions?

Source Target

l(r)

l(x)=x2

* *

( ) (1 ) 1

0.5 ( ) 0.75

f x x x

x C f

Pigou-like Networks

Pigou-like network:2 vertices: s,t2 edges: stRate: r>0Edge #1: General – l(∙)Edge #2: Constant – l(r)

2 free parameters: r, l Main result (informal): Among all networks,

the largest PoA is achieved in a Pigou-like network

Source Target

l(r)

l(∙)

Pigou BoundMinimal cost:

PoA:

Pigou bound (α): For any set L of latency

functions:

Source Target

l(r)

l(∙)

0inf { ( ) ( ) ( )}x rx l x r x l r

0

( )sup

( ) ( ) ( )x

r l r

x l x r x l r

0 0

( )( ) supsupsup

( ) ( ) ( )l L r x

r l rL

x l x r x l r

Main Result – Statement and Outline

Theorem: For every set L of latency functions, and every selfish routing network with latency functions in L, the PoA is at most α(L)

Proof outline:Preliminaries:

Flows in N.E.N.E. existenceSingular cost at N.E

Proof:Freezing edge latencies in N.E.Comparing f* with flow in N.E

Flows in N.E.Clarification: N.E. with respect to pure

strategiesClaim: A flow f feasible for instance (G,r,l) is

at N.E. iffProof: Trivial Corollary: In N.E., for each i, the latency is

the same for all paths: Li(f).

1 2 1 1 2, , : ( ) 0 ( ( )) ( ( ))ii P P P f P l P f l P f

1

( ) ( )k

i ii

C f L f r

N.E. Existence (1)Goal: Min s.t:

Define: and

Assumptions: is differentiable, is

convex

f is a solution iff

Example: Pigou optimal when

( ) ( )e e e e ee E e E

c f l f f

:

i

p iP P

i f r

:

: e Pp P e P

e f f

: 0PP f

' ( )e e

dl l x

dx '( ) '( )P e e

e P

l f l f

1 21 2 1, , : ( ) 0 '( ) '( )i P Pi P P P f P l f l f

el ( )ex l x

1 2' 2 , ' 2e el x l x

N.E. Existence (2)Now, set , change goal to: Min

Same constraints for flows in N.E. and for

convex programOptimal solutions for convex program are precisely flows at N.E. for (G,r,l)!

Corollary: Under same conditions, f* is an optimal flow for (G,r,l) iff it is an equilibrium flow for (G,r,l’)

Interpretation: Optimal flow and latency function ≈ Equilibrium

flow and latency derivative

0

( ) ( )t

e eh x l t dt ( )e ee E

h f

Singular Value at N.E.Claim: If are flows in N.E then

Proof:

The objective function is convex

Otherwise: A convex combination of would

dominate

,f f ( ) ( )C f C f

: ( )e e e ee l f l f ,e ef f

( )i e i eL f L f

1

( ) ( ) ( )k

i ii

C f L f r C f C f

“Freezing” Latency at N.ENotations: Optimal flow: f, N.E. flow: f*We’ve shown:

Now:

, : 0 ( ) ( )i PP P

P P P f l f l f : ( )i P iP P l f L

1 1

( )i

k k

P P i ii P P i

f l f r L

*

1 1

( )i

k k

P P i ii P P i

f l f r L

* ( ) 0e e e ee E

f f l f

How much is f* better than f?Pigou bound:

For each edge e

Set:

Sum for all edges:

: QED

0 0

( )( ) supsupsup

( ) ( ) ( )l L r x

r l rL

x l x r x l r

* * *

( )( )

( ) ( ) ( )e e e

e e e e e e e

f l fL

f l f f f l f

*, ,e e el l r f x f

* * * ( )( ) ( ) ( )

( )e e e

e e e e e e e

f l ff l f f f l f

L

* * *( )( ) ( ) ( )

( )e e e

e e e e e e e

f l ff l f f f l f

L

* *( ) ( )( ) ( )

( ) ( )e e e e

C f C fC f f f l f

L L

*( )

( )C f

LC f

* ( ) 0e e e ee E

f f l f

Interpretation of Main ResultQuestions from earlier:

General graphs?General latency functions?

Result for polynomial latency functions:

Result as d goes to infinity the PoA goes to infinity

DegreeRepresentative

PoA

1ax+b (Affine)4/3

2ax2+bx+c

d

3 3

3 3 2

0

di

ii

a x

1 1

ln( )1 1

d

d

d d d

dd d d

Capacity Augmentation (1)Different comparison from PoAClaim: If f is an equilibrium flow for (G,r,l),

and f* is feasible for (G,2r,l), then: C(f) ≤ C(f*)

Proof:Li: Minimal cost for f in siti path We will define new latency functions

“Close” to current latency functionAllows to lower bound a flow f* with respect to

C(f)

( ) i ii

C f r L ( )l x

Capacity Augmentation (2)Definition:

1)

( )( )

( )e e e

e

e

l f if x fl x

l x otherwise

* * *( ) :e e ee

l f f C f C f

* * * * * *

* * *

( ) ( )

( ) ( )

( )

e ee e e e e ee E e E

e e ee E

e e ee

l f f C f f l f l f

l f f C f

l f f C f C f

Capacity Augmentation (3) Allows to lower bound a flow f* with respect

to C(f)

2)

0P iP

l f L f

* * *

2 2i

P P i PP i P P

i ii

l f f L f f

L f r C f

* * 2 :P PP

l f f C f l

Capacity Augmentation (4)1)

2)

: QED Generalization: If f is N.E flow for (G,r,l) and f* is

feasible for (G,(1+γ)r,l), then: Interpretation: Helpful if we can increase

route/link speed (without resorting to central routing)

* * *( ) ( ) ( )e e ee

l f f C f C f * * 2P P

e E

l f f C f

* * * * *( ) ( ) ( ) ( )

2

e Pe e Pe E P

C f l f f C f l f f C f

C f C f C f

*C f C f2)

1)

*C f C f

Marginal Cost Pricing (1)We can’t always increase route speedWe can (almost) always charge more…Tax Claim: Given (G,r,l), as defined, then:

is an equilibrium flow for (G,r,(l+τ))Reminder: f* is an optimal flow for (G,r,l) iff

it is an equilibrium flow for (G,r,l’)

( ) ( ) 'e e e e e e el l x l x l f f

'e e e el f f

*, ef *f

Marginal Cost Pricing (2)

: Marginal increase caused by a user

: Amount of traffic suffering from the increase

Tax “aligns” the derivative to fit utilitarian goal

Interpretation:PoA is reduced to 1!However, the costs were artificially raised

(“sticks” as opposed to “carrots”). Might cause users to leave.

'e e e el f f

'e el f

ef

Summing UpRealistic problemPoA interpretationMain result – Reduction to Pigou-like networks

Every network is easy to computeFor some cost functions, PoA is arbitrarily high

How to improve the situationChoose specific cost functionsCapacity augmentation (“carrot”) – Make better

roadsMarginal cost pricing (“stick”) – Collect taxes

Questions?

?

BibliographyRoughgarden T, Tardos E – How bad is selfish

routing? J.ACM, 49(2): 236259, 2002.Stanford AGT course by Roughgarden -

http://theory.stanford.edu/~tim/f13/f13.html (Lecture 11)

Nisan, Roughgarden, Tardos, Vazirani - Algorithmic Game Theory, Cambridge University Press. Chapter 18 (routing games) – 461-486.

Cohen J.E., Horowitz P - Paradoxical behavior of mechanical and electrical networks. Nature 352, 699–701. 1991.