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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.