Non-Atomic Selfish Routing
Course: Price of Anarchy
Professor: Michal Feldman
Student: Iddan Golomb
26/02/2014
Talk Outline
 Introduction
 What are non-atomic selfish routing games
 PoA interpretation
 Main result – Reduction to Pigou-like networks
 Pigou-like networks
 Proof of the main result
 Analysis of consequences
 How to improve the situation
 Capacity augmentation
 Marginal 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: i :  f P  ri


 Latency: Function over E: le : R  R
 Non-negative
 Non-decreasing
 Continuous (differentiable)
 Instance: (G,r,l)
PPi
Non-Atomic Selfish Routing (2)
 Utilitarian cost:
 Edges: C  f    le  f e   f e
eE
 Paths: C ( f )   l p ( f )  f P
P
 Non-atomic: Many players, negligible influence each
 Examples – Driving on roads, packet routing over
the internet, etc.
Price of Anarchy Interpretation
 PoA:
Cost ( N.E. flow)
Cost (optimal flow)
 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 regulation
 Optimal flow is instable
 PoA ≥ 1
 The smaller, the better
 If grows with #players  bad sign…
Pigou’s Example
l(r)
 N.E: C(f)=1
 Optimal:
Source
f ( x)  x  (1  x) 1 
2
x*  0.5  C ( f * )  0.75
 PoA=4/3
 Questions:
 General graphs?
 General latency functions?
Target
l(x)=x
Pigou-like Networks
l(r)
 Pigou-like network:
 2 vertices: s,t
 2 edges: st
Source
Target
l(∙)
 Rate: r>0
 Edge #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
Pigou Bound
l(r)
 Minimal cost:
inf {x  l ( x)  ( r  x)  l (r )}
Source
0 x  r
Target
l(∙)
 PoA:


r  l (r )
sup 

x0  x  l ( x)  (r  x)  l (r ) 
 Pigou bound (α): For any set L of latency functions:


r  l (r )

x

l
(
x
)

(
r

x
)

l
(
r
)


 ( L)  supsupsup 
lL
r 0 x0
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. existence
 Singular 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 strategies
 Claim: A flow f feasible for instance (G,r,l) is at N.E.
iff i,P1, P2 Pi : f (P1)  0 l(P1( f ))  l(P2 ( f ))
 Proof: Trivial
 Corollary: In N.E., for each i, the latency is the same
for all paths: Li(f).
 C( f ) 
k
L ( f )  r
i 1
i
i
N.E. Existence (1)
 Goal: Min
 i :
f
PPi
p
 e : f e 
 c ( f )  l ( f )  f
eE
e
e
eE
e
e
e
s.t:
 ri

pP:eP
fP
 P : f P  0
d
 Define: le '  le ( x)
dx
and lP '( f )   le '( fe )
eP
 Assumptions: le is differentiable, x  le ( x)is convex
 f is a solution iff i,P1, P2 Pi : f (P1)  0 lP1 '( f )  lP2 '( f )
 Example: Pigou optimal when le1 '  2x,le2 '  2x
N.E. Existence (2)
t

 Now, set he ( x)  le (t )dt , change goal to: Min
0
h ( f )
eE
e
e
 Same constraints for flows in N.E. and for convex
program
Optimal 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
Singular Value at N.E.
 Claim: If
f , f are flows in N.E then C( f )  C( f )
 Proof:
 The objective function is convex
 
 e : le ( f e )  le f e
 Otherwise: A convex combination of fe , fe would dominate

 
Li ( f e )  Li f e
 C( f ) 
 L ( f )  r  C( f )  C  f 
k
i1
i
i
“Freezing” Latency at N.E
 Notations: Optimal flow: f, N.E. flow: f*
 We’ve shown:
 P, P  Pi : f P  0  lP ( f )  lP ( f )
 P Pi : lP ( f )  Li
 Now:
k

 f
i1 PPi
k
k
P
 lP ( f )  ri  Li
i1
k
*
f
  P  lP ( f )  ri  Li
i 1 PPi
i 1
*
f
  e  f e   le ( f e )  0
eE
How much is f* better than f?


r  l (r )
 Pigou bound:  ( L)  supsupsup 

lL r 0 x0  x  l ( x)  (r  x)  l ( r ) 
 For each edge e
 Set:
l le , r  fe , x  fe*
f l ( f )
e e
e
  (L)  *
*
fe  le ( fe )  ( fe  fe* )  le ( fe )
f l ( f )
*
*
*
 fe  le ( fe )  ( fe  fe )  le ( fe )  e e e
 (L)
f l ( f )
 f  le ( f )  e e e  ( fe*  fe )  le ( fe )
 (L)
*
e
*
e
 
*
 Sum for all edges: C f 

C( f )
  ( L) : QED 
*
C f 
 f
eE
*
e
 f e   le ( f e )  0
C( f )
C( f )
 ( fe*  fe )  le ( fe ) 
 (L)
 ( L)
Interpretation of Main Result
 Questions from earlier:
 General graphs?
 General latency functions?
 Result for polynomial latency functions:
Degree
Representative
PoA
1
ax+b (Affine)
4/3
2
ax2+bx+c
d
d
 ai x
i 0
i
3 3
3 32
 d  1 d d  1
 d  1 d d  1  d

d
ln( d )
 Result  as d goes to infinity the PoA goes to infinity

Capacity Augmentation (1)
 Different comparison from PoA
 Claim: 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 siti path
 C ( f )   ri  Li
i
 We will define new latency functions
l(x)
 “Close” to current latency function
 Allows to lower bound a flow f* with respect to C(f)
Capacity Augmentation (2)
 Definition:
le ( fe ) if x  fe
l e ( x)  
 le ( x) otherwise
*
*
*
l
f

f

C
(
f
)

C
f
e
 :
1)   e  e
e

*
*
*
*
*
*
l
(
f
)

f

C
(
f
)

f

l
f

l
f
e
e
 e e
 e  e  e e 
eE
eE
  le ( fe )  f e  C ( f ) 
eE
*
*
*
l
f

f

C
(
f
)

C
f
e




 e e
e

Capacity Augmentation (3)
 l Allows to lower bound a flow f* with respect to C(f)
*
*
l
f

f
P
   P  2C f  :
2)
P
l  f   L  f 
  l  f   f   L  f   f

P
0
i
*
*
P
P
P
P
i
PPi
i
  2 Li  f   ri  2  C  f 
i
*
P
Capacity Augmentation (4)
1)
*
*
*
l
(
f
)

f

C
(
f
)  C( f )
e
 e e
e
2)
*
*
l
f

f
P



P  2C  f 
eE
1)
C  f *   l e ( fe* )  fe*  C( f )  l P ( f * )  f P*  C( f )
eE
P
 2C f  C f   C f 
*
 C  f   C  f  : QED
 Generalization: If f is N.E flow for (G,r,l) and f* is feasible
for (G,(1+γ)r,l), then:   C  f   C  f * 
 Interpretation: Helpful if we can increase route/link speed
(without resorting to central routing) 
2)
Marginal Cost Pricing (1)
 We can’t always increase route speed
 We can (almost) always charge more…
 Tax  e  le '  f e   f e

 le  le ( x)   e  le ( x)  le '  f e   f e
 Claim: Given (G,r,l), f , e as defined, then: f 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’)
Marginal Cost Pricing (2)
  e  le '  f e   f e
 le '  f e : Marginal increase caused by a user
 f e : 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.
Summing Up
 Realistic problem
 PoA interpretation
 Main result – Reduction to Pigou-like networks
 Every network is easy to compute
 For some cost functions, PoA is arbitrarily high
 How to improve the situation
 Choose specific cost functions
 Capacity augmentation (“carrot”) – Make better roads
 Marginal cost pricing (“stick”) – Collect taxes
Questions?
Bibliography
 Roughgarden 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.