The Price of Routing Unsplittable Flow

1
The Price of Routing Unsplittable Flow

• Yossi Azar
• Joint work with
• B. Awerbuch and A. Epstein

2
Outline

• Game Theory and Selfish Routing
• Price of Anarchy
• Network Model Previous Results
• Network Model Our Results

3
Selfish Routing

• Large networks
• Infeasible to maintain a central authority
• Users are selfish
• Each user tries to minimize its cost
• Each user is aware of network conditions
• Degradation of network performance

4
Nash Equilibrium

• Game Theory
• Study and predict user behavior
• Nash Equilibrium
• Each agent minimizes its cost /maximizes its
  benefit
• No agent has an incentive change its behavior

5
Example-Prisoners Dilemma

• Nash Equilibrium (c,c)

D C
D (-1,-1) (-4,0)
C (0,-4) (-3,-3)
6
Nash Equilibrium

• Every game has randomized Nash equilibrium
• In general a game may not have pure Nash
  equilibrium
• No Deterministic Nash Equilibrium
• Randomized Strategy pi,j 0.5 is in N.E

H T
H (1,-1) (-1,1)
T (-1,1) (1,-1)
7
Parallel Links (Machines) Model

• Two nodes
• m parallel (related) links
• n jobs
• User cost (delay) is proportional to link load
• Global cost (maximum delay) is the maximum link
  load

8
Nash Equilibrium - Example

• 2 identical links
• 4 jobs with weights 1,2,3,4

2
Not a Nash Equilibrium
4
1
3
Nash Equilibrium
m1
m2
9
Nash Equilibrium - Example
1
2
Optimal Solution
4
3
Also Nash Equilibrium
m1
m2
10
Price of Anarchy

• Price of Anarchy (coordination ratio)
• The worst possible ratio between
• Global cost in Nash Equilibrium and
• Global cost in Optimum
• Global cost maximum/total users cost

11
General Network Model

• A directed Graph G(V,E)
• A load dependent latency function fe(.) for each
  edge e
• n users
• Bandwidth request (si, ti, wi) for user i
• Goal route traffic to minimize total latency

12
Example
Latency function f(x)x
1
2
1
2
1
t
s
2
2
2
2
Latency2125
Latency22228
Total latency Se fe(le)le Se le le
62231127
13
Example

• Traffic rate r1
• Nash total latency10111

f(x)1 l0
t
s
f(x)x l1
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Example

• Traffic rate r1
• Optimal total latency11/21/21/23/4
• R4/3

f(x)1 l1/2
t
s
f(x)x l1/2
15
Braesss Paradox

• Traffic rate r1
• Optimal costNash cost2(1/211/21/2)3/2

fl(x)1 l1/2
f(x)x l1/2
v
t
s
f(x)x l1/2
f(x)1 l1/2
w
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Braesss Paradox

• Traffic rate r1
• Optimal cost did not change
• Nash cost1101112
• Adding edge negatively impact all agents

fl(x)1 l0
f(x)x l1
v
t
f(x)0 l1
s
f(x)x l1
f(x)1 l0
w
17
Related Work-General Network

• Roughgarden and Tardos (FOCS 2000)
• Assumption each user controls a negligible
  fraction of the overall traffic
• Results
• Linear latency functions - R4/3
• Continuous nondecreasing functions-bicriteria
  results
• Results hold also for nonnegligible splittable
  case (Roughgarden SODA 2005)
• Without negligibility assumption no general
  results

18
Our Results

• Unsplittable Flow, general demands
• Linear Latency Functions
• For weighted demands the price of anarchy is
  exactly 2.618 (pure and mixed)
• For unweighted demands the price of anarchy is
  exactly 2.5.
• Polynomial Latency Functions
• The price of anarchy - at most O(2ddd1) (pure
  and mixed)
• The price of anarchy - at least O(dd/2)

19
Remarks

• Valid for congestion games
• Approximate computation
• (i.e approximate Nash) has limited affect

20
Routes in Nash Equilibrium

• Pure strategies user j selects single path Q?
  Qj
• Mixed strategies user j selects a probability
  distribution pQ,j over all paths Q? Qj

21
Routes in Nash Equilibrium

• Definition ( Pure Nash equilibrium)
• System S of pure strategies is in Nash
  equilibrium iff
• for every j ?1,...,nand Q ? Qj
• , where
• Qj path associated with request j

22
Example
Latency function f(x)x
Path Q1
1
USER 1 W11
2
1
2
1
Path Q
t
s
2
2
2
2
CQ1,1 2125
CQ,1 2(11)(11)28
23
Routes in Nash Equilibrium

• Definition
• The expected cost C(S) of system S of mixed
  strategies is
• (i.e. the expected total latency incurred by S)

24
Linear Latency Functions

• fe(x)aexbe for each e?E
• Theorem
• For linear latency functions (pure strategies)
  and weighted demands R 2.618
• Proof
• For simplicity we assume f(x)x
• Qj - the path of request j in system S
• -set of requests that are
  assigned to edge e
• - load of
  edge e
• For optimal routes Qj , J(e) , le

25
Weighted Sum of Nash Eq.

• According to the definition of Nash equilibrium
• We multiply by wj and get
• We sum for all j, and get

26
Classification

• Classifying according to edges indices J(e) and
  J(e), yields
• Using ,
  we get

27
Transformation

• Using Cauchy Schwartz inequality, we obtain
• Define and divide by
  
• Then

28
Unweighted Demands

• Theorem
• For linear latency functions, pure strategies
  and unweighted demands R 2.5.
• Proof

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Proof

• As in the previous proof
• Using ,
  we get

30
Proof

• Applying properties
• Then

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Linear Latency Functions

• Theorem
• For linear latency functions and weighted
  demands
• R2.618.
• Proof
• We consider a weighted network congestion game
  with four players

32
Linear Latency Functions
v
Player 1 (u,v, f) Player 2 (u,w, f) Player 3
(v,w, 1) Player 4 (w,v, 1)
0
x
u
x
x
x
0
w
OPTNASH12f2 212 2f4
33
Linear Latency Functions
v
Player 1 (u,v, f) Player 2 (u,w, f) Player 3
(v,w, 1) Player 4 (w,v, 1)
0
x
u
x
x
x
0
w
NASH22(f1)2 2f2 8 f 6
R f12.618
34
Linear Latency Functions

• Theorem
• For linear latency functions and unweighted
  demands
• R2.5.
• Proof
• The same example as in the weighted case with
  unit demands

35
Mixed Strategies

• Definition (Nash equilibrium)
• System S of mixed strategies is in Nash
  equilibrium iff
• for every j ?1,...,nand Q,Q ? Qj, with
  pQ,jgt0
• cQ,j cQ,j where
• XQ,j indicates whether request j is assigned to
  path Q
• - load of edge e

36
Mixed Strategies

• Theorem
• For linear latency functions (mixed strategies)
  and weighted demands R 2.618.
• Proof
• Let pQ,j be the probability distribution of the
  system S.
• The expected latency of user j for assigning his
  request to path Q in S is

37
Step 1

• According to the definition of Nash equilibrium
  for , hence
• We multiply by pQ,jwj and get

38
Step 2

• Sum over all paths and all users and classify
  according to the edges
• Augment to
• Obtain the same inequality as in the pure
  strategies case

39
General Latency Functions

• General functions-no bicriteria results
• Polynomial Latency Functions
• The price of anarchy - at most O(2ddd1) (pure
  and mixed)
• The price of anarchy - at least O(dd/2)

40
Polynomial Latency Functions

• Theorem
• For polynomials of degree d latency functions R
  
• O(dd/2).
• Proof
• We use the construction of Awerbuch et. al for
  the parallel links restricted assignment model.

41
Example
l3
OPT
Group 1
Group 2
Group 3
m0
m0
m0
m0
m0
m1
m0
m1
m1
m1
m1
m1
m2
m2
m2
m3
Group 3
NASH
Group 2
Group 1
m0
m0
m0
m0
m0
m1
m0
m1
m1
m1
m1
m1
m2
m2
m2
m3
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The Construction

• Total ml! links each has a latency function
  f(x)xd
• l1 type of links
• For type k0l there are mkT/k! links
• l types of tasks
• For type k1l there are kmk jobs, each can be
  assigned to link from type k-1 or k
• OPT assigns jobs of type k to links of type k-1
  one job per link.

43
System of Pure Strategies

• System S of pure strategies
• Jobs of type k are assigned to links of type k
• k jobs per link
• Lemma
• The System S is in Nash Equilibrium.

44
The Coordination Ratio

45
Summary

• We showed results for general networks with
  unsplittable traffic and general demands
• For linear latency functions and weighted
  demands R2.618
• For linear latency functions and unweighted
  demands R2.5
• For Polynomial Latency functions of degree d ,
• Rd?(d)

46
Related Work-Machines Model

• Main references
• Koutsoupias and Papadimitriou (STACS 99)
• Mavronicolas and Spirakis (STOC 2001)
• Czumaj and Vocking (SODA 2002)
• Awerbuch, Azar, Richter and Tsur (WAOA 2003)

47
Related Work-Machines Model

• Main results (global cost maximum users cost)
• For m identical links, identical jobs (pure) R1
• For m identical links (pure) R2
• For m identical links (mixed)
• R- Price of Anarchy

48
Mixed Strategies -Example

• Machines model
• (nm)
• Pure strategy Assign job i to link i
• maximum cost1
• Mixed strategy assign jobs to links uniformly
  at random

49
Related Work (Cont)

• Main results
• For 2 related links R1.618
• For m related links / restricted assignment
  (pure)
• For m related links / restricted assignment
  (mixed)
• R- Price of Anarchy