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 Users are selfish
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 Nash Equilibrium
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-Prisoner’s Dilemma
Nash Equilibrium (c,c) D C
(-1,-1)
(-4,0)
(0,-4)
(-3,-3)

6. Nash Equilibrium H T (1,-1) (-1,1)
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
(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 1 4 3
Nash Equilibrium m1 m2

9. Nash Equilibrium - Example1 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 user’s 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 2 1 1 t s 2 2 2 2 Latency=2+1+2=5
Total latency =Σe fe(le)·le= Σe le· le =6·2·2+3·1·1=27

13. Example Traffic rate r=1 Nash total latency=1·0+1·1=1 f(x)=1 l=0 t s
f(x)=x
l=1

14. Example Traffic rate r=1 Optimal total latency=1·1/2+1/2·1/2=3/4 R=4/3
f(x)=1
l=1/2 t s
f(x)=x
l=1/2

15. Braess’s Paradox Traffic rate r=1
Optimal cost=Nash cost=2(1/2·1+1/2·1/2)=3/2
fl(x)=1
l=1/2
f(x)=x
l=1/2 v t s
f(x)=1
l=1/2
f(x)=x
l=1/2 w

16. Braess’s Paradox Traffic rate r=1 Optimal cost did not change
Nash cost=1·1+0·1+1·1=2
Adding edge negatively impact all agents
fl(x)=1
l=0
f(x)=x
l=1 v
f(x)=0
l=1 t s
f(x)=1
l=0
f(x)=x
l=1 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 - R=4/3
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 (pure and mixed)
For unweighted demands the price of anarchy is exactly 2.5.
Polynomial Latency Functions
The price of anarchy - at most O(2ddd+1) (pure and mixed)
The price of anarchy - at least Ω(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,...,n}and Q’  {Qj} :
, where
Qj – path associated with request j

22. Example Latency function f(x)=x Path Q1 1 USER 1 : W1=1 2 2 1 1 Path Q
CQ1,1 =2+1+2=5
CQ,1 =2+(1+1)+(1+1)+2=8

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)=aex+be 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 : Proof :
For linear latency functions, pure strategies and unweighted demands R ≤ 2.5.
Proof :

29. Proof
As in the previous proof
Using , we get

30. Proof
Applying properties
Then

31. Linear Latency Functions
Theorem :
For linear latency functions and weighted demands
R≥2.618.
Proof:
We consider a weighted network congestion game with four players

32. Linear Latency Functionsv
Player 1 : (u,v, φ)
Player 2 : (u,w, φ)
Player 3 : (v,w, 1)
Player 4 : (w,v, 1) x u x x x w
OPT=NASH1=2φ2 + 2·12 = 2φ+4

33. Linear Latency Functionsv
Player 1 : (u,v, φ)
Player 2 : (u,w, φ)
Player 3 : (v,w, 1)
Player 4 : (w,v, 1) x u x x x w
NASH2=2(φ+1)2 + 2·φ2 = 8 φ +6
R= φ+1=2.618

34. Linear Latency Functions
Theorem :
For linear latency functions and unweighted demands
R≥2.5.
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,...,n}and Q,Q’  {Qj}, with pQ,j>0 :
cQ,j ≤ cQ’,j where
XQ,j – indicates whether request j is assigned to path Q
- load of edge e

36. Mixed Strategies Theorem : Proof :
For linear latency functions (mixed strategies) and weighted demands R ≤
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,j·wj 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(2ddd+1) (pure and mixed)
The price of anarchy - at least Ω(dd/2)

40. Polynomial Latency Functions
Theorem :
For polynomials of degree d latency functions R =
Ω(dd/2).
Proof:
We use the construction of Awerbuch et. al for the parallel links restricted assignment model.

41. Example l=3 OPT Group 1 Group 2 Group 3 m0 m0 m0 m0 m0 m0 m1 m1 m1 m1
NASH
Group 2
Group 1 m0 m0 m0 m0 m0 m0 m1 m1 m1 m1 m1 m1 m2 m2 m2 m3

42. The Construction Total m=l! links each has a latency function f(x)=xd
l+1 type of links
For type k=0…l there are mk=T/k! links
l types of tasks
For type k=1…l there are k·mk 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 R=2.618
For linear latency functions and unweighted demands R=2.5
For Polynomial Latency functions of degree d ,
R=dӨ(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 user’s cost)
For m identical links, identical jobs (pure) R=1
For m identical links (pure) R=2
For m identical links (mixed)
R- Price of Anarchy

48. Mixed Strategies -Example
Machines model
(n=m)
Pure strategy – Assign job i to link i
maximum cost=1
Mixed strategy – assign jobs to links uniformly at random

49. Related Work (Cont’) Main results R- Price of Anarchy
For 2 related links R=1.618
For m related links / restricted assignment (pure) :
For m related links / restricted assignment (mixed) :
R- Price of Anarchy