Price of Anarchy BoundsPrice of Anarchy Convergence

Based on Slides by Amir Epstein and by Svetlana Olonetsky

Modified/Corrupted by Michal Feldman and Amos Fiat

Equal Machine Load Balancing = Parallel Links

• 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

Price of Anarchy

• Price of Anarchy:

The worst possible ratio between: - Objective function in Nash Equilibrium and- Optimal Objective function

• Objective function: total user cost, total user utility, maximal/minimal cost, utility, etc., etc.

Identical machines

• Main results (objective function – maximum load)- For m identical links, identical jobs (pure) R=1- For m identical links (pure) R=2-1/(m+1)- For m identical links (mixed)

m

mR

loglog

log

Lower bound – easy : uniformly choose machine with prob. 1/mUpper bound – assume opt = 1, opt = max expected ≤ 2 in NE (otherwise not NE,

NE = expected max ≤ log m / loglog m due to Hoeffding concentration inequality

Related Work (Cont’)

• Main results

- For 2 related links R=1.618

- For m related links (pure)

- For m related links (mixed)

- For m links restricted assignment (pure)

- For m links restricted assignment (mixed)

m

mR

loglog

log

m

mR

logloglog

log

m

mR

loglog

log

m

mR

logloglog

log

• m (=3) machines• n (=4) jobs

• vi – speed of machine i

• wj – weight of job j

v1 = 4 v2 = 2 v3 = 1

1 (4) 2 (4) 2 (2)

1 (2)

L1 = 1 L2 = 3 L3 = 2

• Li – load on machine i

Price of Anarchy: Lower Bound

k! / (k-i)!

Gi

k-i

k !1

k

Gk

k

k-1

k(k-1)

k-2

G0 G1 G2

v=2k-i v=1v=2k

w=2k-iw=2k

v=2w=2

v=2k-1

w=2k-1

Price of Anarchy: Lower Bound

Gi

k-i

k !1

k

Gk

k

k-1

k(k-1)

k-2

G0 G1 G2k! ~ m

k ~ log m / log log m

k! / (k-i)!

v=2k-i v=1v=2k

w=2k-iw=2k

v=2w=2

v=2k-1

w=2k-1

11

Its a Nash Equilibrium

Gi

k-i

k !1

k

Gk

k

k-1

k(k-1)

k-2

G0 G1 G2

k! / (k-i)!

2

v=2k-i v=1v=2k

w=2k-iw=2k

v=2w=2

v=2k-1

w=2k-1

1

Its a Nash Equilibrium

Gi

k-i

k !1

k

Gk

k

k-1

k(k-1)

k-2

G0 G1 G2

k! / (k-i)!

2 4

v=2k-i v=1v=2k

w=2k-iw=2k

v=2w=2

v=2k-1

w=2k-1

1

The social optimum

k! / (k-i)!

Gi

k-i

k !1

k

Gk

k

k-1

k(k-1)

k-2

G0 G1 G2

21

v=2k-i v=1v=2k

w=2k-iw=2k

v=2w=2

v=2k-1

w=2k-1

The social optimum

k! / (k-i)!

Gi

k-i

k !1

k

k

k-1

k(k-1)

k-2

G0 G1 G2

2

v=2k-i v=1v=2k

w=2k-iw=2k

v=2w=2

v=2k-1

w=2k-1

1

Gk

12

The social optimum

k! / (k-i)!

Gi

k-i

k !1

k

k

k-1

k(k-1)

k-2

G0 G1 G2

2

2 22 2

v=2k-i v=1v=2k

w=2k-iw=2k

v=2w=2

v=2k-1

w=2k-1

Gk

Related Machines: Price of Anarchy upper bound

• Normalize so that Opt = 1

• Sort machines by speed

• The fastest machine (#1) has load Z, no machine has load greater than Z+1 (otherwise some job would jump to machine #1)

• We want to give an upper bound on Z

Related Machines: Price of Anarchy upper bound

• Normalize so that Opt = 1• The fastest machine (#1) has load Z, but

Opt is 1, consider all the machines that Opt uses to run these jobs.

• These machines must have load ≥ Z-1 (otherwise job would jump from #1 to this machine)

• There must be at least Z such machines, as they need to do work ≥ Z

Related Machines: Price of Anarchy upper bound

• Take the set of all machines up to the last machine that opt uses to service the jobs on machine #1.

• The jobs on this set of machines have to use Z(Z-1) other machines under opt.

• Continue, the bottom line is that n ≥ Z!, or that Z ≤ log m / log log m

Restricted Assignment to Machines

m0 m0 m0 m0 m0 m1m0 m1 m1 m1 m1 m1 m2 m2 m2 m3

NASH

Group 1

m0 m0 m0 m0 m0 m1m0 m1 m1 m1 m1 m1 m2 m2 m2 m3

Group 2 Group 3

Group 1

Group 2

Group 3

OPT

l=3

Network models (Many models)

• Symmetric (all players go from s to t)– No weights on the players (all bandwidth

requests are one)– Arbitrary monotonic increasing link delay

function – Polynomial time– How bad a solution can this be?

Network models (Many models)

• Asymmetric with weights – Negligible load (one car out of 100,000 cars

traveling from Tel Aviv to Jerusalem) Famouse as Waldrop equilibrium

– Atomic Splitable (the cars are all controlled by one agent, but the agent can split the routes taken by the cars)

– Atomic Unsplitable (all cars / oil / communications must flow through the same path.

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

Example

st

Latency=2+1+2=5

Latency=2+2+2+2=8

Latency function f(x)=x

Total latency =Σe fe(le)·le= Σe le· le=6·2···

Braess’s Paradox – negligible agents

• Traffic rate r=1

• Optimal cost=Nash cost=2(1/2·1+1/2·1/2)=3/2

s t

w

vf(x)=xload=1/2

f(x)=xload=1/2

f(x)=1load=1/2

f (x)=1load=1/2

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

s t

w

vf(x)=xl=1

f(x)=xl=1

f(x)=1l=0

fl(x)=1l=0

f(x)=0l=1

Negligible networks - POA

Roughgarden and Tardos (FOCS 2000)

• Assumption : each user controls a negligible fraction of the overall traffic

• Results : - Linear latency functions - POA=4/3

- Continuous nondecreasing functions-bicriteria results

• Without negligibility assumption : no general results

Azar, Epstein, Awerbuch

• 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(2ddd+1) (pure and mixed)

- The price of anarchy - at least Ω(dd/2)

Remarks

• Valid for congestion games

• Approximate computation

(i.e approximate Nash) has limited affect

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

Example

st

CQ1,1 =2+1+2=5

Latency function f(x)=xPath Q1

USER 1 : W1=1

CQ,1 =2+(1+1)+(1+1)+2=8

Path Q

Linear Latency Functionsfe(x)=aex+be for each eE

Theorem :

For linear latency functions (pure strategies) and weighted demands R ≤ 2.618

Proof:

• For simplicity 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

*

}|{)( jQejeJ

)(eJj

je wl

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

*** )()()()( jjjjjJ Qe

jeQeQe

jeQeQeee

Qe

wlwlll

2

*jje

Qeje

Qe

wwlwljJ

2

*jje

Qejje

Qej

wwlwljJ

Classification

• Classifying according to edges indices J(e) and J*(e), yields

• Using , we get

Ee eJj

jjeEe eJj

je wwlwl)(

2

)( *

d

eeJj

dje

eJjj lwlw *

)(

*

)(e

J(e)jj

**

, ,lw

Ee

eEe

eeEe

e llll2**2

Transformation

• Using Cauchy Schwartz inequality, we obtain

• Define and divide by

• Then

Ee

eEe

eEe

eEe

e llll2*2*22

Eee

Eee

l

l

x2*

2

Ee

el2*

2

531 22

xxx

Linear Latency Functions

Theorem :

For linear latency functions and weighted demands

R≥2.618.

Proof:

We consider a weighted congestion game with four

players

Linear Latency Functions

u

v

w

x

x

0

0

x x

OPT=NASH1=2φ2 + 2·12 = 2φ+4

Player 1 : (u,v, φ)Player 2 : (u,w, φ)Player 3 : (v,w, 1)Player 4 : (w,v, 1)

Linear Latency Functions

u

v

w

x

x

0

0

x x

NASH2=2(φ+1)2 + 2·φ2 = 8 φ +6

R= φ+1=2.618

Player 1 : (u,v, φ)Player 2 : (u,w, φ)Player 3 : (v,w, 1)Player 4 : (w,v, 1)

General Latency Functions

• Polynomial Latency Functions

- The price of anarchy - at most O(2ddd+1) (pure and mixed)

- The price of anarchy - at least Ω(dd/2)

The Construction

• Total m=l! links each has a latency function f(x)=x

• 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.

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.

The Coordination Ratio

)(d OPT

C(S)R Hence

)(d !

1 C(S)

e1!

1

d/2

d/2

1

1

0

l

K

d

l

K

d

kk

kOPT

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)

Summary

• We showed results for general networks with unsplittable traffic and general demands

- For linear latency functions R≤2.618

- For Polynomial Latency functions of degree d ,

R=dӨ(d)