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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)