Price of Anarchy Bounds Price of Anarchy Convergence Based on Slides by Amir Epstein and by Svetlana...
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Transcript of Price of Anarchy Bounds Price of Anarchy Convergence Based on Slides by Amir Epstein and by Svetlana...
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)