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Load Balancing, Multicast routing, Price of Anarchy and Strong Equilibrium. Computational game theory Spring 2008 Michal Feldman. Load Balancing Model: Unrelated Machines. machines. Set of machines M = {M 1 ,…,M m } Set of jobs N = {1,…,n} - PowerPoint PPT Presentation
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Load Balancing, Multicast routing, Price of Anarchy and Strong
Equilibrium
Computational game theorySpring 2008
Michal Feldman
Load Balancing Model: Unrelated Machines
• Set of machines M = {M1,…,Mm}• Set of jobs N = {1,…,n}• Unrelated machines model:
Job (player) i has load wij on machine j
• Strategy: select a machine
• Cost of a job = total load on selected machine
• Objective: minimize makespan (max load)
• Special cases: – Identical machines: wij=wij’ for all j,j’– Related machines: each machine j has a speed sj, and
each job i has load li, and wij=li/sj
M1M2
J157
J223
J341
5
4
3
L1(s)=9M1 M2
L2(s)=3
jobs
machines
(pure) equilibrium existence
• Potential function– Identical machines: sum of squares (why?)– Unrelated machines:
• Does sum of squares work? • No !
• Before migration: 10, after migration: 9, so cost decreased• Yet, sum of squares increased from 102+52 to 92+92
105
1 4
Lexicographic order
• Definition: a vector (l1,…lm) is smaller than (l1’,…,lm’) lexicographically if for some i, li < li’ and lk = lk’ for all k<I
• Definition: A joint action s is smaller than s’ lex. (ss’) if the vector of machine loads L(s), sorted in non-decreasing order, is smaller lex. than L(s’)
s
s’ss’
(Pure) NE Existence• Lemma: if a job i improves its cost by migration, then the
lexicographic order decreases • Proof sketch:
– a job migrating from blue machine to red machine– Only the load on these two machines change (blue decreases, red
increases)– But if the migrating job improves, red (in post-migration) must be
lower than blue (in pre-migration)– Thus after migration, both blue and red are lower than blue prior to
migration– Thus profile decreases lexicographically
• Conclusion 1: load balancing game admit a Nash equilibrium in pure strategies
• Conclusion 2: price of stability of any load balancing game is 1
Price of Anarchy for identical machines
• Theorem: in any load balancing game on identical machines, it holds that
• Proof: – Let s be a NE and let s* be OPT– Let i’ be a machine with highest cost in s, and let j’ be job
with lowest weight on machine i’– wlog, at least 2 jobs on machine i’ (why?), thus w j’≤ ½
cost(s)– Since s is a NE, for any machine i≠I’ (job j’ stays)
• li ≥ li’ – wj’ ≥ cost(s) – ½ cost(s) = ½ cost(s)
122
m
POA
mstm
m
mstst
m
lj
m
wst ji
i
2)(cos)1()1)((cos2
1)(cos*)(cos
122
12
*)(cos)(cos
mmm
stst
Convergence time of best response for identical machines
• Max-weight best response policy: – activate jobs, always activating job of max-weight
among unsatisfied jobs– activated job migrates to its best machines (i.e.,
performs a best-response)• Theorem: for any load balancing game on
identical machines, the max-weight best response policy converges to a NE, after each agent was activated at most once (from any initial profile)
Convergence time of best response for identical machines
• Proof sketch:– Claim: once a job was activated, it never gets unsatisfied again– Proof of claim is based on two observations (for identical
machines):• Job is satisfied IFF assigned to machine with minimal load (other than
itself)• Best response never decreases the minimal load among the machines
(why?)– Thus, a job can become unsatisfied only if another job migrated
to its own machine– Thus, sufficient to show that a migration of a job of lower
weight into one’s machine cannot make it unsatisfied– Proof in class..
• Note: order is crucial. Under “min-weight best response policy”, there may be instances with an exponential number of steps
Price of anarchy for unrelated machines
• POA for unrelated machines is unbounded
1
1
Job 1
Job 2
Machine 1 Machine 2
1 1
Machine 2Machine 1Machine 2
Machine 1
Social optimum Nash equilibrium
makespan= makespan=
PoA=1/
Allowing Coordination in Equilibrium
• Strong Equilibrium [Aumann’59]– No coalition can deviate and strictly improve the
utility of all of its members• very robust concept• may be a better prediction of rational behavior• most games do not admit Strong Eq.
– usually applied to pure Eq with pure deviations
Example 1: Prisoner’s Dilemma
0,5
5,0
cooperate
cooperate
defect
defect
Unique Nash Eq.
Strong Eq? .
Prisoner’s dilemma does not admit any Strong Eq .
Strong Price of Anarchy
• Determining SPoA requires two parts:– Proving existence of Strong Eq– Bounding the worst ratio
• SE NE SPoA ≤ PoA
Price of Anarchy (PoA) [KP00]:
optimum socialmequilibriuNash worst
PoA
Strong Price of Anarchy (SPoA):
optimum socialmequilibriu Strongworst
SPoA
k-Strong Equilibrium• A joint action sS is not resilient to a pure
deviation of a coalition if there is a pure action profile of such that ci(s-,)<ci(s) for any i – e.g., (defect,defect) in Prisoner’s dilemma
• A pure Nash Eq sS is resilient to pure deviation of coalitions of size k if there is no coalition of size at most k such that s is not resilient to a pure deviation by
• A k-Strong Equilibrium is a pure Nash Eq that is resilient to pure deviation of coalitions of size at most k
S=S1x…xSn
Strong Equilibrium Hierarchy
1-SE
2-SE
n-SE
=NE
=SE [Aumann]
Related Work
• Existence of Strong Equilibrium– monotone decreasing congestion games [Holzman+Lev-tov
1997, 2003]– monotone increasing congestion games + correlated SE
[Rosenfeld+Tennenholtz 2006]
• Related solution concepts– Coalition-proof Eq. [Bernheim 1987]– Group-strategyproof mechanisms
[Moulin+Shenker 2001]– Coalitions with transferable utilities
[Hayrapetyan et al 2006]
SECPE
NE
Existence of Strong Equilibrium in load balancing games
• Is every Nash Eq. on identical machines also a Strong Eq ?– NO ! (for m ≥ 3)
5
5
4 4
3 3
10 7 7
s
55
4 433
6 99
s’Coalition: 5,5,3,3
Strong Eq. Existence
• Theorem: in any load balancing game, the lex. minimal joint action s is a k-SE for any k
Recall Lexicographic Order
• Definition: a vector (l1,…lm) is smaller than (l1’,…,lm’) lexicographically if for some i, li < li’ and lk = lk’ for all k<I
• Definition: A joint action s is smaller than s’ lex. (ss’) if the vector of machine loads L(s), sorted in non-decreasing order, is smaller lex. than L(s’)
s
s’ss’
Proof of SE Existence• Suppose in contradiction that s (lex. minimal) is not a SE, and
let be the smallest coalition (deviating to s’).• Claim: the same set of machines are chosen by in s and in s’
(denote it M())– If a job migrates TO some
machine, another jobmigrates FROM it
• else contradicting s is NE – If a job migrates FROM some
machine, another jobmigrates TO it
• else contradicting minimality of • Since all jobs in must benefit, all loads of M() in s’ must be
smaller than max load of M() in s – Contradicting minimality of s
Price of Anarchy (PoA)• Recall: for unrelated machines, PoA may be unbounded
1
1
Job 1
Job 2
Machine 1 Machine 2
Objective: min makespan
Social optimumNash equilibrium Nash equilibrium
PoA=1/
1 1
M1
makespan=
M2
makespan=
M1 M2
Strong equilibrium Strong equilibrium
SPoA=1
Strong Price of Anarchy
• Theorem: for any job scheduling game with m unrelated machines and n jobs, SPoA ≤ m
Proof for SpoA ≤ m
• Claim 1: L1(s) ≤ OPT– else: coalition of all jobs to OPT
M1Mm Mi Mi-1 M1Mm Mi Mi-1
OPT
L1(s)
OPT
L1(s)
Proof for SpoA ≤ m
• Claim 1: L1(s) ≤ OPT– else: coalition of all jobs to OPT
• Claim 2: i Li(s)-Li-1(s) ≤ OPT – else: consider s’, where all jobs on machines i..m go to OPT. For all J
• cJ(s) > Li-1(s) + OPT• cJ(s’) ≤ Li-1(s) + OPT (since all J together add at most OPT)
M1Mm Mi Mi-1 M1Mm Mi Mi-1
>OPT OPT
Lm(s) ≤ m OPT
Li-1(s) L1(s)
Li(s)
Lower Bound (m machines)• Theorem: there exists a job scheduling game with m unrelated
machines for which SPoA ≥ m• Proof:
M1M2M3M4Mm
J111
J212
J313
J414
Jm1m
OPT = 1
makespan=mSE
Identical Machines
• Theorem: there exists a job scheduling game with m identical machines and n jobs, such that
m
SPoA 11
2
12m-1mJ1
Jm
Jm+1
J2m
1
1/m1 m-2 m-1m
OPT
SE
1+1/m
2
Mixed Deviations and Mixed Strong Eq
• Nash Eq – unilateral deviations– pure and mixed deviations are equivalent
• Strong Eq – coordinated deviation– Pure and mixed deviations are not equivalent– Given a mixed deviation, there might be no single pure
deviation which is good
J1
M1 M2
J3
J2Unique Nash Eq
J1
J2
¾ ¼
cJ1=cJ2=15/8
cJ1=cJ2=2
mixed deviation
J1
M1 M2
J3
J2
½½
Mixed Deviations and Mixed Equilibrium
• However, in many cases, allowing mixed deviations by a coalition eliminates all Nash Eq.
• Theorem: for m≥5 identical machines, and n>3m unit jobs, there is no 4-Strong Eq when mixed deviations are allowed– Based on a lemma that shows that the support of any two
“mixing” jobs must be disjoint
Strong equilibrium in multicast routing
Theorem: There exists a multicast routing game that does not posses a strong equilibrium.
Proof:
s
t1 t2
2
1 1
-½3ε-½3ε2+2ε
1-2ε
1+3εUnique NE: c1(S) = c2(S) = 2/2+1=2
deviation: ci(S) < 2
No SE in game
Strong Price of Anarchy
Theorem : The strong price of Anarchy of a multicast routing game with n players is at most H(n).
Proof:• Let S be a SE, and SΓ be the induced profile of players in Γ, and
let S* be OPT• For k=n,…,1, since S is SE, there exists a player
“k” k={1,…,k} that does not benefit from coal. deviation. i.e., ))(())(( )( ),()( 1
**** kkkkk SSScSScSc
kkk
))(()()(
1
SnHcj
cS eEe
eEe
Sn
j
ee
Potential function:
Proof (cont’d)
• We got for every k: • Summing over all players:
)()()(
))(()(
))(())(()(
*
*
**
**
OPTnHnHc
SnHcS
SSSc
See
eSe
e
nNi
i
))(())(()( 1**
kkk SSSc
Lower Bound• OPT: all users use indirect edge, c(OPT)=1+• Unique NE and SE: each user uses direct edge to ti,
c(NE)=c(SE)=H(n) PoA = SPoA = PoS = SPoS = H(n)
21
n
t1 tn-2t3t2 tn-1 tn
s
1 31
11
n n1 1+2
1