worst-case equilibria elias koutsoupias and christos papadimitriou
DESCRIPTION
Worst-case Equilibria Elias Koutsoupias and Christos Papadimitriou. Tight Bounds for Worst-case Equilibria Artur Czumaj and Berthold Vocking. Presenter: Yishay Mansour. Outline. Motivation Model Unit speed links Weighted speed links. Motivation. Internet users: - PowerPoint PPT PresentationTRANSCRIPT
Worst-case Equilibria Elias Koutsoupias and Christos Papadimitriou
Presenter: Yishay Mansour
Tight Bounds for Worst-case Equilibria
Artur Czumaj and Berthold Vocking
Outline
• Motivation
• Model
• Unit speed links
• Weighted speed links
Motivation
• Internet users:– very selfish and spontaneous behavior,
– No one is thinking to achieve the “social optimum”.
• Game theory as an analysis tool: – rational behavior and Nash Equilibrium.
• Nash equilibrium:– no optimization of overall system performance. – design mechanisms that encourage behaviors
close to the social optimum.
Motivation
• Nash Equilibrium versus global optimum
• Many cases: best Nash Equilibrium is global (social) optimal
• Worse case analysis– Compare worse Nash to optimum– How bad can things get
Current Work
• Coordination ratio - the ratio between – the worst possible Nash equilibrium and– social (global) optimum
• This works:– Very simple network model.– Derive upper and lower bounds.– Evaluate the price due to lack of coordination.
Model
• Simple routing model:– Two nodes
– m parallel links with speeds si– n jobs/connection weights wj
• Load model:– The delay of a connection is proportional to
load on link
Cost Measure
• Each job selects a link• Jobs(j) jobs assigned to link j• Cost of jobs assigned to link j
– Lj = j in Jobs(i) wj /sj
• Total cost of a configuration– Maxj {Lj}
• Social optimum– Min Maxj {Lj }
Nash Equilibria
• Each job i assigns a probability p(i,j) to link j– Support(i) = { j : p(i,j) > 0}
– Deterministic: one p(i,j) =1 other p(i,j’)=0
• Expected link j load– E[Lj] = i p(i,j) wi / sj
• Job i view of link j:– Cost(i,j) = wi /sj+ ki p(k,j) wk / sj = E[Lj] + (1-p(i,j))wi
– Cost after job i moves to link j
Nash Equilibria
• For every job i
• Min_cost(i) = MINj cost(i,j)
• For every link j:– IF cost(i,j) > min_cost(i) THEN p(i,j)=0
Example
• Two links, unit speed:– s1 = s2 =1
• Social optimum is hard:– Problem is NP-complete– Partition
• Two trivial lower bounds:– Max weight job: wmax = MAXi {wi}– Average over machines: i wi /m
Example I
• Deterministic Example– 2 jobs of weight 2– 2 jobs of weight one
• Optimum = 3
• Nash = 4
• Coordination ratio 4/3
Example
• Stochastic Example– 2 jobs of weight 2
• Optimum = 2
• Nash: – P(i,j)= ½– Expected Cost = 3
• Coordination ratio 3/2
Upper bound: Deterministic
• Load L1 and L2; L1 > L2
• Difference at most wmax; L1 – L2 = v wmax
• Nash_Cost = L1
– IF L2 > v/2 THEN • OPT_cost L2 + v/2 • Nash cost = L2 + v • Coordination ratio 3/2
– Otherwise • opt_cost wmax & L1 (3/2 )wmax
• Coordination ratio 3/2
Upper Bound: Stochastic
• Contribution probability qi of job i:– Probability that it is in the unique max load link
(assume tie breaker)– Cost = i qi wi
• Collision probability t(i,k) of jobs i and k– Probability they select the same link– Both contribute to social cost only if they
collide:• qi + qk 1+t(i,k)
Upper bound proof
• Lemma: ik t(i,k) wk = min_cost(i) – wi
• Claim:
• Theorem: The coordination ratio for two unit speed links is 3/2
ii i w
m
m
m
w 1 )min_cost(i
Unit speed: many links – DET.
• Lmax = MAX Lj ; Lmin = MIN Lj
• Lmax – Lmin wmax
• IF Lmin wmax THEN
– OPT cost wmax & Lmax 2 wmax
• OTHERWISE: – OPT cost Lmin & Lmax 2 Lmin
• Coordination ratio 2
Unit speed: many links – STOCH.
• Lower bound:– m links m jobs– p(i,j) =1/m– m balls in to m buckets.– Probability of k balls approx. 1/ kk
– Need probability of 1/m– Max load ( log m / log log m)
Unit speed: many links – STOCH.
• Upper bound: – Nash load 2 OPT
– Large deviation bound.
– bound α by log m / log log m
i i wXE
i ii i
eXEX
/][
][)1(Pr
Multiple speeds:
• Each link i has speed si
• Assume s1 ≥ ... ≥ sm
Multiple speeds: Lower bound
• Let K = log m /log log m• K+1 groups of links
– Nj links in group j
• Nk = m
• Nj = (j+1) Nj+1
• N0 = K! m
• Group k has speed 2k
• Assignment:– Each Link in group k has k jobs of weight 2k
Multiple speeds: Lower bound
• Configuration load = K
• OPT load < 2
• System in Nash
• Lower bound for deterministic NASH
Multiple speeds: Upper bound
• c = MAX E[Lj]
• LEMMA:
ms
s
m
mOc 1log,
loglog
logmin OPT
Multiple speeds: Upper bound
• C = E[ MAX{Lj}]
• LEMMA:
cm
mOC
log OPTlog
log OPT
Expected Load I
• Let Jk =r if the least index link with load
less than k*OPT is r+1
• Every link j Jk has load at least k*OPT
• Link Jk+1 has load less than k*OPT
• Let c* = (c-OPT)/OPT• Target: show that J1 > c*!
• Since J1 m then a [log m /log log m] bound.
Expected Load I
• Claim: E[L1] c –OPT
• Proof: By contradiction– consider the most loaded link – Any job J from it can move to link 1– Its running time of link 1 is at most OPT– Job J improves its load.
• Corollary: Jc* 1
Expected Load I
• Lemma: Jk (k+1) Jk+1
• Proof: T are jobs in links 1 to Jk+1
– Claim: OPT can not allocate job from T to link r>Jk
• Jobs in T observe load at least (k+1)*OPT
• Link Jk+1 has load less than k*OPT.
• No job from T wants to move to link Jk+1=u
• Minimum weight in T at least su*OPT
• On any link r>u any job from T will run more than OPT
Expected Load I
– Claim: IF OPT allocates jobs from T to links 1 to Jk
THEN Jk (k+1) Jk+1
• W sum of weights of jobs in T
• W j sj E[Lj] (k+1) OPT j J(k+1) sj
• Since OPT allocate jobs in T in links 1 to Jk
• W OPT j J(k) sj
j J(k) sj (k+1)j J(k+1) sj
• Since link speeds are decreasing claim follows.
Expected Load II
• c=O( log (s1 / sm) )
– CLAIM: for 1 k c-3
– Corollary: sm 2-(c-5)/2 s1
– Or: c 2 log (s1 /sm) + O(1)
11 22 kk JJ ss
Proof
• OPT schedule some job i:– Nash in j in {1 .. Jk+2 }
• cost(i,j) (k+2)*OPT
– OPT in j’ in {Jk+2+1 , ... m}
• wi SJ(k+2)+1OPT
– cost(i,Jk+1) k*OPT + wi/ sJ(k)+1
• Nash implies:– cost(i,j) cost(i,Jk+1)
Expected Maximum Load
• Large deviation result
• Each link near its expectation.
• Separates small and large jobs
• Large jobs: contribution proportional to weight.
• Small jobs: use Hoeffding relative bound.