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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:

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worst case equilibria elias koutsoupias and christos papadimitriou

Worst-case EquilibriaElias Koutsoupias and Christos Papadimitriou

Tight Bounds for Worst-case EquilibriaArtur Czumaj and Berthold Vocking

Presenter: Yishay Mansour

outline
Outline
  • Motivation
  • Model
  • Unit speed links
  • Weighted speed links
motivation
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.
motivation1
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
slide5

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
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
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
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+ ki p(k,j) wk / sj = E[Lj] + (1-p(i,j))wi
    • Cost after job i moves to link j
nash equilibria1
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
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
Example I
  • Deterministic Example
    • 2 jobs of weight 2
    • 2 jobs of weight one
  • Optimum = 3
  • Nash = 4
  • Coordination ratio  4/3
example1
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
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
Upper Bound: Stochastic
  • Contribution probability qi of job i:
    • Probability that it is in the unique max load link (assume tie breaker)
    • Cost = iqi 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
Upper bound proof
  • Lemma: ikt(i,k) wk = min_cost(i) – wi
  • Claim:
  • Theorem: The coordination ratio for two unit speed links is 3/2
unit speed many links det
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
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
    • Needprobabilityof 1/m
    • Max load ( log m / log log m)
unit speed many links stoch1
Unit speed: many links – STOCH.
  • Upper bound:
    • Nash load  2 OPT
    • Large deviation bound.
    • bound α by log m / log log m
multiple speeds
Multiple speeds:
  • Each link i has speed si
  • Assume s1 ≥ ... ≥sm
multiple speeds lower bound
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 groupk has k jobs of weight 2k
multiple speeds lower bound1
Multiple speeds: Lower bound
  • Configuration load = K
  • OPT load < 2
  • System in Nash
  • Lower bound for deterministic NASH
multiple speeds upper bound
Multiple speeds: Upper bound
  • c = MAX E[Lj]
  • LEMMA:
multiple speeds upper bound1
Multiple speeds: Upper bound
  • C = E[ MAX{Lj}]
  • LEMMA:
expected load i
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 i1
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 i2
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 i3
Expected Load I
  • Claim: IF OPT allocates jobs from T to links 1 to Jk

THENJk (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
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)
proof
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
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.