Worst-case%20Equilibria%20Elias%20Koutsoupias%20and%20Christos%20Papadimitriou

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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

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Outline

• Motivation
• Model
• Unit speed links
• Weighted speed links

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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.

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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

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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.

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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

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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

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Nash Equilibria

• Each job i assigns a probability p(i,j) to link j
• Support(i) j p(i,j) gt 0
• Deterministic one p(i,j) 1 other p(i,j)0
• Expected link j load
• ELj ?i p(i,j) wi / sj
• Job i view of link j
• Cost(i,j) wi /sj ?k?i p(k,j) wk / sj ELj
  (1-p(i,j))wi
• Cost after job i moves to link j

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Nash Equilibria

• For every job i
• Min_cost(i) MINj cost(i,j)
• For every link j
• IF cost(i,j) gt min_cost(i) THEN p(i,j)0

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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

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Example I

• Deterministic Example
• 2 jobs of weight 2
• 2 jobs of weight one
• Optimum 3
• Nash 4
• Coordination ratio ? 4/3

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Example

• Stochastic Example
• 2 jobs of weight 2
• Optimum 2
• Nash
• P(i,j) ½
• Expected Cost 3
• Coordination ratio ? 3/2

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Upper bound Deterministic

• Load L1 and L2 L1 gt L2
• Difference at most wmax L1 L2 v ? wmax
• Nash_Cost L1
• IF L2 gt 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

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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 ? 1t(i,k)

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Upper bound proof

• Lemma ?i?k t(i,k) wk min_cost(i) wi
• Claim
• Theorem The coordination ratio for two unit
  speed links is 3/2

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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

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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)

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Unit speed many links STOCH.

• Upper bound
• Nash load ? 2 OPT
• Large deviation bound.
• bound a by log m / log log m

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Multiple speeds

• Each link i has speed si
• Assume s1 ... sm

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Multiple speeds Lower bound

• Let K log m /log log m
• K1 groups of links
• Nj links in group j
• Nk ?m
• Nj (j1) Nj1
• N0 K! ?m
• Group k has speed 2k
• Assignment
• Each Link in group k has k jobs of weight 2k

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Multiple speeds Lower bound

• Configuration load K
• OPT load lt 2
• System in Nash
• Lower bound for deterministic NASH

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Multiple speeds Upper bound

• c MAX ELj
• LEMMA

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Multiple speeds Upper bound

• C E MAXLj
• LEMMA

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Expected Load I

• Let Jk r if the least index link with load
• less than kOPT is r1
• Every link j ? Jk has load at least kOPT
• Link Jk1 has load less than kOPT
• Let c ?(c-OPT)/OPT?
• Target show that J1 gt c!
• Since J1 ? m then a log m /log log m bound.

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Expected Load I

• Claim EL1 ? 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

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Expected Load I

• Lemma Jk ? (k1) Jk1
• Proof T are jobs in links 1 to Jk1
• Claim OPT can not allocate job from T to link
  rgtJk
• Jobs in T observe load at least (k1)OPT
• Link Jk1 has load less than kOPT.
• No job from T wants to move to link Jk1u
• Minimum weight in T at least suOPT
• On any link rgtu any job from T will run more than
  OPT

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Expected Load I

• Claim IF OPT allocates jobs from T to links 1 to
  Jk
• THEN Jk ? (k1) Jk1
• W sum of weights of jobs in T
• W ? ?j sj ELj ? (k1) OPT ?j ?J(k1) sj
• Since OPT allocate jobs in T in links 1 to Jk
• W ? OPT ?j ?J(k) sj
• ?j ?J(k) sj ? (k1)?j ?J(k1) sj
• Since link speeds are decreasing claim follows.

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Expected Load II

• cO( log (s1 / sm) )
• CLAIM for 1 ? k? c-3
• Corollary sm ? 2-(c-5)/2 s1
• Or c ? 2 log (s1 /sm) O(1)

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Proof

• OPT schedule some job i
• Nash in j in 1 .. Jk2
• cost(i,j) ? (k2)OPT
• OPT in j in Jk21 , ... m
• wi ? SJ(k2)1OPT
• cost(i,Jk1) ? kOPT wi/ sJ(k)1
• Nash implies
• cost(i,j) ? cost(i,Jk1)

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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.