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.