Price of Anarchy Bounds Price of Anarchy Convergence

1
Price of Anarchy BoundsPrice of Anarchy
Convergence

• Based on Slides by Amir Epstein and by Svetlana
  Olonetsky
• Modified/Corrupted by Michal Feldman and Amos Fiat

2
Equal Machine Load Balancing Parallel Links

• Two nodes
• m parallel (related) links
• n jobs
• User cost (delay) is proportional to link load
• Global cost (maximum delay) is the maximum link
  load

3
Price of Anarchy

• Price of Anarchy
• The worst possible ratio between
• Objective function in Nash Equilibrium and
• Optimal Objective function
• Objective function total user cost, total user
  utility, maximal/minimal cost, utility, etc.,
  etc.

4
Identical machines

• Main results (objective function maximum load)
• For m identical links, identical jobs (pure) R1
• For m identical links (pure) R2-1/(m1)
• For m identical links (mixed)

Lower bound easy uniformly choose machine
with prob. 1/m Upper bound assume opt 1, opt
max expected 2 in NE (otherwise
not NE, NE expected max log m /
loglog m due to Hoeffding
concentration inequality
5
Related Work (Cont)

• Main results
• For 2 related links R1.618
• For m related links (pure)
• For m related links (mixed)
• For m links restricted assignment (pure)
• For m links restricted assignment (mixed)

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• m (3) machines
• n (4) jobs
• vi speed of machine i
• wj weight of job j

• Li load on machine i

1 (2)
2 (4)
2 (2)
1 (4)
v1 4
v2 2
v3 1
L1 1
L2 3
L3 2
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Price of Anarchy Lower Bound
G0
G1
G2
Gi
Gk
k
k-i
k-1
k-2
k! / (k-i)!
k!
1
k
k(k-1)
8
Price of Anarchy Lower Bound
G0
k! m k log m / log log m
G1
G2
Gi
Gk
k
k-i
k-1
k-2
k!
1
k
k(k-1)
k! / (k-i)!
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Its a Nash Equilibrium
G0
G1
G2
2
1
1
Gi
Gk
k
k-i
k-1
k-2
k!
1
k
k(k-1)
k! / (k-i)!
10
Its a Nash Equilibrium
G0
G1
G2
4
2
1
Gi
Gk
k
k-i
k-1
k-2
k!
1
k
k(k-1)
k! / (k-i)!
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The social optimum
G0
G1
G2
Gi
Gk
k
k-i
k-1
k-2
k! / (k-i)!
k!
1
k
k(k-1)
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The social optimum
G0
G1
G2
Gi
Gk
k
k-i
k-1
k-2
k! / (k-i)!
k!
1
k
k(k-1)
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The social optimum
G0
G1
G2
Gi
Gk
k
k-i
k-1
k-2
k! / (k-i)!
k!
1
k
k(k-1)
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Related Machines Price of Anarchy upper bound

• Normalize so that Opt 1
• Sort machines by speed
• The fastest machine (1) has load Z, no machine
  has load greater than Z1 (otherwise some job
  would jump to machine 1)
• We want to give an upper bound on Z

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Related Machines Price of Anarchy upper bound

• Normalize so that Opt 1
• The fastest machine (1) has load Z, but Opt is
  1, consider all the machines that Opt uses to run
  these jobs.
• These machines must have load Z-1 (otherwise
  job would jump from 1 to this machine)
• There must be at least Z such machines, as they
  need to do work Z

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Related Machines Price of Anarchy upper bound

• Take the set of all machines up to the last
  machine that opt uses to service the jobs on
  machine 1.
• The jobs on this set of machines have to use
  Z(Z-1) other machines under opt.
• Continue, the bottom line is that n Z!, or that
  Z log m / log log m

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Restricted Assignment to Machines
l3
OPT
Group 1
Group 2
Group 3
m0
m0
m0
m0
m0
m1
m0
m1
m1
m1
m1
m1
m2
m2
m2
m3
Group 3
NASH
Group 2
Group 1
m0
m0
m0
m0
m0
m1
m0
m1
m1
m1
m1
m1
m2
m2
m2
m3
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Network models (Many models)

• Symmetric (all players go from s to t)
• No weights on the players (all bandwidth requests
  are one)
• Arbitrary monotonic increasing link delay
  function
• Polynomial time
• How bad a solution can this be?

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Network models (Many models)

• Asymmetric with weights
• Negligible load (one car out of 100,000 cars
  traveling from Tel Aviv to Jerusalem) Famouse as
  Waldrop equilibrium
• Atomic Splitable (the cars are all controlled by
  one agent, but the agent can split the routes
  taken by the cars)
• Atomic Unsplitable (all cars / oil /
  communications must flow through the same path.

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General Network Model

• A directed Graph G(V,E)
• A load dependent latency function fe(.) for each
  edge e
• n users
• Bandwidth request (si, ti, wi) for user i
• Goal route traffic to minimize total latency

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Example
Latency function f(x)x
1
2
1
2
1
t
s
2
2
2
2
Latency2125
Latency22228
Total latency Se fe(le)le Se le le
62231127
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Braesss Paradox negligible agents

• Traffic rate r1
• Optimal costNash cost2(1/211/21/2)3/2

f (x)1 load1/2
f(x)x load1/2
v
t
s
f(x)x load1/2
f(x)1 load1/2
w
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Braesss Paradox

• Traffic rate r1
• Optimal cost did not change
• Nash cost1101112
• Adding edge negatively impact all agents

fl(x)1 l0
f(x)x l1
v
t
f(x)0 l1
s
f(x)x l1
f(x)1 l0
w
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Negligible networks - POA

• Roughgarden and Tardos (FOCS 2000)
• Assumption each user controls a negligible
  fraction of the overall traffic
• Results
• Linear latency functions - POA4/3
• Continuous nondecreasing functions-bicriteria
  results
• Without negligibility assumption no general
  results

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Azar, Epstein, Awerbuch

• Unsplittable Flow, general demands
• Linear Latency Functions
• For weighted demands the price of anarchy is
  exactly 2.618 (pure and mixed)
• For unweighted demands the price of anarchy is
  exactly 2.5.
• Polynomial Latency Functions
• The price of anarchy - at most O(2ddd1) (pure
  and mixed)
• The price of anarchy - at least O(dd/2)

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Remarks

• Valid for congestion games
• Approximate computation
• (i.e approximate Nash) has limited affect

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Routes in Nash Equilibrium

• Pure strategies user j selects single path Q?
  Qj
• Mixed strategies user j selects a probability
  distribution pQ,j over all paths Q? Qj

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Routes in Nash Equilibrium

• Definition ( Pure Nash equilibrium)
• System S of pure strategies is in Nash
  equilibrium iff
• for every j ?1,...,nand Q ? Qj
• , where
• Qj path associated with request j

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Example
Latency function f(x)x
Path Q1
1
USER 1 W11
2
1
2
1
Path Q
t
s
2
2
2
2
CQ1,1 2125
CQ,1 2(11)(11)28
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Routes in Nash Equilibrium

• Definition (Nash equilibrium)
• System S of mixed strategies is in Nash
  equilibrium iff
• for every j ?1,...,nand Q,Q ? Qj, with
  pQ,jgt0
• cQ,j cQ,j where
• XQ,j indicates whether request j is assigned to
  path Q
• - load of edge e

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Routes in Nash Equilibrium

• Definition
• The expected cost C(S) of system S of mixed
  strategies is
• (i.e. the expected total latency incurred by S)

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Linear Latency Functions

• fe(x)aexbe for each e?E
• Theorem
• For linear latency functions (pure strategies)
  and weighted demands R 2.618
• Proof
• For simplicity assume f(x)x
• Qj - the path of request j in system S
• -set of requests that are
  assigned to edge e
• - load of
  edge e
• For optimal routes Qj , J(e) , le

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Weighted Sum of Nash Eq.

• According to the definition of Nash equilibrium
• We multiply by wj and get
• We sum for all j, and get

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Classification

• Classifying according to edges indices J(e) and
  J(e), yields
• Using ,
  we get

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Transformation

• Using Cauchy Schwartz inequality, we obtain
• Define and divide by
  
• Then

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Mixed Strategies

• Theorem
• For linear latency functions (mixed strategies)
  and weighted demands R 2.618.
• Proof
• Let pQ,j be the probability distribution of the
  system S.
• The expected latency of user j for assigning his
  request to path Q in S is

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Step 1

• According to the definition of Nash equilibrium
  for , hence
• We multiply by pQ,jwj and get

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Step 2

• Sum over all paths and all users and classify
  according to the edges
• Augment to
• Obtain the same inequality as in the pure
  strategies case

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UNWEIGHTED DEMANDS

• Theorem
• For linear latency functions, pure strategies
  and unweighted demands R 2.5.
• Proof

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Proof

• Classifying according to edges indices J(e) and
  J(e), yields
• Using ,
  we get

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Proof

• Applying inequalities
• Then

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Linear Latency Functions

• Theorem
• For linear latency functions and weighted
  demands
• R2.618.
• Proof
• We consider a weighted congestion game with four
  players

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Linear Latency Functions
v
Player 1 (u,v, f) Player 2 (u,w, f) Player 3
(v,w, 1) Player 4 (w,v, 1)
0
x
u
x
x
x
0
w
OPTNASH12f2 212 2f4
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Linear Latency Functions
v
Player 1 (u,v, f) Player 2 (u,w, f) Player 3
(v,w, 1) Player 4 (w,v, 1)
0
x
u
x
x
x
0
w
NASH22(f1)2 2f2 8 f 6
R f12.618
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Linear Latency Functions

• Theorem
• For linear latency functions and unweighted
  demands
• R2.5.
• Proof
• The same example as in the weighted case with
  unit demands

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Polynomial Latency Functions

• Theorem
• For polynomials of degree d latency functions R
  
• O(dd/2).
• Proof
• We use the construction of Awerbuch et. al for
  the parallel links restricted assignment model.

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General Latency Functions

• Polynomial Latency Functions
• The price of anarchy - at most O(2ddd1) (pure
  and mixed)
• The price of anarchy - at least O(dd/2)

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The Construction

• Total ml! links each has a latency function
  f(x)x
• l1 type of links
• For type k0l there are mkT/k! links
• l types of tasks
• For type k1l there are kmk jobs, each can be
  assigned to link from type k-1 or k
• OPT assigns jobs of type k to links of type k-1
  one job per link.

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System of Pure Strategies

• System S of pure strategies
• Jobs of type k are assigned to links of type k
• k jobs per link
• Lemma
• The System S is in Nash Equilibrium.

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The Coordination Ratio

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General Latency Functions

• General functions-no bicriteria results
• Polynomial Latency Functions
• The price of anarchy - at most O(2ddd1) (pure
  and mixed)
• The price of anarchy - at least O(dd/2)

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Summary

• We showed results for general networks with
  unsplittable traffic and general demands
• For linear latency functions R2.618
• For Polynomial Latency functions of degree d ,
• Rd?(d)