How Bad is Selfish Routing

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How Bad is Selfish Routing

• A survey on existing models for selfish routing

Professor John Lui, David Yau and Dah-Ming Qiu
presented by Joe W.J. Jiang 2004-05-18
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Outline of my talk

• Introduction to selfish routing
• Preliminaries and Nash equilibrium
• How bad is selfish routing
• Other models on selfish routing related work
• Conclusions and problems

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Introduction to selfish routing

• Routing in the Internet
• RIP (distance vector routing, Bellman-Ford)
• OSPF (link state routing, Dijkstra)
• BGP (exterior gateway protocol)
• These routing metrics of the above protocols are
  generally based on hop counts.
• There is an inherent inefficiency from the users
  perspective bandwidth, latency, jitter.
• There is an incentive for users to choose routes
  themselves.

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Selfish routing in the Internet

• Source routing Nimrod -- route information is
  contained in the header of route request
• Overlay routing Detour or RON routing via peer
  nodes in the overlay network
• Such end-to-end route selection is selfish by
  nature, optimizing their own performance without
  considering others.

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Selfishness in the Internet

• Internet users with a multitude of diverse
  economic interests
• browsers
• routers
• servers
• Selfishness parties will deviate from their
  protocol if it is in their interest.
• How to study these problems
• Algorithmic Game Theory algorithms game
  theory

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Where are you?

• Introduction to selfish routing
• Preliminaries and Nash equilibrium
• How bad is selfish routing
• Other models on selfish routing related work
• Conclusions and problems

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Routing Problems

• Optimization problem
• given a network, a traffic rate between each pair
  of nodes
• latency function of each edge
• objective the total latency is minimized

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Nash Equilibrium

• A Nash Equilibrium is a set of strategies (one
  status) one for each player, such that no player
  has incentive to unilaterally change his action.
• Players are in equilibrium if a change in
  strategies by any one of them would lead that
  player to earn less than current strategy.
• It is well known that Nash equilibria do not in
  general optimize social welfare Prisoners
  Dilemma.

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Braesss Paradox
1
x
1
1/2
0
s
t
1/2
x
1
average latency 10.5 1.5
average latency 11 2

• the price of anarchy
• 2/1.5 4/3!

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Some Algorithmic Issues

• Price of Anarchy
• A measure of degradation of performance caused
  by lack of cooperation (regulation)
  selfishness.
• Mechanism Design
• How to design games so that selfish behaviors
  would lead to desire outcome.
• Coalitional Games
• E.g., how to share costs incurred by a group of
  users.

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Mathematical Models

• A directed graph G(V, E)
• source-sink pairs si, ti for i1,..,k
• rate ri ? 0 of traffic between si and ti for each
  i1,..,k
• set of si-ti paths Pi
• P
• for each edge e, a latency function le()
  nonnegative, differentiable, non-decreasing.

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Mathematical Model Traffic and Flows

• A flow vector specifies a traffic pattern fp
  amount of flow on si-ti path P
• flow of an edge e
• A flow f is said to be feasible if for all i,
• We call triple (G, r, l) an instance.
• The latency of a path P
• cost of all flows C(f) -- total latency

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Flows and game theory

• Flow represents routes of many noncooperative
  agents
• each agent controlling infinitesimally small
  amount
• cars in a highway system
• packets in a network
• The cost (total latency) of a flow represents
  social welfare.
• Agents are selfish in that
• minimize personal latency
• do not care about social welfare

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Flows at Nash equilibrium

• A flow is at Nash equilibrium (or is a Nash flow)
  if no agent can improve its latency by changing
  its path.

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Wardrops Principle
In particular, all paths to which f assigns
positive amount of flow, have equal latency, say
Li(f)
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Optimal Flow

• An optimal flow is a flow that minimizes total
  latency/ average latency.
• Convex programming

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Optimal Flow (Solution)

• If the objective function ce(fe)le(fe)fe is
  convex, global optimal local optimal
• We expect a flow to be locally optimal if and
  only if the marginal benefit of decreasing flow
  along any si-ti path the marginal cost of
  increasing flow along any other si-ti path.

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Beckmans Interpretation
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Existence of Nash Equilibrium
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A good but not optimal upper bound
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A good but not optimal upper bound (cont)
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Where are you?

• Introduction to selfish routing
• Preliminaries and Nash equilibrium
• How bad is selfish routing
• Other models on selfish routing related work
• Conclusions and problems

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A simple bad example
1
½11/2
1
s
t
1
111
½1/21/4
x
the price of anarchy 1/ (3/4) 4/3 !
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Bicriteria Results
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Bicriteria Results (cont)
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Worst-Case Ratio of 4/3 with Linear Latency
Functions
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corollary
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Important results
The most important theorem Theorem If (G, r,
l) has linear latency functions, then ?(G, r,
l)4/3
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Proof of 4/3 coordination ratio
cost of optimal at rate r/2
cost of increasing from optimal at rate r/2 to
optimal at rate r
cost of optimal at rate r


optimal at r/2 C(f/2)1/4 C(f)
At least (r/2) L 1/2 C(f)
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Lemma
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A simple example
1
x
1
0
s
t
x
1
1/2
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Proof of lemma
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Proof of ?(G, r, l)4/3
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Extensions

• Flows at Approximate Nash Equilibrium
• Finitely Many Agents Splittable Flow
• Finitely Many Agents Unsplittable Flow
• Central regulation.

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Where are you?

• Introduction to selfish routing
• Preliminaries and Nash equilibrium
• How bad is selfish routing
• Other models on selfish routing related work
• Conclusions and problems

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Related Papers

• How bad is selfish routing -- Roughgarden
  Tardos
• Worst-case Equilibrium -- Koutsoupias
  Papadimitriou
• The Price of Selfish Routing -- Mavronicolas
  Spirakis
• Realistic Models for Selfish Routing in the
  Internet -- Akella

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KP model (task allocation model)
m servers
Main emphasizes on service cost (routing cost
neglected)
n jobs
Cost service cost
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KP model (cont)
Main emphasizes on service cost (routing cost
neglected)
Routing in a network consisting of parallel links
only
Scheduling-type problems Schedule tasks to
minimize the execution time (cost)
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KP model (cont)

• simple routing model
• two nodes
• m parallel links with speeds si
• (1 i m)
• n jobs with weights wj
• (1 j n)
• service cost
• the delay of a connection is proportional to load
  on link

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Cost measure

• After each job selects a link
• Jobs(j) jobs assigned to link j
• Cost of jobs assigned to link j
• Total weight of jobs assigned to link j over the
  speed of link j
• (Total) cost of a configuration
• maxj Cj
• Social optimum (minimized cost)
• min maxj Cj

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Results

• Koutsoupias and Papadimitriou99
• defined the problem
• solved some of most basic cases
• for 2 identical links price of anarchy 1.5
• for 2 links price of anarchy is ? ¼ 1.618
• for m identical links price of anarchy is
• for m links price of anarchy is

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KPs conjecture

• Koutsoupias-Papadimitriou conjecture
• for m identical links
• price of anarchy is
• most natural behavior (random) is worst
• proved by Mavronicolas Spirakis

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Akellas Model

• Selfish users choose routes that maximize the
  bandwidth available to the flow.
• Bandwidth available to agent i
• Objective function is total bandwidth used by all
  users
• The price of anarchy in a network with n flows
  ban be as large as O(n)

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Where are you?

• Introduction to selfish routing
• Preliminaries and Nash equilibrium
• How bad is selfish routing
• Other models on selfish routing related work
• Conclusions and problems

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Conclusions Problems

• Selfish behaviors would degrade the performance
  of the network.
• However, some simulation results on Internet show
  that selfish routing is close to optimal routing.
  ???
• Other problem route oscillation (Internet/
  overlay network)
• Goal how to design network or design games (what
  information should users know? ) so that selfish
  behavior would lead to desired outcome?

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Thank you for your attention!

• The End