GameTheoretic Analysis of Network QualityofService Pricing

1
Game-Theoretic Analysis of NetworkQuality-of-Serv
ice Pricing
Paris Metro Pricing
Expedited Service
Braess Paradox
General Networks
Introduction
TCP/IP Back-off

• Game Generator
• Object-Oriented Python API
• Takes Network object as input
• Generates AGG file
• Launches AGG solver
• Interprets results
• Restricted to parallel paths or Braess-structured
  networks, with perfect expedited service
• Arbitrary latency functions for each link, L
  fL( of users) ? Real value
• Supports richer QoS measure than just latency
  fL( of users) ? Q where Q is an arbitrary set
  (e.g. vectors of features such as bandwidth,
  latency, probability of packet loss)
• Arbitrary utility functionsf U(Q) ? Real value

Network System Q Does a network provide
good quality of service? A That depends on what
its users want from it.

• Network System
• 2 TCP/IP users, 1 shared link
• Converges to
• Equal division of bandwidth
• Limited congestion

Introduction First class
Economy class Q Why charge different
prices for identical service? A Because theyre
expensive, first-class cars are less
crowded. Same concept applied to highway
traffic Toronto 407s toll is tuned to control
congestion

• Network System
• Q How does a tiered QoS system compare with
  Paris Metro pricing?
• Consider the same network and assumptions as in
  Paris Metro pricing example
• Add Perfect expedited service Expedited
  traffic unaffected by non-expedited

Network System Delay of a path is the sum
of delays of link segments along the path
Game Theoretic Model Same utility and user model
as Paris Metro pricing example

• Game Theoretic Model
• Different users have different values for quality
  of service
• Users experienced QoS (e.g. latency) influenced
  by other users actions (which cause congestion)
• This interdependence means game theory applies.
• Definition Nash equilibrium a stable state
  where no user wants to change their action, given
  the actions of everyone else Nash, 1950

Game Theoretic Model 20 users Each can choose
any path from s to t At equilibrium flow split
between 2 paths

• Game Theoretic Model
• Suppose user 1 hacks his TCP/IP back-off
  implementation
• Converges to
• Unequal share of bandwidth
• More congestion
• Suppose both users hack

• Network System
• Future extensions to network model
• Arbitrary network topology
• Richer models of usage (e.g. bandwidth
  consumption, burstiness)
• Richer models of tiered service (i.e. imperfectly
  expedited service)

Adding a link At equilibrium all users
choose path s,u,v,t All users are worse off

• Network System
• Q Can we use this idea to prevent internet
  congestion? Odlyzko, 1997 Ros Tuffin, 2004
• Linear, additive model of latency
• Delay ?( Usage ) / Bandwidth
• A perfect fair queue of unlimited length

• Game Solver
• Iterate over a range of prices 0.00 to 2.00 in
  0.01 increments
• AGG solver finds usage pattern given costs

• Game Theoretic Model
• Future extensions to user model
• Arbitrary source and destination nodes
• Uncertainty about the types of other agents (i.e.
  Bayesian games)

Pricing Put price on link (u,v) When users have
same values (u,v) useless Q What happens if
users have different values?

• Game Solver
• All proposed model extensions are possible within
  existing AGG framework
• When utility, latency functions have simple
  structure (e.g. path latency sum of link
  latencies, path bandwidth min of link
  bandwidths) even more optimization may be
  possible

• Game Theoretic Model
• 18 low priority users, 2 high priority users
• Linear model of utility
• Utility Delay ValueForTime LinkToll
• Utility measured in (cost-benefit trade-off of
  QoS)

• Game Solver
• Normally impractical Nash equilibria are too
  expensive to compute (O(22n) where n is number of
  users)
• Action Graph Games exploit structure for massive
  speed gain Bhat Leyton-Brown, 2004 Jiang
  Leyton-Brown, 2006
• Anonymity other users behavior affects my QoS,
  not their identities
• Context specific independence my QoS is
  unaffected by traffic on links Im not using
• Can be treated as a black-box
• Input network

• Implications and Conclusions
• Economically efficient when cost gt 0.72 (Cost of
  latency minimized)
• Most profit goes to users.
• No waste Load always uniformly balanced

Reference Cole, Dodis, Roughgarden
(2006) How much can taxes help selfish routing?
Journal of Computer and System Sciences Kearns,
Littman, Singh (2001) Graphical Models for Game
Theory, UAI Monderer (2007) Multipotential
Games, IJCAI Nash (1950) Equilibrium Points in
N-person Games Odlyzko (1997) A Modest Proposal
for Preventing Internet Congestion Ros, Tuffin
(2004) A Mathematical Model of the Paris Metro
Pricing Scheme, Computer Networks

• Game Solver
• AGG solver finds usage pattern given costs

• Related Work
• Network Model
• Ros Tuffin (2004) game-theoretic analysis of
  Paris-Metro Pricing
• Cole et al (2006) analyzes putting taxes on
  links to reduce congestion
• (neither paper modeled users with different
  values for latency)
• Game Representations
• Kearns et al (2001) Graphical Games
• exploits strict independence structure
• cannot compactly represent games here
• Monderer (2006) Player-specific congestion games
  
• Can compactly represent games here
• Did not focus on computation of Nash equilibria

• Game Solver
• Equivalent to prisoners dilemma
• Only equilibrium is for both users to hack

• Game Solver
• Iterate over a range of prices 0.00 to 2.00 in
  0.01 increments
• AGG solver finds usage pattern given costs

• Implications and Conclusions
• Economically efficient between 0.72 and 1.10
  (Cost of latency minimized)
• Most profit goes to network provider
• Significant waste Costly link sits idle while
  users wait in free links queue

• Implications
• Economically efficient between 0.81 and 9.50
• Most profit goes to users
• More efficient than without the link (u,v)

Implications and Conclusions The only equilibrium
is the least economically efficient state.
Fortunately, TCP/IP hacks have a cost to adopt
and hackers have a disincentive to share their
work.

• Implications and Conclusions
• The equilibrium of an AGG would allow us to
  answer questions about the proposed network
• What paths through the network would the users
  choose?
• How much load would occur on each link?
• What is each users utility? (i.e. how happy are
  they with the network?)
• Definition Social welfare sum of all parties
  utilities (users and network providers)
• Definition Economic efficiency maximizing
  social welfare