Efficiency Loss in a Network Resource Allocation Game

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Efficiency Loss in a Network Resource Allocation
Game

• Paper by Ramesh Johari, John N. Tsitsiklis
• 2004 - Informs
• Presented by Gayatree Ganu

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Topics List

• Motivation
• Single Sink Game Problem Definition
• Price Taking Version
• Price Anticipating Version
• Price of Anarchy
• General Networks / Extensions

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Motivation

• Typical networks like the Internet have resources
  shared across multiple users.
• Different end users place different values for
  good performance.
• Congestion Pricing of shared resources has been
  proposed

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Problem Definition

• Single network manager
• Multiple competing users for the network
• Each user has a utility function depending on the
  allocated rate
• Each user submits a bid to the network
• GOAL
• Allocation of network capacity efficiently to
  maximize aggregate utility. User is allocated a
  rate in proportion to his bid and inversely
  proportional to the price of the link

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Problem Definition
Fixed Total Capacity C
Allocated capacity dr at price µ
Bid wr
User r
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More formally

• R users share a communication link
• Total Link capacity C gt 0
• dr is the rate allocated to user r
• r receives a utility equal to Ur(dr)
• Ur(dr) is concave, strictly increasing and
  continuously differentiable with domain drgt0
• wr is the payment that user r is willing to make
  for the link

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Price Taking Version

• Each user acts as a price taker, users do not
  anticipate the effect of their actions on the
  price of the link.
• Main result in Charging and rate control for
  elastic traffic F.P. Kelly 1997

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Price Taking Version

• Goal
• maximize ?r Ur(dr)
• subject to ?r dr lt C
• dr gt 0, r1,,R
• But, utility functions are not available to link
  manager.

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Price Taking Version

• Each user r submits a bid wr to the link manager,
  assume wrgt 0
• Given the vector w(w1,,wr) manager chooses rate
  allocation vector d(d1,,dr)
• Assumptions
• Manager is price indiscriminate, each user is
  charged the same price µ gt0. Hence, drwr/µ
• Manager always seeks to allocate entire link
  capacity C. Hence, ?r (wr/µ) C

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Price Taking Version

• Each user tries to maximize his payoff given by
  the function
• Pr(wrµ) Ur(wr/µ) wr
• Kelly 1997 Competitive equilibrium exists,
  users maximize their payoff and network clears
  the market
• Pr(wrµ) gt Pr(wrµ) for wrgt0, r1,,R
• µ ?r wr / C

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Price Anticipating Version

• Users realize that the price µ is set according
  to their bids, and adjust their bids accordingly.
• This makes it a game between R players.
• Main result in Do greedy autonomous systems
  make for a sensible internet Hajek,
  Gopalakrishnan 2002

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Price Anticipating Version

• If w-r denotes the vector of all bids other than
  the user r, w-r(w1,,wr-1,wr1,wR) then each
  user r wants to maximize the payoff function
• Qr(wrw-r) Ur((wr/?s ws) C) wr, if wrgt0
• Ur(0), if wr0
• The second condition is chosen so that rate
  allocation to user r is zero when wr0, even if
  all other users choose w-r so that ?s?r ws0

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Price Anticipating Version

• This new payoff function is discontinuous, which
  may preclude the existence of a Nash equilibrium.
• Example Suppose there is a single user with
  strictly increasing utility function U. Any
  positive payment results in the entire link being
  allocated to the single user
• Q(w) U(C) w, if wgt0
• U(0), if w0
• Since U is strictly increasing, U(C)gtU(0)
• Nash equilibrium does not exist
• For a bid w0, deviate to any bid 0ltwltU(C)-U(0)
• For a bid wgt0, deviate to any bid 0ltwltw

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Price Anticipating Version

• Hajek, Gopalakrishnan 2002 - Nash Equilibrium
  exists with modified utility function
• maximize ?r Ûr(dr)
• subject to ?r dr lt C
• dr gt 0, r1,,R
• where

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Price of Anarchy

• Measure of how much utility is lost because
  users attempt to game the system?
• Result Price of anarchy is ¾ for single sink
  game with price anticipating users.
• Investigation of Price of Anarchy is used to
  design systems with robustness against selfish
  behavior. Selfish behavior does not degrade
  network performance arbitrarily- efficiency loss
  is at most 25

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Price of Anarchy

• Let dG represent optimal solution in price
  anticipating game and dS represent optimal
  solution in price taking system.
• To prove
• and that the bound is tight, i.e.

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Price of Anarchy

• Lemma Worst case occurs with linear utility
  functions. For any d- (d1-,,dr-) satisfying ?r
  dr- lt C
• Let Ur be linear with Ur(dr) ar dr ,where argt0.
  If dG represents the Nash equilibrium then the
  price of anarchy is given by PoA

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Price of Anarchy

• Without loss of generality assume that maxr ar
  a1 1 and C1.
• Worst case occurs when (d1G ?Rr2 ar dRG) is
  minimum, resulting in the following LP
• Since largest ar 1 and C 1, optimum value of
  objective function gives PoA

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Price of Anarchy

• We need to consider only the first condition ar
  (1 - dRG ) 1 - d1G
• This results in the following reduced LP
• This problem is well defined (i.e. Nash
  equilibrium exists) only if d1G gt 1/R and
  drG(1-d1G)/R-1

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Price of Anarchy

• Substituting for drG ,we have the following LP

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Price of Anarchy

• The previous objective function is decreasing as
  R increases. Worst case price is given with the
  limit R -gt infinity
• PoA is given by the solution to
• The solution is d1G ½, resulting in PoA3/4

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Other Results

• Multiple Link Each user requests service from
  multiple links by submitting a bid to each link.
• Nash equilibrium might not exist due to
  discontinuity in payoff function of individual
  players
• Extended game Each user can request a non-zero
  rate without submitting a positive bid to the
  link, if the total payment made by other users to
  that link is zero.
• Nash equilibrium exists
• Price of Anarchy is 3/4
• A general game where user utility is not a
  function of flow that a user can send.
• Utility is any concave function of vector of
  resources allocated
• Price of Anarchy is 3/4

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Thank you