Efficiency Loss in a Network Resource Allocation Game

Paper by: Ramesh Johari, John N. Tsitsiklis[2004 - Informs]

Presented by: Gayatree Ganu

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

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

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

Problem DefinitionFixed Total Capacity C

Allocated capacity dr

at price μ

User r

Bid wr

More formally… R users share a communication link Total Link capacity C > 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 dr>=0

wr is the payment that user r is willing to make for the link

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]

Price Taking Version Goal:

maximize ∑r Ur(dr)

subject to ∑r dr <= C

dr >= 0, r=1,…,R

But, utility functions are not available to link manager.

Price Taking Version Each user r submits a bid wr to the link

manager, assume wr>= 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 >0. Hence, dr=wr/μ

Manager always seeks to allocate entire link capacity C. Hence, ∑r (wr/μ) = C

Price Taking Version Each user tries to maximize his payoff given

by the function:Pr(wr;μ) = Ur(wr/μ) – wr

Kelly [1997] – Competitive equilibrium exists, users maximize their payoff and network “clears the market”Pr(wr;μ) >= Pr(w’r;μ) for w’r>=0, r=1,…,Rμ =∑r wr / C

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]

Price Anticipating Version If w-r denotes the vector of all bids other than the

user r, w-r=(w1,…,wr-1,wr+1,…wR) then each user r wants to maximize the payoff function:

Qr(wr;w-r) = Ur((wr/∑s ws) C) – wr, if wr>0

= Ur(0), if wr=0

The second condition is chosen so that rate allocation to user r is zero when wr=0, even if all other users choose w-r so that ∑s≠r ws=0

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 w>0= U(0), if w=0

Since U is strictly increasing, U(C)>U(0) Nash equilibrium does not exist:

For a bid w=0, deviate to any bid 0<w’<U(C)-U(0) For a bid w>0, deviate to any bid 0<w’<w

Price Anticipating Version Hajek, Gopalakrishnan [2002] - Nash

Equilibrium exists with modified utility function

maximize ∑r Ûr(dr)

subject to ∑r dr <= C

dr >= 0, r=1,…,R

where

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%

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.

Price of Anarchy Lemma: Worst case occurs with linear

utility functions. For any d- = (d1-,…,dr

-) satisfying ∑r dr

- <= C

Let Ur be linear with Ur(dr) = ar dr ,where ar>0. If dG represents the Nash equilibrium then the price of anarchy is given by: PoA =

Price of Anarchy Without loss of generality assume that maxr ar =

a1 = 1 and C=1. Worst case occurs when (d1G + ∑R

r=2 ar dRG) is minimum, resulting in the following LP

Since largest ar = 1 and C = 1, optimum value of objective function gives PoA

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 >= 1/R and drG=(1-d1G)/R-1

Price of Anarchy Substituting for drG ,we have the following

LP

Price of Anarchy The previous objective function is

decreasing as R increases. Worst case price is given with the limit R -> infinity

PoA is given by the solution to

The solution is d1G = ½, resulting in PoA=3/4

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

Thank you