The Price of Anarchy in a Network Pricing Game (II)

SHI Xingang & JIA Lu

14-05-2008

Outline

• Can We Find a Bound?

• How Can We Find the Bound?

• Let's Prove the Bound

• Let's Prove It Again

• How About Convex Latency

• Conclusion and extension

Can We Find a Bound?

Optimal price

p=0, d=0, f=2, W=0+3/2=3/2

Equilibrium price

p*=1, d*=1, f*=1, W*=1+0=1

W / W*=1.5

[3] has proved that 1.5 is the tight upper bound, using mathematical programming

How Can We Find the Bound?

Linearization

and

Truncation

[2] brings the idea for truncation

Linearized Disutility Function• Lemma : The Nash equilibrium flow and

the price vectors of are the same as the Nash flow and the price vectors of

• Remember the sufficient and necessary condition

Linearized Disutility Function

• Lemma : The Optimal Welfare of is no more than that of

This paper missed this point

• Remember this is an optimization problem

And for a linearized game, d* < d

• Remember the sufficient and necessary condition

Linearized and Truncated Disutility Function

• Lemma : The Nash equilibrium flow and the price vectors of are the same as the Nash flow and the price vectors of

Sufficient and Necessary Condition For

It's also easy to see that the optimal flow and price vectors are the same as

Linearized and Truncated Disutility Function

• Lemma :

• Proof : introduce a truncated utility function , , so the optimization

result is larger

Now we only need to deal with linearized and truncated disutility function!

can decrease no more

than from

Deal with Linearized Truncated Disutility Function

• There cannot exist links used in social optimum that are not used in Nash Equilibrium

Let's Find and • Linear (not truncated)

disutility function• Linear truncated

disutility function

same (d,f) and (d*,f*)

and

and

Let's Prove

Let's Prove

• But and– (we have and )

• there do exist chances that the sum is negative

Let's Prove

• This paper proves by the following way:– restricting , and we can prove

– since is decreasing in [0,1/2]

– we only need to prove the diagonal elements are positive, where

Let's Prove Again

– When there is no unused flow in optimal,

is actually 0 (restricting it by is too loose). We have proved successfully.

Anyway, linearization is a very important step

• The reason it fails – bound is too low

– When there is unused flow in optimal,

using to replace makes the value too small. We are walking on this way.

Convex Latency Function

d

f1l1

l2

p1 p2

f2

b'1

b'2

d+

f1

l1+l2+

p1 p2

f2

dl+

f1

l1l+l2l+

p1 p2

f2

G G+

Gl+

When equilibrium exists, we have

linearization again!

Conclusion and Extension

• Analogy of circuit may give us some interesting ideas

• Linearization is sometimes more simple and more powerful

• Multi-commodity– Multiple source and destination pairs– Different type of sensitivity to latency

References[1] John Musacchio, The Price of Anarchy in a Network Pricing Game,

Presentation at Allerton07.[2] A.Hayrapetyan, E. Tardos and T. Wexler, A Network Pricing Game f

or Selfish Traffic, Twenty-Fourth Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2005)

[3] D. Acemoglu and A.ozdaglar, Competition and Efficiency in Congested markets, Mathematics of Operations Research, 2007

[4] John Musacchio and Shuang Wu, The Price of Anarchy in a Network Pricing Game, The Forty-Sixth Annual Allerton Conference on Communication, Control, and Computing (Allerton07)

[5] S. Boyd and L. Vandenberghe, Convex optimzation, Camebridge University Press, 2004

[6] T. Roughgarden, The Price of Anarchy is Independent of the Network Topology, 34th ACM Symposium on Theory of Computing (STOC 2002)