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Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

The Price of Anarchy of Selfish Routing

Arthur van Goethem & Sk´ uli Arnlaugsson June 8, 2011

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 1/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Overview

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 2/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Introduction

Price of Anarchy

The Price of Anarchy of a non-atomic1 selfish routing game is the ratio between the cost of an equilibrium flow and that of an optimal flow. p(G, r, c) = C(f ) C(f ∗)

◮ f denotes the equilibrium flow, and f ∗ denotes the optimal flow ◮ What is the potential function?

1Very large number of players, each controlling a negligible fraction

• f the overall traffic

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 3/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Introduction

Price of Anarchy

The Price of Anarchy of a non-atomic1 selfish routing game is the ratio between the cost of an equilibrium flow and that of an optimal flow. p(G, r, c) = C(f ) C(f ∗)

◮ f denotes the equilibrium flow, and f ∗ denotes the optimal flow ◮ What is the potential function?

1Very large number of players, each controlling a negligible fraction

• f the overall traffic

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 3/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Proving an upper bound on the PoA for non-atomic selfish routing

◮ PoA depends solely on the (non-)linearity cost function ◮ Assume

xce(x) ≤ γ x ce(y)dy ∀e ∈ E and ∀x ≥ 0 I.e. the cost function for a single edge is smaller that the potential function for this edge.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 4/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Pigou bound

◮ The potential function gives a good, but not optimal bound on the

• PoA. Can we do better?

◮ To show this, we introduce the Pigou bound

Definition 18.18 (Pigou bound)

Let C be a nonempty set of cost functions.The Pigou bound α(C) for C is α(C) = sup

c∈C

sup

x,r≥0

r ∗ c(r) x ∗ c(x) + (r − x) ∗ c(r) with the understanding that 0/0 = 1.

◮ The Pigou bound is a natural lower bound on the PoA based on

“Pigou-like examples”.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 5/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Pigou bound

◮ The potential function gives a good, but not optimal bound on the

• PoA. Can we do better?

◮ To show this, we introduce the Pigou bound

Definition 18.18 (Pigou bound)

Let C be a nonempty set of cost functions.The Pigou bound α(C) for C is α(C) = sup

c∈C

sup

x,r≥0

r ∗ c(r) x ∗ c(x) + (r − x) ∗ c(r) with the understanding that 0/0 = 1.

◮ The Pigou bound is a natural lower bound on the PoA based on

“Pigou-like examples”.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 5/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Pigou bound

◮ The potential function gives a good, but not optimal bound on the

• PoA. Can we do better?

◮ To show this, we introduce the Pigou bound

Definition 18.18 (Pigou bound)

Let C be a nonempty set of cost functions.The Pigou bound α(C) for C is α(C) = sup

c∈C

sup

x,r≥0

r ∗ c(r) x ∗ c(x) + (r − x) ∗ c(r) with the understanding that 0/0 = 1.

◮ The Pigou bound is a natural lower bound on the PoA based on

“Pigou-like examples”.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 5/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Pigou bound, cont’d

Proposition 18.19

Let C be a set of cost functions that contains all of the constant cost

• functions. Then the price of anarchy in non-atomic instances with cost

functions on C can be arbitrarily close to α(C).

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 6/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Variational Inequality

◮ It is also shown that Pigou bound is an upper bound on the PoA in

general multi-commodity flow networks

◮ To show this we need the Variational inequality characterization

Proposition 18.20 (Variational inequality characterization)

Let f be a feasible flow for the non-atomic instance (G, r, c). The flow is an equilibrium flow if and only if

• e∈E

ce(fe)fe ≤

• e∈E

ce(fe)f ∗

e

for every flow f ∗ feasible for (G, r, c).

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 7/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Variational Inequality

◮ It is also shown that Pigou bound is an upper bound on the PoA in

general multi-commodity flow networks

◮ To show this we need the Variational inequality characterization

Proposition 18.20 (Variational inequality characterization)

Let f be a feasible flow for the non-atomic instance (G, r, c). The flow is an equilibrium flow if and only if

• e∈E

ce(fe)fe ≤

• e∈E

ce(fe)f ∗

e

for every flow f ∗ feasible for (G, r, c).

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 7/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Variational Inequality

◮ It is also shown that Pigou bound is an upper bound on the PoA in

general multi-commodity flow networks

◮ To show this we need the Variational inequality characterization

Proposition 18.20 (Variational inequality characterization)

Let f be a feasible flow for the non-atomic instance (G, r, c). The flow is an equilibrium flow if and only if

• e∈E

ce(fe)fe ≤

• e∈E

ce(fe)f ∗

e

for every flow f ∗ feasible for (G, r, c).

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 7/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Tightness of the Pigou bound

◮ We can now show that the Pigou bound is tight (bounding from

above and below)

Theorem 18.21 (Tightness of the Pigou bound)

Let C be a set of cost functions and α(C) be the Pigou bound for C. If (G, r, c) is a non-atomic instance with cost functions in C, then the price

• f anarchy of (G, r, c) is at most α(C).

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 8/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Tightness of the Pigou bound

◮ We can now show that the Pigou bound is tight (bounding from

above and below)

Theorem 18.21 (Tightness of the Pigou bound)

Let C be a set of cost functions and α(C) be the Pigou bound for C. If (G, r, c) is a non-atomic instance with cost functions in C, then the price

• f anarchy of (G, r, c) is at most α(C).

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 8/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Largest possible PoA

◮ Determining the largest possible PoA in Pigou-like examples is a

tractable problem in many cases

◮ When C is a set of affine cost functions it is precisely 4 3 ◮ When C is a set of polynomials with degree at most p and

non-negative coefficients: (1 − p(p + 1)−(p+1)/p)−1 ≈

p lnp

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 9/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Largest possible PoA

◮ Determining the largest possible PoA in Pigou-like examples is a

tractable problem in many cases

◮ When C is a set of affine cost functions it is precisely 4 3 ◮ When C is a set of polynomials with degree at most p and

non-negative coefficients: (1 − p(p + 1)−(p+1)/p)−1 ≈

p lnp

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 9/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Largest possible PoA

◮ Determining the largest possible PoA in Pigou-like examples is a

tractable problem in many cases

◮ When C is a set of affine cost functions it is precisely 4 3 ◮ When C is a set of polynomials with degree at most p and

non-negative coefficients: (1 − p(p + 1)−(p+1)/p)−1 ≈

p lnp

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 9/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Braess’s Paradox discussion

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 10/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Proving an upperbound on the PoA for Atomic Selfish Routing

◮ Atomic? Each player controls a non-negligible flow ◮ Potential function bound? ◮ But we can prove:

PoA ≤ 3 + √ 5 2 ≈ 2, 618

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 11/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Proving an upperbound on the PoA for Atomic Selfish Routing

◮ Atomic? Each player controls a non-negligible flow ◮ Potential function bound? ◮ But we can prove:

PoA ≤ 3 + √ 5 2 ≈ 2, 618

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 11/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Proving an upperbound on the PoA for Atomic Selfish Routing

◮ Atomic? Each player controls a non-negligible flow ◮ Potential function bound? ◮ But we can prove:

PoA ≤ 3 + √ 5 2 ≈ 2, 618

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 11/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Proving an upperbound.. cont’d

PoA ≤ 3 + √ 5 2 (18.23)

• e∈Pi

[aefe + be] ≤

• e∈P∗

i

[ae(fe + ri) + be] (18.24) C(f ) ≤ C(f ∗) +

• e∈E

aefef ∗

e

(18.25) C(f ) C(f ∗) − 1 ≤

• C(f )

C(f ∗) (p 18.23)

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 12/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Proving an upperbound.. cont’d

PoA ≤ 3 + √ 5 2 (18.23)

• e∈Pi

[aefe + be] ≤

• e∈P∗

i

[ae(fe + ri) + be] (18.24) C(f ) ≤ C(f ∗) +

• e∈E

aefef ∗

e

(18.25) C(f ) C(f ∗) − 1 ≤

• C(f )

C(f ∗) (p 18.23)

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 12/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Proving an upperbound.. cont’d

PoA ≤ 3 + √ 5 2 (18.23)

• e∈Pi

[aefe + be] ≤

• e∈P∗

i

[ae(fe + ri) + be] (18.24) C(f ) ≤ C(f ∗) +

• e∈E

aefef ∗

e

(18.25) C(f ) C(f ∗) − 1 ≤

• C(f )

C(f ∗) (p 18.23)

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 12/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Proving an upperbound.. cont’d

PoA ≤ 3 + √ 5 2 (18.23)

• e∈Pi

[aefe + be] ≤

• e∈P∗

i

[ae(fe + ri) + be] (18.24) C(f ) ≤ C(f ∗) +

• e∈E

aefef ∗

e

(18.25) C(f ) C(f ∗) − 1 ≤

• C(f )

C(f ∗) (p 18.23)

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 12/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Reducing the Price

◮ The PoA can be high when cost functions are highly non-linear ◮ Can we design or modify a selfish routing network to minimize the

inefficiency?

◮ Without imposing an optimal solution

◮ Two methods

◮ Marginal cost pricing ◮ Capacity augmentation Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 13/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Reducing the Price

◮ The PoA can be high when cost functions are highly non-linear ◮ Can we design or modify a selfish routing network to minimize the

inefficiency?

◮ Without imposing an optimal solution

◮ Two methods

◮ Marginal cost pricing ◮ Capacity augmentation Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 13/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Reducing the Price

◮ The PoA can be high when cost functions are highly non-linear ◮ Can we design or modify a selfish routing network to minimize the

inefficiency?

◮ Without imposing an optimal solution

◮ Two methods

◮ Marginal cost pricing ◮ Capacity augmentation Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 13/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Marginal Cost Pricing

Intuitive idea

Impose a marginal tax on each edge to a network user for the additional cost it’s presence causes for the other users of that edge.

Sound familiar?

This is similar to the Clarke Pivot Rule from chapter 9.3.4 ”In other words, the payments make each player internalize the externalities that he causes.”

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 14/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Marginal Cost Pricing

Intuitive idea

Impose a marginal tax on each edge to a network user for the additional cost it’s presence causes for the other users of that edge.

Sound familiar?

This is similar to the Clarke Pivot Rule from chapter 9.3.4 ”In other words, the payments make each player internalize the externalities that he causes.”

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 14/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Marginal Cost Pricing, cont’d

◮ Formally we give each edge a non-negative tax, τe ◮ Denoted as (G, r, c + τe) ◮ According to definition 18.1, an equilibrium flow for such an instance

is when all traffic traveling on routes that minimize the sum of the edge cost and edge taxes

◮ We can denote the instance as (G, r, cτ), where the cost function cτ

is a shifted as: ce : cτ

e (x) = ce(x) + τe, for all x ≥ 0.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 15/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Marginal Cost Pricing, cont’d

◮ Formally we give each edge a non-negative tax, τe ◮ Denoted as (G, r, c + τe) ◮ According to definition 18.1, an equilibrium flow for such an instance

is when all traffic traveling on routes that minimize the sum of the edge cost and edge taxes

◮ We can denote the instance as (G, r, cτ), where the cost function cτ

is a shifted as: ce : cτ

e (x) = ce(x) + τe, for all x ≥ 0.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 15/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Marginal Cost Pricing, cont’d

◮ The principle of MCR asserts that for a feasible flow f , the tax, τe,

should be τe = fe ∗ c′

e(fe) where c′ e is the derivative of ce (assumes

∀c ∈ C are differentiable)

Theorem 18.27

Let (G,r,c) be a non-atomic instance such that for every edge e, the function x ∗ ce(x) is convex and continuously differentiable. Let f ∗ be an

• ptimal flow for (G,r,c) and let τe = f ∗

e ∗ c′ e(f ∗ e ) denote the marginal cost

tax for edge e with respect f ∗. Then f ∗ is an equilibrium flow for (G,r,c + τ).

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 16/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Marginal Cost Pricing, cont’d

◮ The principle of MCR asserts that for a feasible flow f , the tax, τe,

should be τe = fe ∗ c′

e(fe) where c′ e is the derivative of ce (assumes

∀c ∈ C are differentiable)

Theorem 18.27

Let (G,r,c) be a non-atomic instance such that for every edge e, the function x ∗ ce(x) is convex and continuously differentiable. Let f ∗ be an

• ptimal flow for (G,r,c) and let τe = f ∗

e ∗ c′ e(f ∗ e ) denote the marginal cost

tax for edge e with respect f ∗. Then f ∗ is an equilibrium flow for (G,r,c + τ).

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 16/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Capacity Augmentation

◮ We can bound the inefficiency of equilibrium flows in non-atomic

selfish routing games with arbitrary cost functions

◮ Note that PoA is unbounded here (Non-linear Pigou)

◮ By moderately increasing link speed we can offset the inefficiency ◮ See example 18.28

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 17/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Capacity Augmentation, cont’d

Theorem 18.29

If f is an equilibrium flow for (G,r,c) and f ∗ is feasible for (G,2r,c), then C(f ) ≤ C(f ∗).

◮ This theorem can be interpreted such as the benefit of centralized

control is equaled or exceeded by a sufficient improvement in link technology.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 18/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Capacity Augmentation, cont’d

Theorem 18.29

If f is an equilibrium flow for (G,r,c) and f ∗ is feasible for (G,2r,c), then C(f ) ≤ C(f ∗).

◮ This theorem can be interpreted such as the benefit of centralized

control is equaled or exceeded by a sufficient improvement in link technology.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 18/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Summary

◮ We have seen how non-linearity can affect the PoA ◮ Potential function as an upper bound ◮ Pigou bound provides a better bound ◮ Atomic routing games with non-negative affine cost functions have

an upper bound

◮ How to reduce the PoA

◮ Marginal cost pricing ◮ Capacity augmentation Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 19/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

Thank you!

Questions?

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 20/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

References I

Tim Roughgarden. Selfish routing with atomic players. In Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, SODA ’05, pages 1184–1185, Philadelphia, PA, USA, 2005. Society for Industrial and Applied Mathematics. Tim Roughgarden. On the severity of braess’s paradox: Designing networks for selfish users is hard. Journal of Computer and System Sciences, 72(5):922 – 953, 2006. Special Issue on FOCS 2001. Tim Roughgarden. Algorithmic Game Theory, chapter 18, pages 461–486. Cambridge University Press, 2007.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 21/21

Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary

References II

Tim Roughgarden and ´ Eva Tardos. How bad is selfish routing?

• J. ACM, 49:236–259, March 2002.

Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 22/21