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The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid 27th International Colloquium on Structural Information and Communication Complexity (SIROCCO) June 29 - July 1, 2020 The Value of Information in Selfish

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The Value of Information in Selfish Routing

Simon Scherrer, Adrian Perrig, Stefan Schmid 27th International Colloquium on Structural Information and Communication Complexity (SIROCCO) June 29 - July 1, 2020

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Network-based path selection

A2 A3 A6 A5 A4 A1 A7 p

e1 e2

Suboptimal paths No robustness to failures

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Source-based path selection

A2 A3 A6 A5 A4 A1

e1 e2

A7 p1 p2 p3

Suboptimal paths No robustness to failures Best path for use case Fast rerouting

  • n failure

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Network-based path selection: Network operator view

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Source-based path selection: Network operator view

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Revisit selfish-routing concepts to investigate two issues arising in emerging path-aware Internet architectures: ▪ Impact of information: What network state information should be shared with end-hosts? ▪ Impact on network operators: What is the impact

  • f selfish routing on the cost of network operators?

Goals of our work

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Price of Anarchy: Three components

C

Social cost function

Fopt Social optimum Feq

Equilibrium

PoA = C(Feq) C(Fopt)

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Adapted Wardrop model of source-based path selection

A2 A1 A3

α β γ

d = (d1,2, d3,4) = (1, 1)

e3 e2 e1 e4

F = (Fα, Fγβ, Fβ, Fαγ) f = (fα, fβ, fγ) cα(fα) = 1 cβ(fβ) = fβ

2

cγ(fγ) = fγ Cπ(F) = Σℓ ∈ π cℓ

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

End-host cost function: C* = Σend-hosts Σpaths flow on path · path cost (classic)

= Σπ ∈ Π Fπ · Cπ(F) = Σℓ ∈ L fℓ · cℓ(fℓ)

Total cost functions and social optima

Network-operator cost function: C# = Σlinks link cost = Σℓ cℓ(fℓ) End-host optimum:

F* = argminF C*(F)

Network-operator optimum:

F# = argminF C#(F)

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Characterizing social optima: Suboptimal path flow pattern

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Cπ(F) Fπ

Cα(Fα) Cβ(Fβ)

d = (d1,2) = (1) F = (Fα, Fβ)

Cα

C(F) = Cα(Fα) + Cβ(Fβ)

Cβ ΔCα

+

ΔCβ

  • ∃ δ. |ΔCα

+| < |ΔCβ

  • |

⇒ C can be reduced

1 δ δ

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Characterizing social optima: Optimal path flow pattern

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Cπ(F) Fπ

Cα(Fα) Cβ(Fβ) Cα Cβ ΔCα

+

ΔCβ

  • 1

δ δ

m = ∂C/∂Fα m = ∂C/∂Fβ

∀δ. |ΔCα

+| > |ΔCβ

  • |

⇒ C cannot be reduced ⇒ C is optimal ∂C/∂Fα = ∂C/∂Fβ ⇒ ∀δ. |ΔCα

+| > |ΔCβ

  • |

⇒ C is optimal

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Socially optimal marginal costs

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∂C(F)/∂Fπ is the marginal cost of path π given path-flow pattern F A path-flow pattern F is optimal w.r.t. a cost function C ∈{C*,C#} if for every origin-destination pair: Fα, …, Fρ > 0 Fσ, …, Fω = 0 = … = ≤ ≤ … ≤ ∂C(F) ∂Fα ∂C(F) ∂Fρ ∂C(F) ∂Fσ ∂C(F) ∂Fω

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Social optimum: Comparison (Example)

F# = (Fα, Fβ, Fγ) = (½, 0, ½)

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F* = (Fα, Fβ, Fγ) = (⅔, ⅓, 0) Different optima! Network operators prefer usage

  • f links with little variable cost (here: γ)

A1 A2

e1 e2

α β Cα(Fα) = Fα

2

Cβ(Fβ) = 2 Fβ γ Cγ(Fγ) = Fγ

+ 2

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Price of Anarchy: Where are we?

C

Total cost function

Fopt Social optimum Feq

Equilibrium

PoA = C(Feq) C(Fopt)

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Equilibrium with latency-only information (LI equilibrium)

A1 A2

e1 e2

α β F = (Fα, Fβ) = (1, 0) d = (d1,2) = (1) Cα = Cβ ⇒ F = (1,0) is an LI equilibrium

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Cα = 2 Cβ = 2

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Characterizing the LI equilibrium

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A path flow pattern F is an LI equilibrium if for every origin-destination pair: Fα, …, Fρ > 0 Fσ, …, Fω = 0 Cα(F) = … = Cρ(F) ≤ Cσ(F) ≤ … ≤ Cω(F)

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Equilibrium with perfect information (PI equilibrium)

A1 A2

e1 e2

α: cα(fα) = fα β: cβ(fβ) = 2 F(1)= (Fα, Fβ) d(1) = (d1,2) = (1) fα = Fα + 1 fβ = Fβ + 1 Minimize selfish cost C(1)(F(1)) = Fα·(Fα+1) + Fβ·2 ⇒ (Fα, Fβ) = (⅔, ⅓) is a PI equilibrium

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Characterizing the PI equilibrium

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A path flow pattern F is a PI equilibrium if for every origin-destination pair of any end-host e: Fα, …, Fρ > 0 Fσ, …, Fω = 0 = … = ≤ ≤ … ≤ ∂C(e)(F) ∂Fα ∂C(e)(F) ∂Fρ ∂C(e)(F) ∂Fσ ∂C(e)(F) ∂Fω

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Capturing the value of information

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Information assumption Latency-only Information (LI) Perfect Information (PI) Equilibrium F0 F+ Price of Anarchy PoA0 = a PoA+ = a C(F0) C(Fopt) C(F+) C(Fopt) Δ = Value of Information (VoI)

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The benefits of information

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VoI > 0

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The benefits of information: Network of parallel links

(cf. Roughgarden 2003)

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A1 A2

e1

eT

α β cα(fα) = 1 cβ(fβ) = fβ

eK

...

Σk dk,T = 1 F = (F1α, F1β, ... FKα, FKβ,) EH Opt: F* s.t. fβ = 1/2 LI Eq: F0 s.t. fβ = 1 PI Eq: F+s.t. fβ = K/(K+1) NO Opt: F# s.t. fβ = 0

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The benefits of information: Network of parallel links

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LI equilibrium PI equilibrium End-host perspective Network-

  • perator

perspective

PoA*0 = 4/3 PoA*+ = (K2+K+1)/(K2+2K+1) · 4/3 ≤ PoA*0 PoA#+ = 1+ K/(K + 1) ≤ 2 = PoA#0 PoA#0 = 2

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The benefits of information: Network of parallel links

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LI equilibrium PI equilibrium End-host perspective Network

  • perator

perspective

PoA*0 = 4/3 PoA*+ = (K2+K+1)/(K2+2K+1) · 4/3 ≤ PoA*0 PoA#+ = 1+ K/(K + 1) ≤ 2 = PoA#0 PoA#0 = 2

PI equilibrium cheaper than LI equilibrium

VoI > 0

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The drawbacks of information

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VoI < 0

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The drawbacks of information: Ladder network

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A11

d = (d11,12, d21,22) = (1, 1)

A12 A21 A22

h1 ch1(fh1) = fh1

2

h2 ch2(fh2) = fh2

2

v1 cv1(fv1) = fv1 cv2(fv2) = fv2 v2 F→ = (1, 0, 1, 0)

e21 e11 e12 e22

Direct-only F→ is universally optimal: F→ = F* = F#

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The drawbacks of information: Ladder network

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A11 A12 A21 A22

h1 ch1(fh1) = fh1

2

h2 ch2(fh2) = fh2

2

v1 cv1(fv1) = fv1 cv2(fv2) = fv2 v2 F→ = (1, 0, 1, 0)

e21 e11 e12 e22

C1H(F→) = 1 = C1V(F→) ⇒ F→ = F0 (LI equilibrium is optimal)

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The drawbacks of information: Ladder network

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A11 A12 A21 A22

h1 ch1(fh1) = fh1

2

h2 ch2(fh2) = fh2

2

v1 cv1(fv1) = fv1 cv2(fv2) = fv2 v2 F→ = (1, 0, 1, 0)

e21 e11 e12 e22

C(1)(F→) = 1 > C(1)(F~) = 0.87 ⇒ F→ ≠ F+

(PI equilibrium is suboptimal)

F~ = (0.9, 0.1, 1, 0)

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The drawbacks of information: Ladder network

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A11 A12 A21 A22

h1 ch1(fh1) = fh1

2

h2 ch2(fh2) = fh2

2

v1 cv1(fv1) = fv1 cv2(fv2) = fv2 v2 F→ = (1, 0, 1, 0)

e21 e11 e12 e22

C(1)(F→) = 1 > C(1)(F~) = 0.87 ⇒ F→ ≠ F+

(PI equilibrium is suboptimal)

C(1)(F) = F1H · C1H(F) + F1V · C1V(F) F~ = (0.9, 0.1, 1, 0) PI equilibrium more costly than LI equilibrium

VoI < 0

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

The drawbacks of information: Generalized ladder network

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h1 ch1(fh1) = fh1

p

h2 v11 cv11(fv11) = t·fv11 cv12(fv12) = t·fv12 v12 hH-1 hL vH-1,1 vH-1,2 H

Upper bound on PoA for network

  • perators:

PoA#+ ≤ 1 + p.

≤ 1 + p

2(H-1) 3H 2 3

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Drawback of information: Abilene Topology Case Study

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| | The Value of Information in Selfish Routing Simon Scherrer, Adrian Perrig, Stefan Schmid @ SIROCCO 2020

Questions

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Thank you for your attention! Happy to answer questions in the chat forum! Or by email: Simon Scherrer simon.scherrer@inf.ethz.ch