Download presentation

Presentation is loading. Please wait.

Published byLucia Linger Modified over 2 years ago

1
Course: Price of Anarchy Professor: Michal Feldman Student: Iddan Golomb 26/02/2014 Non-Atomic Selfish Routing

2
Talk Outline Introduction What are non-atomic selfish routing games PoA interpretation Main result – Reduction to Pigou-like networks Pigou-like networks Proof of the main result Analysis of consequences How to improve the situation Capacity augmentation Marginal cost pricing Summing-up

3
Motivation

4
Non-Atomic Selfish Routing (1) Directed graph (network): G(V,E) Source-target vertex pairs: (s 1,t 1 ),…, (s k,t k ) Paths: P i from s i to t i Flow: Non-negative vector over paths. Rate: Total flow. f is feasible for r if: Latency: Function over E: Non-negative Non-decreasing Continuous (differentiable) Instance: (G,r,l)

5
Non-Atomic Selfish Routing (2) Utilitarian cost: Edges: Paths: Non-atomic: Many players, negligible influence each Examples – Driving on roads, packet routing over the internet, etc.

6
Price of Anarchy Interpretation PoA: Pure N.E. (non-atomic) In our case, we will show: N.E. exists All N.E. flows have same total cost Examples when PoA is interesting: Limited influence on starting point (“in the wild”) Limited traffic regulation Optimal flow is instable PoA ≥ 1 The smaller, the better If grows with #players bad sign…

7
Pigou’s Example N.E: C(f)=1 Optimal: PoA=4/3 Questions: General graphs? General latency functions? SourceTarget l(r) l(x)=x

8
Pigou-like Networks Pigou-like network: 2 vertices: s,t 2 edges: s t Rate: r>0 Edge #1: General – l(∙) Edge #2: Constant – l(r) 2 free parameters: r, l Main result (informal): Among all networks, the largest PoA is achieved in a Pigou-like network SourceTarget l(r) l( ∙ )

9
Pigou Bound Minimal cost: PoA: Pigou bound ( α ): For any set L of latency functions: SourceTarget l(r) l( ∙ )

10
Main Result – Statement and Outline Theorem: For every set L of latency functions, and every selfish routing network with latency functions in L, the PoA is at most α (L) Proof outline: Preliminaries: Flows in N.E. N.E. existence Singular cost at N.E Proof: Freezing edge latencies in N.E. Comparing f* with flow in N.E

11
Flows in N.E. Clarification: N.E. with respect to pure strategies Claim: A flow f feasible for instance (G,r,l) is at N.E. iff Proof: Trivial Corollary: In N.E., for each i, the latency is the same for all paths: L i (f).

12
N.E. Existence (1) Goal: Min s.t: Define: and Assumptions: is differentiable, is convex f is a solution iff Example: Pigou optimal when

13
N.E. Existence (2) Now, set, change goal to: Min Same constraints for flows in N.E. and for convex program Optimal solutions for convex program are precisely flows at N.E. for (G,r,l)! Corollary: Under same conditions, f* is an optimal flow for (G,r,l) iff it is an equilibrium flow for (G,r,l’) Interpretation: Optimal flow and latency function ≈ Equilibrium flow and latency derivative

14
Singular Value at N.E. Claim: If are flows in N.E then Proof: The objective function is convex Otherwise: A convex combination of would dominate

15
“Freezing” Latency at N.E Notations: Optimal flow: f, N.E. flow: f* We’ve shown: Now:

16
How much is f* better than f? Pigou bound: For each edge e Set: Sum for all edges: : QED

17
Interpretation of Main Result Questions from earlier: General graphs? General latency functions? Result for polynomial latency functions: Result as d goes to infinity the PoA goes to infinity DegreeRepresentativePoA 1ax+b (Affine)4/3 2ax 2 +bx+c d

18
Capacity Augmentation (1) Different comparison from PoA Claim: If f is an equilibrium flow for (G,r,l), and f* is feasible for (G,2r,l), then: C(f) ≤ C(f*) Proof: L i : Minimal cost for f in s i t i path We will define new latency functions “Close” to current latency function Allows to lower bound a flow f* with respect to C(f)

19
Capacity Augmentation (2) Definition: 1)

20
Capacity Augmentation (3) Allows to lower bound a flow f* with respect to C(f) 2)

21
Capacity Augmentation (4) 1) 2) : QED Generalization: If f is N.E flow for (G,r,l) and f* is feasible for (G,(1+ γ )r,l), then: Interpretation: Helpful if we can increase route/link speed (without resorting to central routing) 2) 1)

22
Marginal Cost Pricing (1) We can’t always increase route speed We can (almost) always charge more… Tax Claim: Given (G,r,l), as defined, then: is an equilibrium flow for (G,r,(l+ τ )) Reminder: f* is an optimal flow for (G,r,l) iff it is an equilibrium flow for (G,r,l’)

23
Marginal Cost Pricing (2) : Marginal increase caused by a user : Amount of traffic suffering from the increase Tax “aligns” the derivative to fit utilitarian goal Interpretation: PoA is reduced to 1! However, the costs were artificially raised (“sticks” as opposed to “carrots”). Might cause users to leave.

24
Summing Up Realistic problem PoA interpretation Main result – Reduction to Pigou-like networks Every network is easy to compute For some cost functions, PoA is arbitrarily high How to improve the situation Choose specific cost functions Capacity augmentation (“carrot”) – Make better roads Marginal cost pricing (“stick”) – Collect taxes

25
Questions? ?

26
Bibliography Roughgarden T, Tardos E – How bad is selfish routing? J.ACM, 49(2): 236259, 2002. Stanford AGT course by Roughgarden - http://theory.stanford.edu/~tim/f13/f13.html (Lecture 11) Nisan, Roughgarden, Tardos, Vazirani - Algorithmic Game Theory, Cambridge University Press. Chapter 18 (routing games) – 461-486. Cohen J.E., Horowitz P - Paradoxical behavior of mechanical and electrical networks. Nature 352, 699– 701. 1991.

27
Extension – Approximate N.E. Agents distinguish between paths that differ in latencies by at least Claim: f is at ε -approximate N.E. iff Another claim: If f is at ε -approximate N.E. with ε <1 for (G,r,l), and f* is feasible for (G,2r,l) then

28
Braess’ Paradox SourceTarget 1 x E2 E1 x 1 0

29
Braess’ Paradox – Physical Examples

Similar presentations

OK

Balázs Sziklai Selfish Routing in Non-cooperative Networks.

Balázs Sziklai Selfish Routing in Non-cooperative Networks.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on resources and development Ppt on different types of computer languages Ppt on autotrophic mode of nutrition Ppt on political parties and electoral process in nigeria Ppt on needle stick injury report Ppt on regulated power supply Ppt on acute coronary syndrome protocol Pdf to ppt online convertor Download ppt on biomass power plant Ppt on national education day essay