Non-Atomic Selfish Routing

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

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: (s1,t1),…, (sk,tk) • Paths: Pi from sito ti • 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? l(r) Source Target 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 l(r) Source Target l(∙)

9. Pigou Bound • Minimal cost: • PoA: • Pigou bound (α): For any set L of latency functions: l(r) Source Target 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: Li(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 

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: • Li: Minimal cost for f in siti 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:

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

21. Capacity Augmentation (4) • : 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) 1) 2) 

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.