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