How Much Can Taxes Help Selfish Routing?

1. How Much Can Taxes Help Selfish Routing? Richard Cole Yvgeniy Dodis Tim Roughgarden Presented By: Omry Tuval Some examples are taken form Tim Roughgarden’s slides

2. Selfish Routing • a directed graph G=(V,E) • Source vertex s, sink vertex t • Each edge has a latency function • Infinitely large population • Traffic is selfish – minimize latency • Social goal – total latency

3. Selfish Routing - Example

4. Flows • the fraction of traffic flowing on path P from s to t. • The flow function is the routing of the traffic • Interested in the stable state considering selfish traffic. (Nash Equilibrium)

5. Nash Flow – Def. • A flow is a “Nash Flow” if all traffics is routed along minimum cost paths given current edge congestion • Existence and Uniqueness [wardrope, beckmann 1950s]

6. Price Of Anarchy – How Bad Is Selfish Routing? • POA is unbounded • POA in linear latency networks is 4/3

7. Economic Incentives - Taxes • Encouraging desired behavior • Each edge has usage tax: • Users selfishly try to minimize personal cost = personal latency + personal tax • Hopefully will result in better global cost

8. Marginal Cost Taxes [Beckmann et al. 1956] • Each user should pay a tax equal to the additional delay other users experience because of its presence • With regard to a flow • Formal:

9. MCT Theorem [beckmann 1959] • Let be the optimal flow of original network. • Let be the MCT with regard to • The Nash flow in the taxed network is • MCT minimized total latency • Social injustice • Total cost = total latency + total tax • Total cost might be higher then original network • Should study taxation with regard to total cost!

10. Let’s get Mathematical… • G=(V,E) directed graph with source s and sink t • P: the set of simple s-t paths in G • Flow is a function • Given a flow we define • Given a traffic amount r, a flow f is feasible if

11. Let’s get more Mathematical ... • Each edge has a continuous, non negative, non decreasing latency function • Each edge has a non negative tax • We denote the cost of a feasible flow • We denote an instance of the game by and a taxed instance by

12. Nash Flow properties • Claim 1: let be a Nash flow for then there is a constant such that: • Proof: straight from definition. • We will denote that constant as • Claim 2: is non decreasing in r • Proof: [Hall 1978]

13. MCT can be bad for you • Result: In all linear latency networks, using MCT can only increase the cost of the Nash flow • Theorem: let be an instance with linear latency functions and be the corresponding marginal cost taxes. Let be the Nash flow for and be the Nash flow for then: • Proof

14. Taxes – How good can it get? • For linear latency networks, taxes can improve Nash cost (at best) by a factor of 4/3 • Proof • For general latency networks, taxes can improve Nash cost at best by a factor of n/2 • Proof: extension of [Roughgarden 2001]

15. Taxes vs. Edge Removal • A different approach to improve selfish routing • Studied in [Roughgarden FOCS 2001] • Taxes at least as powerful as Edge Removal • Infinitely large tax means practically removing the edge • When are taxes actually better than Edge Removal?

16. Taxes vs. Edge RemovalLinear networks • Theorem: for an instance with linear latency functions, the optimal tax is a tax • Proof sketch • For linear networks, taxes never improve over edge removal.

17. Taxes vs. Edge RemovalGeneral networks • Result: for each n, exists a selfish routing instance where taxes are better than edge removal by a factor of n/2. • Proof

19. Taxes can be usefulComputability? • We’ve seen that in general latency networks, taxes can improve the Nash cost, and even improve on edge removal • Can the optimal tax be computed efficiently? • The problem is NP-hard • What about approximations?

20. “Forget about it” theorem • If there is NO • approximation algorithm for linear latency networks • approximation algorithm for general latency networks • The trivial algorithm (return T=0 as approx.) is: • approximation algorithm for linear latency networks • approximation algorithm for general latency networks

21. Conclusion

22. the end

23. MCT can be bad for youproof • Let be the optimal min. latency flow for • Linear latency: • MCT: • Define: • For each edge: • Therefore is Nash not only of but also of and with the same cost

24. MCT can be bad for youproof • is the Nash flow of • Path cost with regard to flow in is identical to path cost with regard to flow in • Therefore is the Nash flow in and • QED

25. Taxes in linear networks • For linear latency networks, taxes can improve Nash cost (at best) by a factor of 4/3 • Let be an instance with linear latency with Nash flow and optimal flow . • POA for linear latency is 4/3

26. Taxes in linear networks • Let be the optimal taxes, and the Nash flow for QED

27. Taxes vs. Edge RemovalLinear networks • Theorem: for an instance with linear latency functions, the optimal tax is a tax • Assume false, look at minimal counterexample • Look at counterexample optimal tax that has the smallest sum (existence if proven by minimality) • Understand how Nash flow change under local changes in the tax (linear equations) • Perturbing to a smaller tax will increase cost • Opposite perturbation lower costs (contradiction) • QED

28. Taxes vs. Edge RemovalGeneral networks • Theorem: • For each integer there is an instance such that for all subgraphs of it holds that but for some tax it holds that • for simplicity n is even, and n=2k+2 • Our network will be the Braess graph,

29. Kth Braess Graph • Vertices • Edges: • Type A: • Type B: • Type C:

30. Latency functions Saturation!

31. Lemma Saturation! • If is a subgraph of and a Nash flow saturates an edge in then

32. s-t paths in the graph • k paths Pi of the form: • k-1 paths Qi of the form: • Q1 is the path: • QK is the path:

33. Taxes are good • Consider • Suppose we tax each edge with 1 unit of tax, and 0 elsewhere • The following flow is Nash flow: • 1 unit of flow on Pi • 1/(k+1) units of flow on Qi • This shows

34. Edge Removal is bad • We now must show that for every subgraph of • Suppose • Nash Flow • 1+1/k units of flow on Pi • Also true if removes only type B edges

35. Type C removal • Suppose removes a type C edge, say • How much flow can leave Swithout saturation? • At Most • An edge adjacent to S is saturated!

36. Type A removal • Suppose removes some type A edge, say • How much flow can leave Swithout saturation? • At most • An edge in the graph is saturated! QED