Mechanism Design without Money

1. Mechanism Design without Money Lecture 4

2. Price of Anarchy simplest example X • G is given • Route 1 unit from A to B, through AB,AXB • OPT– route ½ on AB and ½ on AXB (check!) • NASH – route 1 on AXB • Ratio is 4/3 0 x A B 1

3. Price of Anarchy: general functions X c(x) • G is given • Route 1 unit from A to B, through AB,AXB • OPT – choose y to minimize yC (y) + (1 – y)C (1) • Nash – route 1 on AXB • Ratio is • Also called the Pigou Bound 0 A B c(1)

4. Can things be worse? • No! • For a set of cpu functions C, define • In our examples before r was 1 • Theorem: For any routing game G, if the cost per unit functions come from C, the Price of Anarchy is at most a(C)

5. Another property of Nash flows • Theorem: Let f be a Nash flow. For any other flow f* which routes the same amount, we have • Note: The cost per unit of every edge is constant, and we just want to route the flow.

6. Proof • Define Note that • Therefore, we need to prove that H(f*,f) ≥ H(f,f) • This follows by using that if a path has any flow in it in a Nash flow, its cost is minimal =

7. Proof that a(C) is the bound on PoA Let C be the set of functions on the edges. Let f* be the optimal solution, and f be the Nash. And in particular setting r=feand y=f*e ) - =

8. How big can a(C) be? • Theorem: If C is a set of affine functions, a(C) is at most 4/3 • Proof: Do this at home. Hint: compute the derivative. You should get x = r/2

9. Fighting selfishness • Braess paradox shows that selfish agents can improve their situation if an edge is removed from the graph • Given a graph, which edges should be removed? • We are looking at

10. Fighting selfishness is (computationally) hard • Problem: Which edges should I cut to improve the worst Nash? • It is trivial to get a 4/3 approximation – just cut nothing. The worst Nash for a subgraph is always worse than OPT for the original graph, and the PoA with linear cost functions is 4/3 • Thm: It is NP hard to approximate better than 4/3

11. Proof • Reduction from 2DPP: Given a graph G, two sources s1,s2 and two targets t1,t2 are there two vertex disjoint paths s1t1 and s2t2 • If there are no two disjoint paths you will always have a path s2t1 G s1 t1 1 x s t 1 s2 t2 x

12. What happens for non linear cost functions? • We said price of anarchy can grow, but what about fighting selfishness? • Thm: There exists a graph with n vertices and non linear cost functions, such that removing edges improves the worst Nash by a factor of n/2

13. The Graph

14. Bad Nash flow

15. After edge removal

16. Atomic flows • Multiple equilibria (remember the examples) • Sometimes there is no pure equilibrium • Weaker bounds, different techniques

17. No pure Nash • P1 routes 1 unit from s to t • P2routes 2 units from s to t

18. Price of anarchy example • Not all paths in the equilibrium have the same cost • In the example: PoA of 5/2 • This is the worst case for affine functions if allplayers have the sameamount of flow • We will prove a weaker boundwhen players control different amounts of flow s1,s2 t1,s3 ,t4 0 V U x 0 x x x W t2,t3s4

19. Atomic flow for affine functions • Edge e has CPU aex + be • Player i sends ri • Let f be Nash, f* be OPT. We have for every player i:

20. Summing over the players:

21. Manipulations • You get • Solving x2-3x+10 gives (3 + 51/2)/2

22. Questions?

23. Extra Slides

24. Chicken

25. Road example • 50 people want to get from A to B • There are two roads, each one has two segments. One takes an hour, and the other one takes the number of people on it 1 hour N minutes A B 1 hour N minutes

26. Nash in road example • In the Nash equilibrium, 25 people would take each route, for a travel time of 85 minutes 1 hour N minutes A B 1 hour N minutes

27. Braess’ paradox • Now suppose someone adds an extra road which takes no time at all. Travel time goes to 100 minutes 1 hour N minutes A B Free 1 hour N minutes