1. Algorithms and Economics of Networks Abraham Flaxman and Vahab Mirrokni, Microsoft Research

2. Outline Network Congestion Games Congestion Games  Rosenthal’s Theorem: Congestion games are potential Games: PoA for Congestion Games. Market Sharing Games. Network Design Games.

3. Network Congestion Games A directed graph G=(V,E) with n users, Each edge e in E(G) has a delay function f e, Strategy of user i is to choose a path A j from a source s i to a destination t i, The delay of a path is the sum of delays of edges on the path, Each user wants to minimize his own delay by choosing the best path.

4. Example: Network Congestion Game s1t1 t3t2 s2 s3

5. Example: Network Congestion Game s1t1 t3t2 s2 s3 Agent 2 path 1 Agent 2 path 2

6. Congestion Games n players, a set of facilities E, Strategy of player i is to choose a subset of facilities (from a given family of subsets T i ), Facility i have a cost (delay) function f e which depends on the number of players playing this facility, Each player minimizes its total cost,

7. Example: Congestion Games Picture from Kapelushnik Lior f1f2f3f4F5F6

8. Example: Congestion Games f1f2f3f4F5F6

9. Example: Congestion Games f1f2f3f4F5F6

10. Example: Congestion Games f1f2f3f4F5F6

11. Congestion Games: Pure NE Rosenthal’s Theorem (1979): Any congestion game is an exact potential game. Proof is based on the following Potential Function

12. Classes of Congestion Games Every network congestion game is a congestion game Symmetric and Asymmetric Players Network Design Games. Maximizing Congestion Games: Each player wants to maximize his payoff (instead of minimizing his delay)  Market Sharing Games. Generalizations:  Weighted Congestion Games  Player-specific Congestion Games

13. Congestion Games: Social Cost Two social Cost functions: Consider a pure Strategy A = (A 1, A 2, …, A n ). Defintion 1: Defintion 2:

14. Congestion Games: PoA PoA for two social Cost functions: Defintion 1: Defintion 2: We Prove bounds for

15. Congestion Games: PoA PoA for congestion game with linear delay functions is at most 5/2. Proof: Lemma 1: for a pair of nonnegative integers a,b: Proof: …

16. Congestion Games: Lower Bound S1 S2 S3 t1 t2 t3

17. Congestion Games: PoA for mixed NE Theorem: PoA for mixed Nash equilibria in congestion games with linear latency function is 2.61. Theorem: PoA for mixed Nash equilibria of weighted congestion games with linear latency function is 2.61. Theorem: PoA for polynomial delay functions of constant degree is a constant.

18. Other Variants Atomic Congestion Games: Many infinitesimal users. The load of each user is very small. Theorem: PoA for non-atomic congestion games with linear latency functions is 4/3. Splittable Network Congestion Games

19. Market Sharing Games Congestion Game Facilities are Markets. Cost function  Profit Function. Players share the profit of markets (equally). Each player has some packing constraint for the set of markets he can satisfy. PoA: 1/2.

20. Network Design Games Players want to construct a network. They share the cost of buying links in the network. Known Results: Price of Stability, Convergence…

21. Next Lecture Coordination Mechanism Design