Network Congestion Games

1. Network Congestion Games Assistant Professor Texas A&M University College Station, TX EvdokiaNikolova TX Dallas College Station Austin Houston

2. Best route depends on others Network Congestion Games

3. Travel time increases with congestion • Highway congestion costs were$115 billion in 2009. • Avg. commuter travels 100 minutes a day. Network Congestion Games

4. Example: Inefficiency of equilibria Delay is 1.5 hours for everybody at the unique Nash equilibrium 1/2 x hours 1 hour Town B Town A 1 hour x hours 1/2 Suppose drivers (total 1 unit of flow) leave from town A towards town B. Every driver wants to minimize her own travel time. What is the traffic on the network? In any unbalanced traffic pattern, all drivers on the most loaded path have incentive to switch their path. Network Congestion Games

5. Example: Inefficiency of equilibria Delay is 2 hours for everybody at the unique Nash equilibrium 1 x hours 1 hour Town B Town A 0 hours 1 hour x hours A benevolent mayor builds a superhighway connecting the fast highways of the network. What is now the traffic on the network? No matter what the other drivers are doing it is always better for me to follow the zig-zag path. Network Congestion Games

6. Example: Inefficiency of equilibria 1 1 hour 1 hour x hours x hours vs B A 1/2 1 hour 1 hour x hours x hours Adding a fast road on a road-network is not always a good idea! Braess’s paradox B A In the RHS network there exists a traffic pattern where all players have delay 1.5 hours. 1/2 Price of Anarchy: measures the loss in system performance due to free-will Network Congestion Games

7. Game model • Directed graph G = (V,E) Multiple source-dest. pairs (sk,tk), demand dk • Players (users): nonatomic(infinitesimally small) • Strategy set: paths Pk between (sk,tk) for all k Players’ decisions: flow vector Sometimes will use for path flow. • Edge delay (latency) functions: typically assumed continuous and nondecreasing. Network Congestion Games

8. Outline • Wardrop Equilibrium • Social Optimum • Price of Anarchy Network Congestion Games

9. Outline • Wardrop Equilibrium • Social Optimum • Price of Anarchy Network Congestion Games

10. Wardrop’s First Principle • “Travel times on used routes are equal and no greater than travel times on unused routes.” • Definition: A flow x is a Wardrop Equilibrium (WE) if for every source-dest. pair k and for every path with positive flow between this pair, where Also called User Equilibrium or Nash Equilibrium. • Equilibrium flow is called Nash flow. Network Congestion Games

11. Outline • Wardrop Equilibrium • Social Optimum • Price of Anarchy Network Congestion Games

12. Wardrop’s Second Principle • “The average [total] journey time is minimum.” • The cost of flow x is defined as the “total journey time”: • Denote , assumed convex. Network Congestion Games

13. Wardrop’s Second Principle • “The average [total] journey time is minimum.” • Definition: A flow x is a Social Optimum if it minimizes total delay: flow constraints Network Congestion Games

14. Social Optimum • “The average [total] journey time is minimum.” • Definition: A flow x is a Social Optimum if it minimizes total delay: Network Congestion Games

15. Social Optimum • Definition: A flow x is a Social Optimum if it solves • Lemma: A flow vector x is locally optimal if for each path p with positive flow and each path p’, where Proof sketch: marginal benefit of marginal cost of reducing traffic on p increasing traffic on p’ Network Congestion Games

16. Social Optimum (SO) • Definition: A flow x is a Social Optimum if it solves • Lemma: A flow vector x is locally optimal if for each path p with positive flow and each path p’, • Corollary 1: If costs are convex, local opt is a global opt, and lemma gives equivalent defn of Social Optimum. Network Congestion Games

17. Social Optimum (SO) • Definition: A flow x is a Social Optimum if it solves • Lemma: A flow vector x is locally optimal if for each path p with positive flow and each path p’, • Corollary 2: If costs are convex, SO is an equilibrium with respect to modified latencies Network Congestion Games

18. Social Optimum (SO) Mechanism Design Interpretation: If users value time and money equally, imposing tolls per unit flow on each edge will cause selfish players to reach the Social Optimum! • Definition: A flow x is a Social Optimum if it solves • Lemma: A flow vector x is locally optimal if for each path p with positive flow and each path p’, • Corollary 2: If costs are convex, SO is an equilibrium with respect to modified latencies Network Congestion Games

19. Computing Social Optimum (SO) • Definition: A flow x is a Social Optimum if it solves • Corollary 3: If costs are convex, SO exists and can be found efficiently by solving convex program above. Network Congestion Games

20. Outline • Revisit Wardrop Equilibrium • Social Optimum • Price of Anarchy Network Congestion Games

21. Equilibrium existence • WE Definition: A flow x is a Wardrop Equilibrium if for every source-dest. pair k and for every path with positive flow between this pair, compare with: • SO Definition:A flow vector x is a Social Optimumfor every source-dest. pair k and for every path with positive flow between this pair, Network Congestion Games

22. Equilibrium existence • WE Definition: A flow x is a Wardrop Equilibrium if for every source-dest. pair k and for every path with positive flow between this pair, where • SO Definition:A flow vector x is a Social Optimumfor every source-dest. pair k and for every path with positive flow between this pair, Network Congestion Games

23. Equilibrium existence • WE Definition: A flow x is a Wardrop Equilibrium if for every source-dest. pair k and for every path with positive flow between this pair, where • Alternative SO Definition: A flow vector x is a Social Optimum if it solves: Network Congestion Games

24. Equilibrium existence • Alternative WE Definition: A flow vector x is a Wardrop Equilibrium if it solves: where • Alternative SO Definition: A flow vector x is a Social Optimum if it solves: Network Congestion Games

25. Equilibrium existence • Alternative WE Definition: A flow vector x is a Wardrop Equilibrium if it solves: where • Alternative SO Definition: A flow vector x is a Social Optimum if it solves: Network Congestion Games

26. Equilibrium existence • Alternative WE Definition: A flow vector x is a Wardrop Equilibrium if it solves: • Theorem: A Wardrop Equilibrium exists and can be computed in polynomial time. Also, if program above is strictly convex, equilibrium is unique, up to same flow cost. Network Congestion Games

27. Outline • Wardrop Equilibrium • Social Optimum • Price of Anarchy Network Congestion Games

28. Example: Inefficiency of equilibria Delay is 1.5 hours for everybody at the unique Nash equilibrium 1/2 x hours 1 hour Town B Town A 1 hour x hours 1/2 Suppose drivers (total 1 unit of flow) leave from town A towards town B. Every driver wants to minimize her own travel time. What is the traffic on the network? In any unbalanced traffic pattern, all drivers on the most loaded path have incentive to switch their path. Network Congestion Games

29. Example: Inefficiency of equilibria Delay is 2 hours for everybody at the unique Nash equilibrium 1 x hours 1 hour Town B Town A 0 hours 1 hour x hours A benevolent mayor builds a superhighway connecting the fast highways of the network. What is now the traffic on the network? No matter what the other drivers are doing it is always better for me to follow the zig-zag path. Network Congestion Games

30. Example: Inefficiency of equilibria 1 1 hour 1 hour x hours x hours vs B A 1/2 1 hour 1 hour x hours x hours Adding a fast road on a road-network is not always a good idea! Braess’s paradox B A In the RHS network there exists a traffic pattern where all players have delay 1.5 hours. 1/2 Price of Anarchy: measures the loss in system performance due to free-will Network Congestion Games

31. Price of Anarchy • Cost of Flow: total user cost • Social optimum: flow minimizing total user cost • Price of anarchy: (Koutsoupias, Papadimitriou ’99) Network Congestion Games

32. Variational Inequality representation of equilibria Theorem: Equilibria in nonatomic games are solutions to the Variational Inequality (VI) where VI Solution exists over compact convex set with ℓ(x) continuous [Hartman, Stampacchia ‘66]. ∎ • VI Solution unique if ℓ(x) is monotone: (ℓ(x)-ℓ (x’))(x-x’) ≥ 0. [Exercise: verify] Proof: Flow x is an equilibrium if and only if ℓ(x).x <= ℓ(x).x’ . Proof: (=>) Equilibrium flow routes along minimum-cost paths ℓ(x). Fixing path costs at ℓ(x), any other flow x’ that assigns flow to higher-cost paths will result in higher overall cost ℓ(x).x’. (<=) Suppose x is not an eq. Then there is a flow-carrying path p with ℓp(x) > ℓp’(x). Shifting flow from p to p’ will obtain flow x’ with ℓ(x).x’ < ℓ(x).x, contradiction. Network Congestion Games

33. Price of Anarchy with linear latencies Theorem**: The price of anarchy (PoA) is 4/3 in general graphs and latencies i.e. where x is WE and x* is SO flow. Pf: Network Congestion Games **References: Roughgarden, Tardos ’02; Correa, Schulz, Stier-Moses ‘04, ‘08

34. Take-away points • Equilibrium and Social Optimum in nonatomic routing games exist and can be found efficiently via convex programs. • Social optimum is an equilibrium with respect to modified latencies = original latencies plus toll. • Price of anarchy: 4/3 for linear latencies, can be found similarly for more general classes of latency functions. Network Congestion Games

35. References • Wardrop ‘52, Beckmann et al. ’56, … • A lot of work in AGT community and others. Surveys of recent work: • AGT Book Nisan et al. ‘07 • Correa, Stier-Moses ’11 Network Congestion Games

36. Some open questions • What is the price of anarchy with respect to other Social Cost functions? • Dynamic (time-changing) latency functions? • Uncertain delays? Network Congestion Games