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

2. Best route depends on others
Evdokia Nikolova
Network Congestion Games

3. Travel time increases with congestion
Highway congestion costs were $115 billion in 2009.
Avg. commuter travels 100 minutes a day.
Evdokia Nikolova
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 A
Town B
1/2
Suppose drivers (total 1 unit of flow) leave from town A towards town B.
(Ex): Verify that the split is the unique Nash equilibrium of the system shown above.
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.
Evdokia Nikolova
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 A
Town B
0 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.
Evdokia Nikolova
Network Congestion Games

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

7. Network Congestion Games
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.
Evdokia Nikolova
Network Congestion Games

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

9. Network Congestion Games
Outline
Wardrop Equilibrium
Social Optimum
Price of Anarchy
Evdokia Nikolova
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.
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
Network Congestion Games

11. Network Congestion Games
Outline
Wardrop Equilibrium
Social Optimum
Price of Anarchy
Evdokia Nikolova
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.
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
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
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
Network Congestion Games

14. Network Congestion Games
Social Optimum
“The average [total] journey time is minimum.”
Definition: A flow x is a Social Optimum if it minimizes total delay:
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
Network Congestion Games

15. Network Congestion Games
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
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Proof sketch: marginal benefit of marginal cost of
reducing traffic on p increasing traffic on p’
Evdokia Nikolova
Network Congestion Games

16. Network Congestion Games
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.
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
Network Congestion Games

17. Network Congestion Games
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
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
Network Congestion Games

18. Network Congestion Games
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
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
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.
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
Network Congestion Games

20. Network Congestion Games
Outline
Revisit Wardrop Equilibrium
Social Optimum
Price of Anarchy
Evdokia Nikolova
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 Optimum for every source-dest. pair k and for every path with positive flow between this pair,
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
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 Optimum for every source-dest. pair k and for every path with positive flow between this pair,
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
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:
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
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:
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
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:
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
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.
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Evdokia Nikolova
Network Congestion Games

27. Network Congestion Games
Outline
Wardrop Equilibrium
Social Optimum
Price of Anarchy
Evdokia Nikolova
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 A
Town B
1/2
Suppose drivers (total 1 unit of flow) leave from town A towards town B.
(Ex): Verify that the split is the unique Nash equilibrium of the system shown above.
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.
Evdokia Nikolova
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 A
Town B
0 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.
Evdokia Nikolova
Network Congestion Games

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

31. Network Congestion Games
Price of Anarchy
Cost of Flow: total user cost
Social optimum: flow minimizing total user cost
Price of anarchy: (Koutsoupias, Papadimitriou ’99)
One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each.
Evdokia Nikolova
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.
One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each.
Evdokia Nikolova
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:
One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each.
Evdokia Nikolova
Network Congestion Games
**References: Roughgarden, Tardos ’02; Correa, Schulz, Stier-Moses ‘04, ‘08

34. Network Congestion Games
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.
One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each.
Evdokia Nikolova
Network Congestion Games

35. Network Congestion Games
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
One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each.
Evdokia Nikolova
Network Congestion Games

36. Network Congestion Games
Some open questions
What is the price of anarchy with respect to other Social Cost functions?
Dynamic (time-changing) latency functions?
Uncertain delays?
One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem
All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems
Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each.
Evdokia Nikolova
Network Congestion Games