Bottleneck Routing Games in Communication Networks

1
Bottleneck Routing Games in Communication
Networks

• Ron Banner and Ariel Orda
• Department of Electrical Engineering
• Technion- Israel Institute of Technology

2
Selfish Routing

• Often (e.g., large-scale networks, ad hoc
  networks) users pick their own routes.
• No central authority.
• Network users are selfish.
• Do not care about social welfare.
• Want to optimize their own performance.
• Major Question how much does the network
  performance suffer from the lack of global
  regulation?

3
Selfish Routing Quantifying the Inefficiency

• A flow is at Nash Equilibrium if no user can
  improve its performance.
• May not exist.
• May not be unique.
• The price of anarchy The worst-case ratio
  between the performance of a Nash equilibrium
  and the optimal performance.
• The price of stability The worst-case ratio
  between the performance of a best Nash
  equilibrium and the optimal performance.

4
Cost structures of flows

• Additive Metrics (path performance sum of link
  performances)
• E.g., Delay, Jitter, Loss Probability.
• Considerable amount of work on related routing
  games Orda, Rom Shimkin, 1992
  Korilis, Lazar Orda, 1995 Roughgarden
  Tardos, 2001 Altman, Basar, Jimenez
  Shimkin, 2002 Kameda, 2002 La
  Anantharam, 2002 Roughgarden, 2005
  Awerbuch, Azar Epstein, 2005
  Even-Dar Mansour, 2005
• Bottleneck Metrics (path performance worst
  performance of a link on a path).
• No previous studies in the context of networking
  games!

5
Bottleneck Routing Games (examples)

• Wireless Networks
• Each user maximizes the smallest battery lifetime
  along its routing topology.
• Traffic bursts
• Each user maximizes the smallest residual
  capacity of the links they employ.
• Traffic Engineering
• Each user minimizes the utilization of the most
  utilized buffer
• Avoids deadlocks and packet loss.
• Each user minimizes the utilization of the most
  utilized link.
• Avoids hot spots.
• Attacks
• usually aimed against the links or nodes that
  carry the largest amount of traffic.
• Each user minimizes the maximum amount of traffic
  that a link transfers in its routing topology.

6
Model

• A set of users Uu1, u2,, uN.
• For each user, a positive flow demand ?u and a
  source-destination pair (su,tu).
• For each link e, a performance function qe().
• qe() is continuous and increasing for all links.
• Routing model
• Splittable
• Unsplittable

7
Model (cont.)

• User behavior
• Users are selfish.
• Each minimizes a bottleneck objective
• Social objective
• Minimize the network bottleneck

8
Questions

• Is there at least one Nash Equilibrium?
• Is the Nash equilibrium always unique?
• How many steps are required to reach equilibrium?
• What is the price of anarchy?
• When are Nash equilibria socially optimal?

9
Existence of Nash Equilibrium

• Theorem An Unsplittable Bottleneck Game admits a
  Nash equilibrium
• Very simple proof.
• Theorem A Splittable Bottleneck Game admits a
  Nash Equilibrium.
• Complex proof.
• Splittable bottleneck games are discontinuous!
• why
• Hence, standard proof techniques cannot be
  employed!

10
Questions

• Is there at least one Nash Equilibrium?
• Yes!
• Is the Nash equilibrium unique?
• How many steps are required to reach equilibrium?
• What is the price of anarchy?
• When are Nash equilibria socially optimal?

11
Non-uniqueness of Nash Equilibria
p1
g 1
qe(fe)fe for each e in E.
e1
e3
s
t
e2
p2

• (fp11, fp20) (fp10, fp21) are Unsplittable
  Nash flows.
• (fp10.5, fp20.5) (fp10.25, fp20.75) are
  Splittable Nash flows.
• I.e. at least two different Nash flows for each
  routing game.

12
Questions

• Is there at least one Nash Equilibrium?
• Yes!
• Is the Nash equilibrium always unique?
• No!
• How many steps are required to reach equilibrium?
• What is the price of anarchy?
• When are Nash equilibria socially optimal?

13
Convergence time (unsplittable case)

• Theorem the maximum number of steps required to
  reach Nash equilibrium is
• For O(1) users, convergence time is polynomial.

14
Unbounded convergence time (splittable case)
g 2
g 2
S2
S1
qe(fe)fe for each e in E
T1
T2
15
Questions

• Is there at least one Nash Equilibrium?
• Yes!
• Is the Nash equilibrium always unique?
• No!
• How many steps are required to reach equilibrium?
• Unsplittable
• Splittable 8
• What is the price of anarchy?
• When are Nash equilibria socially optimal?

16
Unbounded Price of Anarchy (unsplittable case)
gA g
gB 2g
Price of anarchy
Nash flow
Optimal flow
Network Bottleneck
17
Unbounded Price of Anarchy (splittable case)
gA g
S1
qe(fe)2fe for each e in E.
gBg
S2
T2
Price of anarchy
Optimal flow
Nash flow
T1
Network Bottleneck
18
Questions

• Is there at least one Nash Equilibrium?
• Yes!
• Is the Nash equilibrium always unique?
• No!
• How many steps are required to reach equilibrium?
• Unsplittable
• Splittable 8
• What is the price of anarchy?
• 8
• When are Nash equilibria socially optimal?

19
Optimal Nash Equilibria (unsplittable case)

• Theorem The price of stability is 1.
• Good news
• Selfish users can agree upon an optimal solution.
• Such solutions can be proposed to all users by
  some centralized protocol.
• Bad news
• We prove that finding such an optimal Nash
  equilibrium is NP-hard.

20
Optimal Nash Equilibria (splittable case)

• Theorem A Nash flow is optimal if all users
  route their traffic along paths with a minimum
  number of bottlenecks.

gA 1
S1
gB 1
S2
T2
T1
qe(fe)fe for each e in E.
21
Questions

• Is there at least one Nash Equilibrium?
• Yes!
• Is the Nash equilibrium always unique?
• No!
• How many steps are required to reach equilibrium?
• Unsplittable
• Splittable 8
• What is the price of anarchy?
• 8
• When Nash equilibriums are socially optimal?
• Unsplittable each best Nash equilibrium (though
  NP-hard to find).
• Splittable each Nash equilibrium with users that
  exclusively route over paths with a minimum
  number of bottlenecks.

22
Some more results

• Unsplittable link performance functions of
  qe(x)xp
• Price of anarchy is O(Ep).
• This result is tight!
• Splittable Nash equilibrium with users that
  exclusively route over paths with minimum number
  of bottlenecks.
• The average performance (across all links) is E
  times larger than the minimum value.
• This result is tight!

23
Conclusions

• Bottleneck games emerge in many practical
  scenarios.
• (yet, they haven't been considered before).
• A Nash equilibrium in a bottleneck game
• Always exists
• Can be reached in finite time with unsplittable
  flows
• Might be very inefficient.

24
Conclusions (cont.)

• BUT, by proper design, Nash equilibria can be
  optimal!
• Unsplittable any best equilibrium.
• Splittable any equilibrium with users that route
  over paths with minimum number of bottlenecks.
• With these findings, it is possible to optimize
  overall network performance.
• Steer users to choose particular Nash equilibria.
• Unsplittable propose a stable solutions to all
  users.
• Splittable provide incentives (e.g., pricing)
  for minimizing the number of bottlenecks.

25
Questions?
26
Splittable bottleneck games are discontinuous!
qe(fe)fe2
e1
g 1
S
T
e2
qe(fe)fe
Flow configuration
Cost