A Game Theoretic Approach to
Network Coding
Jason R. Marden
California Institute of Technology
Michelle Effros
California Institute of Technology
2009 Information Theory and Applications Workshop
February 12, 2009
A Game Theoretic Approach to Network Coding
Marden and Effros
STATUS QUO
ACHIEVEMENT DESCRIPTION
MAIN ACHIEVEMENT:
IMPACT
Introduced game theory as a distributed tractable
mechanism to obtain good network performance
Global Objective: Efficiently use
network using network coding
Approach: Centralized solutions.
(e.g., opportunistic coding) Fix paths,
use coding opportunities if available
Approach provides guarantees
independent of network
structure.
Guarantees existence of an
equilibrium that achieves a
system cost of at most 50%
higher than the optimal.
This offers an improvement over
opportunistic coding.
NEW INSIGHTS
•  Model interactions as a non-cooperative game
- players (unicast flows)
- actions (available paths)
•  Assign each player a “cost” function
•  Analyze efficiency of equilibrium behavior
What about distributed solutions?
What if flows were allowed to select
path in response to local “cost”?
ASSUMPTIONS AND LIMITATIONS:
•  Limited form of network coding (reverse carpool)
Goal: Let users create coding
opportunities to improve efficiency
•  Players have knowledge of available paths
•  Players equilibrate faster than network changes
NEXT-PHASE GOALS
HOW IT WORKS:
Understand the potential
of game theory in
network coding problems
Establish desirable distributed
learning algorithms with good
convergence rates
Extend game theoretic approach
to more general network coding
problems
Game theory is an applicable tool for distributed optimization in network coding
Example: Network coding
Features:
-
large common network
large # of users
different network demands
Global objective: Allocate users efficiently over network (utilizing network coding)
-
maximize throughput
minimize # of transmissions
Challenges:
-
centralized optimization is not feasible
network coding capacity is unsolved
Problem setup
Multiple unicast flows in shared network environment
s2
s1
t2
t3
t1
s3
possible transmissions highlighted by edges on graph
Cost of allocation = number of transmission
Reverse carpooling
Limited form of network coding: “Reverse carpooling”
opportunity for network coding arises when
two unicasts traverse the same node in opposite directions
x
y
y
x
without network coding - 4 transmissions required
x
x+y
y
with network coding - 3 transmissions required
Setup
Approach: Model interactions as a non-cooperative game
•
Players (unicast flows):
•
Actions (available paths):
{1, 2, ..., n}
(si , ti )
ai ∈ Ai
A = A1 × ... × An
•
System cost:
C(a) =
!
max{#0e (a), #1e (a)}
e
Goal: Design local players’ cost functions so that equilibrium behavior is desirable
Ji (a) = Ji (ai , a−i )
Cost functions:
Equilibrium behavior: Pure Nash equilbrium
∗ ∗
Ji (ai , a−i )
≤
∗
Ji (ai , a−i )
Efficiency
Let
E(G) := {a ∈ A : a is a Nash equilibrium of game G}
aopt ∈ arg min C(a)
a∈A
Price of Anarchy
Price of Stability
C(a)
P OA = sup max
opt )
C(a
a∈E(G)
G
C(a)
P OS = sup min
opt )
C(a
a∈E(G)
G
worst case performance of any NE
worst case performance of best NE
(independent of network structure or demands)
Cost design: Wonderful life design
Decision Makers
Ji (ai , a−i )
Wonderful Life:
design
Ji (a) = C(a) −
Positives
local
NE exists (minimizer C)
Global Behavior
C(a) =
!
max{#0e (a), #1e (a)}
e
0
C(ai , a−i )
Negatives
POA unbounded
POS = 1
s1
t2
t1
s2
Extending wonderful life
Select any global cost
φ:A→R
Ji (a) = φ(a) −
Positives
0
φ(ai , a−i )
Negatives
NE exists (minimizer φ )
local
φ(a) = C(a)
POS = 1
POA unbounded
Can we choose a cost that gives us better equilibrium efficiency?
Cost design: Extended WL
Consider
New design:
φ(a) = (α − 1)C(a) +
Ji (a) = φ(a) −
=
Old design:
i
|ai |
0
φ(ai , a−i )
(>)
αNi (a)
Ji (a) = C(a) −
=
!
+
(≤)
Ni (a)
0
C(ai , a−i )
(>)
Ni (a)
α≥0
Cost design: Extended WL
t3
s2
s1
1
1
1
1
1
1
t2
1
t1
1
1
s3
What is the cost of this assignment for player 1?
Ji (a) = φ(a) −
=
0
φ(ai , a−i )
(>)
αNi (a)
+
(≤)
Ni (a)
= 2α + 3
Main result
Price of Anarchy and Price of Stability in Reverse Carpooling Game
System Cost at NE over Optimal System Cost
10
Ji (a) =
9
(>)
αNi (a)
+
(≤)
Ni (a)
8
7
6
5
PoA
4
Is this good? Yes and No!
POS = 3/2
POA = 2
Price of Anarchy
CanPrice
we of
doStability
better?
3
No!
2
PoS
1
0
0
1
2
3
4
5
6
Cost Coefficient: !
7
8
9
10
Conclusions
Recap:
-
Derived “optimal” cost functions for a simple network coding problem (WL)
The results on efficiency have implications beyond simple setup
Goals:
-
Illustrate potential of game theory as mechanism for distributed optimization
Results stem for recent research connecting “potential games” and
cooperative control problems
J. Marden, G. Arslan, and J. Shamma, “Connections between cooperative control and potential games,” 2008.
Left unsaid:
-
How do players reach equilibrium in a distributed fashion?
Existing literature on learning in games provides some algorithms