A Short Course on the Application of Game Theory to Wireless Networks

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A Short Course on the Application of Game Theory
to Wireless Networks

• James Neel

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Homework 4 Solution

• Apply Backwards Induction to the centipede game
  shown
• below.

1
1
C
2,4
4,6
3,1
0,2
5,3
S
S
1,0
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Homework 4 Solution
2. Find the strategic form equivalent to (note
this is 3 stages)
Player 1 strategies
2,4
S,(S,S), S,(S,C), S,(C,S),
S,(C,C) C,(S,S), C,(S,C), C,(C,S),
C,(C,C)
Player 2 strategies
(S,S), (S,C), (C,S), (C,C)
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Homework 4 Solution
(C,C)
(C,S)
(S,S)
(S,C)
S,(S,S)
S,(S,C)
S,(C,S)
S,(C,C)
C,(S,S)
C,(S,C)
C,(C,S)
C,(C,C)
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Homework 4 Solution

• 3. Express the traditional Prisoners Dilemma as
  an extensive form game.

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Homework 4 Solution
Formulate a call admission process for a network
with a capacity of 3 and 4 possible subscribers
as an extensive form game.
M 1,2,3,4
O set of subscribers out of network
Cannot join if N 3
N set of subscribers in network
Should simultaneous calls increase N gt 3,
player with highest player number is blocked.
k is the number of stages a subscriber
has remained in the network
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Lecture 6

• Mixed Strategies
• Wireless Networks as Games

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Mixed Strategies Wireless Networks as Games

• Mixed Strategies
• NE Existence in Mixed Strategies
• Wireless Networks as Games
• Power Control
• Adaptive Interference Avoidance
• Network Formation
• Node Migration (Participation for Vivek)

Skipping today
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Wireless Networks and Games
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There are three primary types of wireless
networks.
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Adaptive wireless network considerations.
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Game theory can be used to analyze distributed
power control algorithms.
Solution of games Nash equilibriums yields
information on networks convergence, fairness,
and optimality
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Common Types of Adaptive (and Uncoordinated)
Behavior in Wireless Networks

• Physical Layer Adaptations (MAC Strategies)
• Ad-hoc Network Formation
• Node Mobility (Participation)

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Physical Layer Adaptations
Nodes are free to alter some aspect of their
physical layer behavior (power level or
waveform). Interaction occurs through
interference caused at nodes of interest.
Interference is a function of both power and the
waveforms selected by each node. Simplified
algorithms may consider one adaptation or the
other separately.
Examples Distributed Power Control (like in
reverse link in cellular or in an
ad-hoc)
Adaptive MAC strategies (alter link in response
to perceived
changes in environment channel,
noise,
interference)
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Physical Layer Model
M the set of decision making radios Ei the
set of possible energy levels available to radio
i ei the energy level chosen by i e - the
tuple of chosen energy levels of all radios in
the network ?i the set of signature waveforms
available to radio i ?i the chosen waveform
of i ? - the tuple of chosen waveforms of all
radios in the network Ni noise power at node
i ?ij - the correlation between the signature
waveform sequences of radios i and j. Note that
?ij necessarily equals ?ji.
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Radio Model (2)
vi node of interest for node i
?i,j path loss from i to j
v1
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Network Formation
Basic Idea Wireless nodes change which nodes
they are communicating with. Some cost
associated with forming a link (function of path
loss and waveform), benefit from minimal number
of hops to all other nodes in the network.
Example routing table generation
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Network Formation Model
M 1,2,3,,n the set of decision making
radios ?i,j path loss from i to j li,j lj,i a
link from i to j ci(li,j) cost of a link from i
to j L A particular network formed from links
li,j ui(L) benefit to i from network L
Actions are ultimately a yes-no decision to each
link.
May be single-sided or two-sided decision process.
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Node Mobility
Basic Idea Wireless nodes change which
network they are participating in. This can be
triggered through node mobility or through
changes in network composition. Performance seen
by a node is generally a function of the
specific network the node is participating in,
the number of nodes in each network, and channel
conditions. Examples Co-existing networks,
handoffs
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Node Mobility Model
M 1,2,3,,m set of nodes
N n1,n2,n3,,nn - set of networks
Ni n1,n2,n3,,nn - set of networks available
to node i
?i network chosen by node i
? network choice vector
? ?i Ni network choice space
- channel seen by node i in network ?i
Value that node i receives from network choice
vector given
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Quick Example
Distributed Power Control
22 Mobiles, uncoordinated power control with base
station. Each adjusts its power level to achieve
a target BER. Rationale For many applications
there is break-point where BER gt target yields
unacceptable QoS, while BER lt target provides
little additional benefit. Mobiles would like
to achieve target BER with minimum transmit
power so as to conserve battery life. Power is
updated mobile decides increase or decrease of
its power level would bring it closer to target
BER.
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Model
Single Cell, 22 devices (Player set 1,2,,22)
No sectorization
Reverse Link (No Synchronization)
DS-SS N 63
at base-station (Node 0)
AWGN
Mobile Power Range (-?, 20 dBm (Action Set)
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A target BER scheme is both convergent and fair.
Distributed algorithm
NE given by maximizer of
where
Network Behavior
Value to Nodes
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Comments

• Modeled System is a (Ordinal) Potential Game
• Network which is a (Ordinal) Potential Game
  requires minimal overhead to support convergence
  of adaptive behavior.
• Note weve not commented on efficiency but if
  measured in terms of system capacity, then
  performance is optimal (equals coordinated
  capacity).

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Extra Slides
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Mixed Strategies
The cliffhanger
Jody, whats mixed strategies?
Mixed Extension to Strategic Form Game
A particular probability assignment
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Mixed Strategies Stories

• Objects of Choice
• Stochastic Steady State
• Pure Strategies in an Extended Game
• Uncertainty in Other Players Payoffs

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N - player set
Ai (Finite) Set of (Pure) Strategies (Actions)
Available to Player i
?i A
Ai (Finite) Set of Actions Available to Player
i
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Mixed Strategy Games
Components
N Set of Players Ai Finite Set of
Actions Available to Player i p ui Set of
Individual Objective Functions
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Nash Equilibrium
Mozart-Mahler
a2
b2
1
a1
1,1
0, 0
b1
0, 0
3, 3
0
1
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1
0
1
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Support and Dominance