Game Theory
 Game theory is a branch of mathematics. It was
first devised by John Von Neumann. It provides
tools for predicting what may happen when
stakeholders with conflicting interests interact.
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Slide 1
Introduction
 Before we comment anymore on game theory lets
consider a soccer game.
 A group of players playing against another group to
score goal(s)
 Do all the players on a group have the same strategies?
 The individual strategies of each player converges to attain
the objective of group.(why?)
How is soccer game different from a two player
game like squash?
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Slide 2
 Three basic components:
 Players
 Strategies
 Payoff (preference relationship)
 Let‘s now think of very basic players, strategies and
payoffs in telecommuncation.
 Players (Nodes, Users, Operators, New/Handover calls)
 Strategies (modulation scheme, amount of bandwidth,
transmit power etc)
 Payoff( Revenue, QoS, Call admission, lower BER etc)
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Slide 3
Game Theory
 Does the decision of one player affect the
decision of other?
 Let‘s consider a game, where a man(buyer) with
limited budget is to buy some groceries at a grocery
store, the decision of setting the price dervies the
selection of grocery item, meaning thereby payoff
of one player is dependent on the payoff of other
player…..
Can you think of any such scenario in mobile
services…. Suggest players and their payoffs?
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Slide 4
Analogy of example in Telecom
Provider – 4
Provider – 3
Provider – 2
Provider – 1
User Pool
Service Area
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Both Providers & Users can be modelled as
buyers / Grocery store.
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Slide 5
Consider the figure, if you are to start
your journey at point „s“ and terminate
at „t“ and you are provided with two
routes.
What are the parameters that you evaluate to choose the
one of the two routes?
1. Distance
2. Others using the same route
Cost Function
Think of a similar scenario in Wireless Communication…
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Slide 6
A simple Game theory example
- It is a two player game (row palyer and column player)
- Player-1 chooses the row and player-2 chooses column
- The values in each cell represent utilities of players
- First number in the cell is utility of player-1
- Second number in the cell is utility of player-2
Prisonner‘s Dilemma
C = Cooperator – dont testify
D = Defect – testify
Let‘s now mathematically define a strategic game………….
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Slide 7
Formal Game Definition
 Normal form (strategic) game
 a finite set N of players
 a set strategies Si for each player i  N
ui (s)
 payoff function
for each player i  N
 where s  S   jN S j is the set of strategies chosen
by all players
 s  S is a set of strategies chosen by players

ui : S  
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Slide 8
Common Games
 Zero Sum Games
They are true games of conflict, Any gain of my side
comes at the expense Of my opponents. E.g.
Matching pennies game. Player-1 gets a Euro from
Player-2 if both choose the same strategy or
otherwise loses a Euro
 Battle of Sexes
A couple wants to spend evening together, wife(P-1)
wants to go to Opera and husband(P-2) wants to go
football match
(1,-1)
(-1,1)
(-1,1)
(1,-1)
(2 , 1)
(0,0)
(0,0)
(1,2)
Normal form game is one instance of repeated game played between
large populations of P-1s & P-2s
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Slide 9
Dominated Strategy
Let‘s consider P-2
• Is M better than R?
• yes (R-dominated)
• No (M-dominated)
Knowing this P-1 will ..
M
• B dominated?
What do we observe?
M
R
4,3
5,1
6,2
2,1
8,4
3,6
3,0
9,6
2,8
T
• T dominated?
• M dominated?
L
B
Do we have a unique strategy profile that both players agree to play?
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Slide 10
Nash Equilibrium
 In the last slide we observed that Neither player has a unilateral
incentive to change its strategy (Nash Equilibrium)
 In any strategic game given by
G ( N , Si , ui )
A strategy profile
is a Nash Equilibrim, such that for
s*there
 S exists
every player
iN
ui (s *i , s *i )  ui (s *i , si )
Few Natural Questions:
1. Do Nash Equilibrium always Exist?
2. Are Nash Equilibrium unique?
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Slide 11
Solving the Game (min-max algorithm)
Player 2
Player 1
A
B
C
D
A
4
-10
7
0
7
B
3
2
5
8
8
C
2
0
1
-4
2
D
5
-1
3
-5
5
2
-10
1
-5
 choose minimum entry in each
 choose maximum entry in each
row
column
 choose the maximum among
 choose the minimum among
these
these
 this is maximin value
 this is the minimax value
 if minimax == maximin, then this is the Nash point of game
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Slide 12
Multiple Nash Equilibriums
 In general, game can have multiple saddle points
Player 2
A
Player 1
B
C
D
A
3
B
2
C
2
D
5
2
5
8
8
-10
2
0
2
0
2
-4
2
-1
3
-5
5
2
-10
2
-5
 Same payoff in every Nash strategy
 unique value of the game
 Strategies are interchangeable
 Example: strategies (A, B) and (C, C) are Nash Strategies
then (A, C) and (C, B) are also Nash Strategies
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Slide 13
Cooperative Games
 In cooperative games coalitions are formed among the players
and all the players then strive to increase the payoff of
coalition. Coalition represents an agreement between players
in the set coalition.
 The Coalition value in quantifies the worth of coalition in a
game. A coalition game is defined as ( N , v )
 The most common form of coalition game is characteristic
form, whereby the value of coalition depends on members of
that coalition with no dependence how the players of set
other than coaltion is structured. The characteristic function
of coalition quantifies the gain of S.
 The characteristic function of empty coalition is zero and
satisfy the superadditive property.
v( S )  v(T )  v( S´U ´T )
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Slide 14
Core
 The solution to coalition games is core
 Given a grand coalition N, a payoff verctor x  R
for dividing v(N ) is a group rational if  x  v( N ) . A
vector is individually rational if every player can
obtain a benefit no less than acting alone i.e. xi  v({i})
An imputation is payoff vector satisfying the above
two conditions. Thus core is defined as
iN
i
cTu  {x :  xi v( N ), xi  v( S )}
iN
iS
Go through TU, NTU cooperative games
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Slide 15
Bargaing Problems
 Bargaining problems refer to the negotiation
process (which is modeled using game theory
tools) to resolve the conflict that occurs when
there are more than one course of actions for all
the players in a situation, where players involved
in the games may try to resolve the conflict by
committing themselves voluntarily to a course
of action that is beneficial to all of them.
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Slide 16
Definition & Axioms
 Bargaining problem is modelled as as pair (F; d), where
F represents theset of all feasible utility pairs and d is
the disagreement point. Players will not form coalition
if the utility that they receive is lesser than disagreement
point. The most common solutions that exist for
bargaining solutions include Nash Bargaining solutions,
Kalai-smorodinsky bargaining solutions etc. All such
solutions have to satisfy few axioms namely
i) individual rationality ii) pareto optimality
iii) independence of irrelevent alternative / individual
monotonicity iv) Symmetry.
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Slide 17
Application Of Game Theory
 Application of Bargaining theory to the
problem of resource allocation and call
admission in heterogeneous wireless network in
our contributed works.
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Slide 18
Game Theory
 A form of mathematics which attempts to
predict behavior in any sort of "strategic"
environment
For more on GAME THEORY…
Kindly watch
the Movie solution concepts for
 It develops
proveable
„The Beautiful Mind“….
negotiating
in situation of conflict of interests….

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Slide 19
Bargaining
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Slide 20
Online Bargaining
Bargaining turns out to
be a daily activity in
most areas…….
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Slide 21
Bargaining
Players will gain if they agree on a solution,
otherwise they will go back to their status quo.
Different solutions have been proposed for bargaining
problems e.g. Nash Bargaining solution, Kalai-Smorodinsky
(varient of Nash)
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Slide 22
Bargaining Problem
Bargaining Problem = (S, d)
U2(x)
Convex and Compact
S
d2
d1
U1(x)
S = feasible set
Which value is the correct solution
in Feasibility set………………..
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d = disagreement point
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Slide 23
Axioms of Bargaining Solution
1. Pareto Optimal
A solution is pareto optimal if it is not possible to find another solution
that leads to a strictly superior advantage for all players simultaneously
U2(x)
S
S2
S1
U1(x)
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Slide 24
Axioms of Bargaining Solution
1. Pareto Optimal
2. Affine Transformation
3. Symmetry
4. Indedependence of Irrelevant Alternatives
4. Individual Monotonicity
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Slide 25
Bankruptcy
General Bankruptcy
1.
Our Bankruptcy Equ.
n
b
0  E   ci
0  rpa (q) 
where C = (c1,…,cn)
E = Requested Bandwidth
wa
w
(q)
i 1
n
x
2.
i 1
i
C = Pre-defined offered bandwidth by
Acc. Tech
 Question…………….
E
n
*
a
x

r
 i p (q)
How should
the
resource be allocated………??
(where x =
allocation)
i 1
(where x* = allocation)
Given that
0  E  ci
(rpa (q), B a (q))
(E, C)
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Slide 26
Resource Bankruptcy as Bargaining
To define bargaining problem
associated with bankruptcy
problem, we define a convex and
compact feasibility set for resource
allocation problem.
S (rpa (q), B a (q))  {x a   n | x a  B a (a),  x wa  rpa (q)}
wW
The disagreement point in our problem formuation is
influenced by cooperation
among
a
a different access
S ((rp (qto),one
B operator.
(q)),0)
technologies belonging
Disagreement Point = 0
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Slide 27
Bargaining Problem
0-associated Bargaining problem
[Proof ommitted]
Proportional Distribution Rule

F P r (( rpa ( q ), B a ( q )) 
(q)
w(W W a )

a

(
q
)

r

 p (q)
w(W W
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a
a A
)
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Slide 28
Bandwidth Request
b a (q)
b a (q)
WLAN
b a (q)
Allocation w.r.t Pre-defined offered
bandwidth…??
Any access technology getting
into congestion will not be able
to offer predefined offered
bandwidth….
b a (q)
a
bba (q(q) )
b a (q)
So we define the term „Offered BW“
WiMAX
UMTS
a3
WiMAX
UMTS
a4
a1
Offered BW  Pre-defined Offered BW
a2
WLAN
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Slide 29
Offered Bandwidth
Offered BW = Tuned by congestion factor
UMTS
y
=
lw
C w  lw
,therefore
 w (q) = bw (q)y
WLAN
Offered BW = Scaled by Loadsharing
 w (q)
WiMAX
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where

.
=
cw

Ca
= b
k
(q )(1  y )
kW a
Slide 30
Offered Bandwidth
So the allocation is…….


X wa rp (q)  b   w (q)
where b is proportionate factor
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Slide 31
Putting things together…


X wa rp (q)  b   w (q)
B (q)
=
Wa
if
a
b
WiMAX
UMTS
rp (q )
a3
0
a4
rp (q )
otherwise
Ca
a1
a2
WLAN
 w (q)
=
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bw (q)y
if
w W a
cw

Ca
if
w W
a
c wa
if
Wa
0
lw
Cw lw
y
=e
b (q)(1  y )
= 
k
kW a
.
Slide 32
CAC, Mobility Algorithms


X wa rp (q) =
X
ao ac
r (q) =
r p (q )    w (q)
r p (q )
w(W W a )
0
r p (q )
p
Otherwise
If
rpac (q) 

w  (W  W
0
a
)
Otherwise

cwao (t  s)  cwao (t )  X ao rp (q)
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 (q )
.

Slide 33
Simulation Scenario
- Area a1 is considered here.
-Calls for different applications
generated using poisson
distribution with mean 7
-Call holding time: infinite
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Slide 34
Mobility simulation
-Simulated for Mobility between areas
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Slide 35
Comparison
-Our approach compared
against the different approaches
e.g. Best Fit, Worst Fit etc.
[comparison paper –
D. Mariz, I. Cananea, D. Sadok,
and G. Fodor, “Simulative analysis
of access selection algorithms for
multi-access networks,”]
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Slide 36
Thanks
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Slide 37