Game Theory In Telecommunications

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Game Theory In Telecommunications

• Manzoor Ahmed Khan
• (manzoor-ahmed.khan_at_dai-labor.de)

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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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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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• Three basic components
• Players
• Strategies
• Payoff (preference relationship)
• Lets 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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Game Theory

• Does the decision of one player affect the
  decision of other?
• Lets 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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Analogy of example in Telecom
Provider 4
Service Area
Provider 3
Provider 2
Provider 1
User Pool
Both Providers Users can be modelled as buyers
/ Grocery store.
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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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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

Prisonners Dilemma C Cooperator dont
testify D Defect testify
Lets now mathematically define a strategic
game.
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Formal Game Definition

• Normal form (strategic) game
• a finite set N of players
• a set strategies for each player
• payoff function for each player
• where is the set
  of strategies chosen by all players
• is a set of strategies chosen by
  players

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Common Games

• Zero Sum Games

(1,-1) (-1,1)
(-1,1) (1,-1)
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

(2 , 1) (0,0)
(0,0) (1,2)
A couple wants to spend evening together,
wife(P-1) wants to go to Opera and husband(P-2)
wants to go football match
Normal form game is one instance of repeated game
played between large populations of P-1s P-2s
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Dominated Strategy

• Lets consider P-2
• Is M better than R?
• yes (R-dominated)
• No (M-dominated)
• Knowing this P-1 will ..
• T dominated?
• M dominated?
• B dominated?

L M
R
4,3 5,1 6,2
2,1 8,4 3,6
3,0 9,6 2,8
T M B
What do we observe? Do we have a unique strategy
profile that both players agree to play?
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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
• A strategy profile is a Nash
  Equilibrim, such that for every player
  there exists

Few Natural Questions

1. Do Nash Equilibrium always Exist?
2. Are Nash Equilibrium unique?

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Solving the Game (min-max algorithm)
Player 2
A B C D
A 4 3 2 5
B -10 2 0 -1
C 7 5 1 3
D 0 8 -4 -5
2
-10
1
-5
Player 1
7 8 2 5

• choose maximum entry in each column
• choose the minimum among these
• this is the minimax value

• choose minimum entry in each row
• choose the maximum among these
• this is maximin value

• if minimax maximin, then this is the Nash
  point of game

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Multiple Nash Equilibriums

• In general, game can have multiple saddle points

Player 2
A B C D
A 3 2 2 5
B 2 -10 0 -1
C 5 2 2 3
D 8 0 -4 -5
2
-10
2
-5
Player 1
8 2 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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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
• 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.

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Core

• The solution to coalition games is core
• Given a grand coalition N, a payoff verctor
  
• for dividing is a group rational if
  . A vector is individually rational if
  every player can obtain a benefit no less than
  acting alone i.e.
• An imputation is payoff vector satisfying the
  above two conditions. Thus core is defined as

Go through TU, NTU cooperative games
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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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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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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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Game Theory

• A form of mathematics which attempts to predict
  behavior in any sort of "strategic" environment
• It develops proveable solution concepts for
  negotiating in situation of conflict of
  interests.

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Bargaining
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Online Bargaining
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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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Bargaining Problem
Bargaining Problem (S, d)
S feasible set
d disagreement point
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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
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Axioms of Bargaining Solution
1. Pareto Optimal
2. Affine Transformation
3. Symmetry
4. Indedependence of Irrelevant Alternatives
4. Individual Monotonicity
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Bankruptcy
where C (c1,,cn)
Question. How should the resource be
allocated??
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Resource Bankruptcy as Bargaining

• To define bargaining problem associated with
  bankruptcy problem, we define a convex and
  compact feasibility set for resource allocation
  problem.

The disagreement point in our problem formuation
is influenced by cooperation among different
access technologies belonging to one operator.
Disagreement Point 0
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Bargaining Problem
Proof ommitted
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Bandwidth Request
Allocation w.r.t Pre-defined offered bandwidth??
Any access technology getting into congestion
will not be able to offer predefined offered
bandwidth.?
So we define the term Offered BW
WLAN
WiMAX
UMTS
Offered BW Pre-defined Offered BW
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Offered Bandwidth
Offered BW Tuned by congestion factor
,therefore

l
w

l
C
w
w
WiMAX
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Offered Bandwidth
So the allocation is.
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Putting things together
WiMAX
UMTS
a3
a1
a4

a2
WLAN
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CAC, Mobility Algorithms

0
Otherwise
If

0
Otherwise
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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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Mobility simulation

• Simulated for Mobility between areas

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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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• Thanks