Title: Game Theory In Telecommunications
1Game Theory In Telecommunications
- Manzoor Ahmed Khan
- (manzoor-ahmed.khan_at_dai-labor.de)
2Game 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.
3Introduction
- 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?
4- 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)
5Game 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?
6Analogy 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.
7Consider 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?
- Distance
- Others using the same route
Cost Function
Think of a similar scenario in Wireless
Communication
8A 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.
9Formal 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 -
10Common 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
(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
11Dominated 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?
12Nash 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
- Do Nash Equilibrium always Exist?
- Are Nash Equilibrium unique?
13Solving 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
14Multiple 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
15Cooperative 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.
16Core
- 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
17Bargaing 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.
18Definition 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.
19Application Of Game Theory
- Application of Bargaining theory to the problem
of resource allocation and call admission in
heterogeneous wireless network in our contributed
works.
20Game 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.
21 Bargaining
22Online Bargaining
23Bargaining
- 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)
24Bargaining Problem
Bargaining Problem (S, d)
S feasible set
d disagreement point
25Axioms 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
26Axioms of Bargaining Solution
1. Pareto Optimal
2. Affine Transformation
3. Symmetry
4. Indedependence of Irrelevant Alternatives
4. Individual Monotonicity
27Bankruptcy
where C (c1,,cn)
Question. How should the resource be
allocated??
28Resource 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
29Bargaining Problem
Proof ommitted
30Bandwidth 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
31Offered Bandwidth
Offered BW Tuned by congestion factor
,therefore
l
w
l
C
w
w
WiMAX
32Offered Bandwidth
So the allocation is.
33Putting things together
WiMAX
UMTS
a3
a1
a4
a2
WLAN
34CAC, Mobility Algorithms
0
Otherwise
If
0
Otherwise
35Simulation Scenario
- - Area a1 is considered here.
- Calls for different applications generated using
poisson distribution with mean 7 - Call holding time infinite
36Mobility simulation
- Simulated for Mobility between areas
37Comparison
-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,
38