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GAME THEORY
PRESENTED BY:
AKANKSHA SHARMA
AKANSHA
BHARGAWA
ANKITA DHEER
ANUSHKA KAPOOR
PRAJAL
RITURAJ SINGH
Game theory
• Developed by Prof. John Von Neumann
and Oscar Morgenstern in 1928 game
theory is a body of knowledge that deals
with making decisions.
• The approach of game theory is to seek,
to determine a rival’s most profitable
counter-strategy to one’s own best moves.
Competitive situations (Games Theory)
Pure Strategy (Saddle Point
exist)Mixed Strategy
2*2 Strategies Game
(Arithmetic Method)
2*n or 2*m strategies game
(Graphical Method)
M*n strategies (Linear
Programming Method)
Classification
• Two-Person Game – A game with 2 number of players.
• Zero-Sum Game – A game in which sum of amounts won by all winners is equal to sum of amounts lost by all losers.
• Non-Zero Sum Game – A game in which the sum of gains and losses are not equal.
• Pure-Strategy Game – A game in which the best strategy for each player is to play one strategy throughout the game.
• Mixed-Strategy Game – A game in which each player employs different strategies at different times in the game.
(1) Saddle point method:
• At the right of each row, write the row minimum and underline the largest of them.
• At the bottom of each column, write the column maximum and underline the smallest of them.
• If these two elements are equal, the corresponding cell is the saddle point and the value is value of the game.
Example: The pay off matrix of a two person zero sum
game is:-
Solution:
(2) Dominance method
It states that if the strategy of a player dominates over the
other strategy in all condition, the later strategy can be
ignored.
• Rule 1: If all the elements in a row of a pay-off matrix are “<” or “=” to the corresponding elements of other row then comparative row will be deleted
• Rule 2: If all elements in a column in a pay-off matrix are “>” or “=” to the corresponding elements of other column then comparative column will be deleted.
Example: consider a game with a pay-off
matrix:b1 b2 b3 b4
A1 42 72 32 12
A2 40 30 25 10
A3 30 8 -10 0
A4 45 10 0 15
Solution :b1 b2 b3 b4
A1 42 72 32 12
A2 40 30 25 10
A3 30 8 -10 0
A4 45 10 0 15
=>
B3 B4
A1 32 12
A2 0 15
B3 B4 1 11 111
A1 32 12 20 15
15/35
A2 0 15 15 20
20/35
1 32 3
11 3 32
111 3/35 32/35
Applying odoment
method:
Value of game= 32x15 + 0x20 = 96/7
35
Graphical method
• It is helpful in finding out which of the two
strategies can be used.
point area
Case (a): mx2 mini max
Case(b): 2xn maxi min
Example: consider a game with a pay-off
matrixB1 B2 b3 B4 B5
A1 2 -4 6 -3 5
A2 -3 4 -4 1 0
Solution: By applying dominance rule we can cut of the following columns:
B1 B2 b3 B4 B5
A1 2 -4 6 -3 5
A2 -3 4 -4 1 0
• So we have: B1 B4
A1 2 -3
A2 -3 1
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
• Applying oddoment method:
B1 B2
A1 2 -3
A2 -3 1
1 5 4
11 4 5
111 4/9 5/9
1 11 111
5 4 4/9
4 5 5/9
Value of game = 2x(4/9) – 3x(5/9)
= 7/9
Algebraic method:
• This method is used for 2*2 games which
do not have any Saddle Point. As it does
not have any saddle point so mixed
strategy has to be used.
• Players selects each of the available
strategies for certain proportion of time
i.e., each player selects a strategy with
some probability.
Example: consider a game with a pay-off matrix
B1 B2
A1 1 3
A2 7 -5
Let, p= probability that A uses strategy A1,
q= probability that B uses strategy B1
So, 1-p= probability that A uses strategy A2,
1-q= probability that B uses strategy B2
V=px1+ (1-p)x7------------------------(1)
V=px3+ (1-p)x(-5)---------------------(2)
V=qx1+(1-q)x3-------------------------(3)
V=qx7+(1-q)x(-5)----------------------(4)
Solution:
From equation (1) and (2) we get :
p= 6/7 & (1-p)= 1/7
Strategy of A is 6/7
1/7
From equation (3) and (4) we get :
q= 4/7 & (1-q)= 3/7
Strategy of B is 4/7
3/7
Value of game :
V= 6/7x1 + 1/7x7= 13/7
Limitations of game theory:
• The assumptions that each player has the knowledge about his own pay-offs and pay-off’s of the opponent is not practical
• The method of solution becomes complex with the increase in no. of players
• In the game theory it is assumed that both the players are equally wise and they behave in a rational way ,this assumption is also not possible.