This document provides an overview of game theory, which was developed in 1928 to analyze competitive situations. It describes various types of games, such as zero-sum, non-zero-sum, pure-strategy, and mixed-strategy games. Methods for solving different types of games are presented, including the saddle point method for 2x2 games, dominance method, graphical method, and algebraic method. Limitations of game theory in assuming perfect information and rational behavior are also noted.

1. GAME THEORY
PRESENTED BY:
AKANKSHA SHARMA
AKANSHA
BHARGAWA
ANKITA DHEER
ANUSHKA KAPOOR
PRAJAL
RITURAJ SINGH

2. 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.

3. 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)

4. 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.

5. (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.

6. Example: The pay off matrix of a two person zero sum
game is:-
Solution:

7. (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.

8. 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

9. 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

10. 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

11. Example: consider a game with a pay-off
matrix
B1 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 column
B1 B2 b3 B4 B5
A1 2 -4 6 -3 5
A2 -3 4 -4 1 0

12. • 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

13. • 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

14. 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.

15. 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

16. 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.