Game theory is used to analyze competitive situations where multiple decision makers try to maximize their own payoffs. It was developed by Von Neumann in 1928 to determine the best course of action for firms considering their rivals' responses. A competitive situation with multiple players choosing strategies simultaneously is called a game. Game theory focuses on two-person zero-sum games where one player's gain equals the other's loss. These games can be solved using pure or mixed strategies to find a saddle point equilibrium. Techniques like dominance rules and algebraic methods are used to solve games of varying complexity.

1. GameTheory
By
Asif Uddin.

2. • In any business organization the term ‘game’ referred to a situation of
conflict and competition in which two or more competitors are
involved in decision making. In such situation decision made by one
decision maker affects the decision made by one or more of the
remaining decision maker.
• Applicable in competitive situations where two or more Individuals ,
organizations,playerstrytomakedecisionsunderconditionsofconflictand
competition
• It determines the ‘best course of action’ for a firm in view of the expected
countermovesoftherivals

3. It was developed in 1928 by Von Neumann. In 1944 Von Neumann and
Oscar Morgenstern published “Theory of Games and Economic
Behavior”
A Competitive situation is called a Competitive Game, the following are
the properties
• There are finite number of participants.
• The number of participants are n>=2
• If n=2, the game is two-person game
• If n>2, the game is n-person game
• Each participant has finite number of possible courses of action
• Each participant must know all the courses of action available to
others, but must not know which of these will be chosen

4. Cont.…
• A play of the game is said to occur when each player chooses one of
his courses of action.
• The choices are assumed to be made simultaneously, so that no
participant knows the choice of other until he has decided his own
• After all participants have chosen a course of action, their respective
gains are finite.
• The gain of participants depends upon his own action as well as those
of others.

5. UsefulTerminology
• Each participant is called a player
• A play of the game results when each player has chosen a course of
action
• After each play of the game, one player pays the other an amount
determined by the courses of action chosen
• The decision rule by which a player determines his course of action is
called strategy.To reach the decision regarding which strategy to use,
neither player needs to know the other’s strategy.
• Optimal Strategy : In this strategy player optimizes his gains or losses,
without knowing the competitor’s strategy.The expected outcome
per play, when player follow their optimal strategy, is called the value
of the game.

6. Cont.….
▪ Pure strategy: If a player decides to use only one particular course of action
during every play, he is said to use a pure strategy. A pure strategy is usually
represented by a number with which the course of action is associated
▪ Mixed strategy: Course of action that are to be selected on a particular
occasion with some fixed probability are called mixed strategy. Thus, there
is a probabilistic situation and objective is to maximize their expected gains
and minimize expected losses.
▪ Two person zero sum game : a gain of one player is loss of other player. In
such a game the interest of two players are opposed so that the sum of
their net gains are zero.
▪ Also called as rectangular games

7. We mainly focus on two person zero
sum game
• Only 2 players participate
• Each player has finite number of
strategies
• Each specific strategy results in a
payoff
• Total payoff of each player at the
end of each play is zero
• Payoff is a table showing the
amounts received by the player
named at the left hand side .
1 2 3 . . . n
1 a11 a12 a13 . . . a1n
2 a21 a22 a23 . . . a2n
3 a31 a32 a33 . . . a3n
.
.
.
.
.
.
.
.
m am1 am2 am3 . . . amn
Player B
Player A

8. Decision-
Making
environments
Uncertainty Risk Certainty
Non-
Competitive
Situation
Competitive
Situation
Mixed
Strategy
2 x 2 order
payoff matrix
Algebraic method
Analytical method
Matrix method
Short-cut method
2 x n or m x 2
order payoff
matrix
Graphical
method
M x n order
payoff matrix
LP method
Pure strategy
Minimax and
Maxmini
Principal
(Saddle point)

9. PURE STRATEGIES:GAMESWITH SADDLE POINT
 The aim of the game is to how the player must select their respective
strategies such that the pay off is optimized.This decision making is
referred to as the minimax-maximin principle to obtain the best
possible selection of a strategy for the players.
Pure strategies are those strategies where minimax equals maximin
.i.e.., where saddle point occurs
 Saddle point : It is the payoff value that represents both minimax
and maximin value of the game.

10. Two person sum game with saddle point
P Q
L -3 3
M -2 4
N 2 3
Player B
Player A
Minimum of the row
-3
-2
(2) Maximin
Maximum of column (2) 4
minimax

11. MIXED STRATEGIES:GAME WITHOUT SADDLE POINT
• In certain cases, there is no pure strategy solution for a game, i.e.no saddle
point exists . In all such cases, to solve games both the players must
determine an optimal mixture of strategies to find a saddle(equilibrium)
point .
• The optimal strategy mixture for each player may be determined by
assigning to each strategy it’s probability of being chosen. The strategies so
determined, are called mixed strategies.
Mix strategy method can be solved by different solution methods such as:
• Algebraic method
• Matrix method
• Linear programming method
• Analytical or calculus method
• Graphical method

12. Two person zero sum game without saddle
point
A B
P 0 -3
Q -1 0
Using arithmetic method
 Game value ofArmy A
= (0)(1/4)+(-1)(3/4)
= -3/4
 Game value of army B
= (-1)(3/4)+(0)(1/4)
= -3/4
 Army A: (1/4, 3/4)
 Army B: (3/4, 1/4)
Army A
Army B
Oddments ofA
1/4
3/4
Oddments of B 3/4 1/4
1
3
3 1

13. Law of dominance: Law of dominance is used to reduce the size of the
payoff matrix
1. For player B who is assumed to be a
loser if each element in a column ,
say Cr is greater than or equal to the
corresponding element in another
column say Cs in the payoff matrix,
then the column Cr is said to be
dominated by column Cs and
therefore, Cr can be deleted from
matrix.
2. For player A , who is assumed to be
gainer, if each element in a row ,say
Rr is less than or equal to the
corresponding element in another
row say Rs.Then Rr is said to be
dominated by Rs and Rr can be
deleted from pay-off matrix.
B1 B2 B3 B4
A1 5 -10 9 0
A2 8 7 15 1
A3 3 4 -1 4
Player B
PlayerA
B3 B4
A3 15 1
A4 -1 4

14. In case the law of dominance does not work
B1 B2 B3
A1 8 15 1
A2 3 -1 4
▪ Trial and error method
▪ compare(B1+B2)/2 and B3
(8+15)/2 =11.5 > 1
(3-1)/2 = 1 < 4
Compare (B2+B3)/2 and B1
[(15+1)/2 = 8] <= 8
[(-1+4)/2 = 1.5] <= 3
So delete column B1

15. Limitations
1. Both the players should be from same industry. In other words
business of other fields cannot be compared.
2. Only two of them can be taken at a time.
3. The assumption that players have the knowledge about their own
pay-offs and the pay-offs of others is not practical.
4. The technique of solving games involving mixed strategies
particularly in case of large pay-offs matrix is very complicated.
5. All the competitive problems cannot be analyzed with the help of
game theory.