Game Theory: Operations research

1. Operations Research
MBA-024

2. GAME THEORY
UNIT III

3. • This is a competitive model.
• In LP we have been dealing with a situation
known as Internal Conflict.
• Theory of Games is a technique of dealing with
external conflict.
• There are a number of firms operating in an
industry producing different brands of the
product.
• The objective of each firm is to have the
maximum slice of the market.

4. Game
• A competitive struggle among n players for a
common objective.
• In an ordinary game, 2 teams are pitted
against one another; both employ different
strategies to achieve their objective.
• However, in business setting there may be
more than 2 firms.
• The objective of a “Game” is to ‘win without
cheating.”

5. Important Characteristics of a
Game
1. There are a finite number of participants.
2. Each participant has only a finite number of
strategies available.
3. Each player knows the strategies available to
the adversary, but he does not know which
strategy will the adversary adopt.
4. All possible outcomes are calculable.

6. • A strategy is a decision rule by which a player
chooses his course of action.
• When a player spends all his time or resources
on a particular course of action, he is said to
have employed a pure strategy.
• When a player spends his time or resources on 2
or more courses of action, he is said to have
employed a mixed strategy.

7. • The outcome which is quantified when players
adopt a specific course of action is a payoff.
• The element denoting this payoff is a payoff
element.
• Payoff matrix is the matrix constituted by the
payoff elements.
• The value of the game is the average winning of
a player per play over a series of plays.

8. Two Person Zero Sum Game
• Algebraic sum of the gains and losses of the
players is zero.
• That is, the gain of one player is the loss of the
other.

9. • The game is given in the form of a payoff matrix:
• Rows represent the strategies available to A.
• Columns represent the strategies available to B.
• Positive payoff elements mean gain to A.
• Negative payoff elements mean loss to A.
B
b1 b2 b3 b4
A a1 60 -30 -
150
-
110
a2 70 10 90 50
a3 -30 0 -50 80

10. • The game is played on the assumption that both
players are conservative in nature.
• Objective of A is to ensure for himself a
minimum gain which cannot be reduced by
whatever B does. (irreducible gain)
• Object of B is to ensure that he does not lose
more than a certain amount.

11. • Strategy recommended for A: Maximin.
• Strategy recommended for B: Minimax.
• Maximin: maximum of the row minima.
• Minimax: minimum of the column maxima.
B
b1 b2 b3 b4
A a1 60 -30 - -
a2 70 10 90 50

12. Principle of Dominance
• If all the elements in the ith
row are less than or
equal to the corresponding elements of the jth
row, then the ith
strategy is said to be
dominated by the jth
strategy.
• If all the elements in the rth
column are greater
than or equal to the corresponding elements
of the sth
column, then the rth
strategy is said to
be dominated by the sth
strategy.

13. Odds Method
• Subtract the smaller payoff in each row from
the larger one and the smaller payoff in each
column from the larger one.
• Interchange each of these pairs of subtracted
numbers found in previous step.
• Put each of the interchanged numbers over
the sum of the pair of numbers.
• Simplify the fraction to obtain the required
strategies.

14. Graphical Method
Player B
B1 B2 ... B3 Probabili
ty
Player A1 a11 a12 ... a1n p1
A2 A21 a22 ... A2n P2
Probabilit q1 q2 … qn