Game theory in economics for managers

1. GAME THEORY

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
A2
42 72 32
40 30 25
12
10 => A1
B3
32
B4
12
A3 30 8 -10 0 A2 0 15
A4 45 10 0 15

9. B3 B4 1 11 111
A1 32 12 20 15
15/35
A2 0 15 15 20
20/35
1
11
32 3
3 32
Applying odoment
method:
111 3/35 32/35
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 colum
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 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
0 0
-1 -1
-2 -2
-3 -3
-4 -4
-5 -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
From equation (1) and (2) we get :
p= 6/7 & (1-p)= 1/7
Strategy ofA 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

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.