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.