GAME THEORY

Game theory may be defined as – “ a body of Game theory may be defined as – “ a body of

knowledge that deals with making decisions knowledge that deals with making decisions

when two or more intelligent and rational when two or more intelligent and rational

opponents are involved under conditions of opponents are involved under conditions of

conflict and competition.” conflict and competition.”

Every game must have a character of Every game must have a character of

competition and two or more players involved in competition and two or more players involved in

it with some predetermined rules.it with some predetermined rules.

The game results either in victory of one or the The game results either in victory of one or the

other or sometimes a draw.other or sometimes a draw.

Therefore, game represents a conflict between Therefore, game represents a conflict between

two parties or countries or persons.two parties or countries or persons.

Y pays Rs. 80 to X X pays Rs. 60 to Y

X pays Rs. 60 to Y Y pays Rs. 80 to X

Player Y

Player XH

T

H T

Thus, Thus, A competitive situation = GameA competitive situation = Game

Gain of one player is the loss of other playerGain of one player is the loss of other player

Sum of gains to both the players is bound to be Sum of gains to both the players is bound to be

zerozero

Depicted by Rectangular Pay-off MatrixDepicted by Rectangular Pay-off Matrix

A strategy is a course of action taken by one of A strategy is a course of action taken by one of

the participants in a game and the pay-off is the the participants in a game and the pay-off is the

result or outcome of the strategy.result or outcome of the strategy.

An Example:An Example:

10,10 100, -30

-20, 30 140, 35

Firm 2

Firm 1

No Price Change

Price Increase

No Price Change

Price Increase

Players adopts pessimistic attitude and plays Players adopts pessimistic attitude and plays

safe.safe.

Players decides to play that strategy which Players decides to play that strategy which

corresponds to the maximum of the minimum corresponds to the maximum of the minimum

gains for his different courses of action.gains for his different courses of action.

Similarly, player B wants to play safe.

Then he selects that strategy which corresponds

to the minimum of the maximum losses.

Course of action or strategy which puts the player Course of action or strategy which puts the player

in the most preferred position, irrespective of the in the most preferred position, irrespective of the

strategy of his competitors.strategy of his competitors.

Any deviation from this strategy results in a Any deviation from this strategy results in a

decreased pay-off for the player.decreased pay-off for the player.

Expected pay-off of the play when all the players Expected pay-off of the play when all the players

of the game follow their optimal strategies.of the game follow their optimal strategies.

Fair – if the value of the game is zero.Fair – if the value of the game is zero.

Unfair- if the value of the game is non-zero.Unfair- if the value of the game is non-zero.

(Saddle Point Exists)

Arithmetical Method

Graphical Method

Linear Programmin

g Method

With saddle point…….

It is an Two person Zero –sum game.

It uses pay-off Matrix.

It involves Maximin principle and Minimax

principle.

Its objective is to bring out Optimal strategies

for both players.

To derive Value of the Game.

• The maximizing player arrives at his optimal strategy on the

basis of the maximin criterion, while the minimizing player

strategy is based on minimax value. The game is solved

when the maximin value equals minimax value. And when

they both equalize that particular point is called as saddle

point.

• Develop the payoff- matrix.

• Identify row minimums and select the largest of these as player

one’s maximin strategy.

• Identify column maximums and select the smallest of these as

the opponents minimax strategy.

• If the maximin value equals minimax value, the game is a pure

strategy game and that value is saddle point.

• The value of the game of player one is the maximin value and to

player two , the value is the nagative of minimax value.

According to the principle the size of the game’s pay-off matrix can be reduced by eliminating a course of action that is so inferior to another that it can never be used.

Such a course of action is said to be dominated by others.

A dominant strategy is the one that is optimal no matter what the opponent does.

In general the following rules of dominance are used to reduce the size of the pay-off

If all the elements in the ith row of the pay-off matrix are less than or equal to the corresponding elements of the other row (say the jth row) then the ith strategy is dominated by the jth strategy.

If all the elements in the rth column of the pay-off matrix are greater than or equal to the corresponding elements of the other column (say the sth column) then the rth strategy is dominated by the sth strategy.

Player A

7 6 8 9

-4 -3 9 10

3 0 4 2

10 5 -2 0

B1 B2 B3 B4

A1

A2

A4

Player B

A1 gives more gain than A3 in all conditions (for all strategies of B) i.e. A1 dominates A3.

Thus the effective pay off matrix shall become :

Player B

Player A

7 6 8 9

-4 -3 9 10

10 5 -2 0

B1 B2 B3 B4

A1

A2

A4

10,5 15,0

6,8 10,2

Firm B

Firm A

Advertise Don’t Advertise

Advertise

Don’t Advertise

Payoff matrix for Advertising game

Firm A and B sell competing products and are deciding whether to undertake advertising or not

10,5 15,0

6,8 20,2

Firm B

Firm A

Advertise Don’t Advertise

Advertise

Don’t Advertise

Modified Advertising Game

However, not every game has a dominant strategy for each player.

Following is an example for the same:

Suppose there are two competitors, X and Y, planning to sell soft drinks on a beach. They both sell the same soft drinks at the same price.

The beach is 200 yards long, and the sunbathers are spread evenly across its length.

Where on the beach should they locate?

0 200A

Ocean

Beach

The “beach location game” can help us understand a variety of phenomena.

For e.g. it explains why two or three petrol pumps, or several roadside restaurants, or several car dealers are located close to each other on a two- or three- mile stretch of road.

(Games Without Saddle Point)

Two breakfast food manufacturing firms A & B are competing for an increased market share. To

improve its market share both the firms decide to launch the following strategies :

A1, B1 = Give Coupons

A2, B2 = Decrease Price

A3, B3 = Maintain Present Strategy

A4, B4 = Increase Advertising

The pay-off matrix describes the Increase in market share for firm A & decrease in market hare

for firm B.

Firm B

EXAMINE THE OPTIMAL SRTATEGIES FOR EACH FIRM & THE VALUE OF THE

GAME

Firm A

B1 B2 B3 B4

A1 35 35 25 5

A2 30 20 15 0

A3 40 50 0 10

A4 55 60 10 15

B1 B2 B3 B4

A1 35 35 25 5

A2 30 20 15 0

A3 40 50 0 10

A4 55 60 10 15

Firm B

Firm A

STEPS:

1. Search For Saddle Point. There is no saddle point.2. Observe if pay-off can be reduced in size by rules of dominance.

We note 2nd row is dominated by 1st row because pay-offs are lessattractive for firm A.

Thus deleting 2nd row reduced matrix becomes :

Firm B

Each element of 2nd column is more than the corresponding elements

in 1st column

Therefore

2nd column is dominated by 1st column because pay-offs are less attractive

for B. (Delete 2nd column)

B1 B2 B3 B4

A1 35 35 25 5

A3 40 50 0 10

A4 55 60 10 15

Firm A

Thus deleting 2nd column reduced matrix becomes :

Further comparing row 2 & 3 , then column 1 & 2 , delete less attractive row

column’s from A’s & B’s point of view.

The reduced pay off matrix is as shown :

Firm B

B1 B3 B4

A1 35 25 5

A3 40 0 10

A4 55 10 15

Firm A

Firm B

B3 B4

A1 25 5

A4 10 15

Firm A

Prob. p1p2

Prob. q1 q2

No saddle point, so use mixed strategies.

For firm A :

Let p1 & p2 be prob. of selecting strategy A1 (Give coupons) & A4( IncreaseAdvertising) respectively.

Expected gain should be equal

25p1 +10(1-p1) = 5p1 + 15(1-p1)

We get p1=1/5 & p2 =1-p1 = 4/5

Player A would play first strategy A1 with prob. 1/5 & A2 with prob. 4/5

B’s strategy Expected Pay-off to firm A

B1 25p1 +10(1-p1)

B2 5p1 + 15(1-p1)

For Firm B :

Let q1 & q2 be prob. of selecting strategies B3 ( Maintaining present strategy) &

B4 (Increasing Advertising)

Expected loss to firm B when firm A uses its A1 & A4 strategies :

By Equating

25q1 +5(1-q1) = 10q1 + 15(1-q1)

We get q1 = 2/5 & q2 (1-q1) = 3/5

A’s strategy Expected Pay-off to player B

A1 25q1 +5(1-q1)

A2 10q1 + 15(1-q1)

Optimal strategy for both manufacturers :

Firm A should adopt strategy A1 ( Give Coupons) & strategy A4 ( Increasing

Advertising) 20% time. (p1)

While firm B should adopt strategy B3 (Maintaining present strategy) & strategy

B4 ( Increasing Advertising) 40% time.(q1)

The Value of Game = Expected gain to firm A

(25 X 1/5) + (10 X 4/5) = 13

(5 X 1/5) + (15 X 4/5) = 13

Value of Game = Expected loss to firm B

(25 X 2/5) + (5 X 3/5) = 13

(10 X 2/5) + (15 X 3/5) = 13

Pepsi calculated the market share of two products, Pepsi and Mountain Dew, against its major competitor Coca Cola’s three products, Coca Cola, Fanta and Sprite and tried to find out the effect of additional advertisement in any of its products against the other.

Pepsi/Coca Cola Sprite Fanta Coca Cola

Mountain Dew 15 6 7

Pepsi 10 12 20

Pepsi/Coca Cola

Sprite Fanta Coca Cola Minimum

Mountain Dew 15 6 7 6

Pepsi 10 12 20 10

Maximum 15 12 20

Maximin= 10 & Minimax= 12

i.e. Maximin is not equal to Minimax=> No saddle point.

Pepsi has two products, Pepsi and Mountain Dew, with probability of their getting selected for advertisement equal to P1 and P2, respectively, such that:

P1 + P2= 1

or P2= 1 – P1. & P1, P2 either > or = 0.

For each of the pure strategies available to Coca cola, i.e. its three products (Coca Cola, Fanta and Sprite), expected pay-off of Pepsi can be represented by plotting straight lines.

Coca cola’s Product Pepsi’s pay-off(market share)

Sprite 15p2 + 10p1

Fanta 6p2 + 12p1

Coca Cola 7p2 + 20p1

Pepsi/Coca Cola Sprite Fanta Coca Cola

Mountain Dew 15 6 7

Pepsi 10 12 20

Pepsi/ Coca cola Sprite Fanta Coca Cola

Mountain Dew 15 6 7

Pepsi 10 12 20

•Coca Cola’s strategy is to yield worst result to Pepsi.

•Pay-offs to Pepsi are represented by lower boundary.

•Pepsi’s strategy is to maximize its expected gain, i.e. market share.

•Maximum pay-off is at highest point on this lower boundary.

Thus maximum gain is found at P, at the intersection of two lines, representing the pay-offs corresponding to Sprite and Fanta.

Pepsi/ Coca Cola Sprite Fanta

Mountain Dew 15 6

Pepsi 10 12

Coca cola’s Product Pepsi’s pay-off(market share)

Sprite 15p2 + 10p1

Fanta 6p2 + 12p1

& solution is found at the intersection of the following two lines:

Pay-off corresponding to Sprite = Pay-off corresponding to Fanta

=> 15p2 + 10p1 = 6p2 + 12p1

Since, p1 + p2 = 1Putting p2 = 1 – p1 and solving… Gives p1 = 9/11 or 81.81% p2 = 2/11 or 18.18%

Which means, Pepsi should advertise Mountain Dew 18.18% times andPepsi 81.81% times of total advertisement in order to obtain optimum result irrespective of rival product’s strategy.

Substituting p1 and p2:

We get, Value of the game = 120/11.

The non-zero-sum games refer to a

situation where there exists a jointly

preferred outcome. Existence of a jointly

preferred outcome means that both players

may be able to increase their pay-offs

through some form of an operation or

agreement concerning actions to be chosen.

Cooperative games : Players are assumed to be equal to realize that it

is mutually advantageous to cooperate on any & every one which is

likely to benefit at least one of players without affecting them

adversely.

Non cooperative games: There is no communication between

participants & there is no way to reach enforcement agreements.

Most popular form of non-cooperative game is

‘Prisoners Dilemma’

Not Confess Confess

Not

Confess

No Prison Term

for both

15 years prison term for 1;

Suspended sentence for 2

Confess Suspended sentence

for 1;

15 years prison term

for 2

8 years prison term for both

Suspect 2

Suspect 1