Game Theory

GAME THEORY



Game theory may be defined as – “ a body

of

knowledge

that

deals

with

making decisions when two or more intelligent and rational opponents are involved under conditions of conflict and competition.”



Every game must have a character of competition and two or more players involved in it with some predetermined rules.



The game results either in victory of one or the other or sometimes a draw.



Therefore, game represents a conflict between two parties or countries or persons.

Player Y H H Player X T

T

Y pays Rs. 80 to X

X pays Rs. 60 to Y

X pays Rs. 60 to Y

Y pays Rs. 80 to X

Thus, A competitive situation = Game



Gain of one player is the loss of other player



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



Depicted by Rectangular Pay-off Matrix



A strategy is a course of action taken by one of the participants in a game and the pay-off is the result or outcome of the strategy.



An Example: Firm 2 No Price Change

Firm 1

No Price Change Price Increase

Price Increase

10,10

100, -30

-20, 30

140, 35



Players adopts pessimistic attitude and plays safe.



Players decides to play that strategy which corresponds to the maximum of the minimum 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 in the most preferred position, irrespective of the strategy of his competitors.



Any deviation from this strategy results in a decreased pay-off for the player.



Expected pay-off of the play when all the players of the game follow their optimal strategies.



Fair – if the value of the game is 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 B B2

B1

B3

B4

7

6

8

9

A2

-4

-3

9

10

A4

3

0

4

2

10

5

-2

0

A1

Player A

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 B2

B1

B3

B4

7

6

8

9

A2

-4

-3

9

10

A4

10

5

-2

0

A1

Player A

Payoff matrix for Advertising game Firm A and B sell competing products and are deciding whether to undertake advertising or not Firm B Advertise

Advertise

Don’t Advertise

10,5

15,0

6,8

10,2

Firm A Don’t Advertise

Modified Advertising Game However, not every game has a dominant strategy for each player. Following is an example for the same: Firm B Advertise

Advertise

Don’t Advertise

10,5

15,0

6,8

20,2

Firm A Don’t Advertise

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? Ocean 0

A

Beach

200

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, A2, A3, A4,

B1 B2 B3 B4

= = = =

Give Coupons Decrease Price Maintain Present Strategy Increase Advertising

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

Firm B

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

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 less attractive for firm A. Firm B

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

Thus deleting 2nd row reduced matrix becomes : Firm B

Firm A

B1

B2

B3

B4

A1

35

35

25

5

A3

40

50

0

10

A4

55

60

10

15

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)

Thus deleting 2nd column reduced matrix becomes : Firm B

Firm A

B1

B3

B4

A1

35

25

5

A3

40

0

10

A4

55

10

15

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 Firm A

Prob.

B3

B4

A1

25

5

A4

10

15

q1

q2

Prob. p1 p2

No saddle point, so use mixed strategies. For firm A : Let p1 & p2 be prob. of selecting strategy A1 (Give coupons) & A4( Increase Advertising) respectively.

B’s strategy

Expected Pay-off to firm A

B1

25p1 +10(1-p1)

B2

5p1 + 15(1-p1)

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

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 :

A’s strategy

Expected Pay-off to player B

A1

25q1 +5(1-q1)

A2

10q1 + 15(1-q1)

By Equating 25q1 +5(1-q1) = 10q1 + 15(1-q1) We get q1 = 2/5 & q2 (1-q1) = 3/5

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

Mountain Dew Pepsi Pepsi/Coca Cola Mountain Dew Pepsi Maximum

Fanta

15 10 Sprite 15 10 15 Maximin= 10 12

Coca Cola

6

7

12 Fanta

Cola

6 12

7

6

20

12 &

Coca

20 Minimum

20 Minimax=

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

10

 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.

Pepsi/Coca Cola Mountain Dew Pepsi

Sprite 15 10

Coca cola’s Product

Fanta 6 12

Coca Cola 7 20

Pepsi’s payoff(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

•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

& solution is found at the intersection of the following two lines: Coca cola’s Product

Pepsi’s payoff(market share)

Sprite

15p2 + 10p1

Fanta

6p2 + 12p1

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

15p2 + 10p1 = 6p2 + 12p1

Since, Putting

p1 + p2 = 1 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 and Pepsi 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’

Suspect 2 Not Confess Not No Prison Term Confess for both Suspect 1

Confess

Confess 15 years prison term for 1; Suspended sentence for 2

Suspended sentence 8 years prison term for both for 1; 15 years prison term for 2