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