Game Theory

By Asif Uddin.

• In any business organization the term ‘game’ referred to a situation of conflict and competition in which two or more competitors are involved in decision making. In such situation decision made by one decision maker affects the decision made by one or more of the remaining decision maker.

• Applicable in competitive situations where two or more Individuals , organizations , players try to make decisions under conditions of conflict and competition

• It determines the ‘best course of action’ for a firm in view of the expected countermoves of the rivals

It was developed in 1928 by Von Neumann. In 1944 Von Neumann and Oscar Morgenstern published “Theory of Games and Economic Behavior”

A Competitive situation is called a Competitive Game, the following are the properties

• There are finite number of participants.

• The number of participants are n>=2

• If n=2, the game is two-person game

• If n>2, the game is n-person game

• Each participant has finite number of possible courses of action

• Each participant must know all the courses of action available to others, but must not know which of these will be chosen

Cont.…

• A play of the game is said to occur when each player chooses one of his courses of action.

• The choices are assumed to be made simultaneously, so that no participant knows the choice of other until he has decided his own

• After all participants have chosen a course of action, their respective gains are finite.

• The gain of participants depends upon his own action as well as those of others.

Useful Terminology

• Each participant is called a player

• A play of the game results when each player has chosen a course of action

• After each play of the game, one player pays the other an amount determined by the courses of action chosen

• The decision rule by which a player determines his course of action is called strategy. To reach the decision regarding which strategy to use, neither player needs to know the other’s strategy.

• Optimal Strategy : In this strategy player optimizes his gains or losses, without knowing the competitor’s strategy. The expected outcome per play, when player follow their optimal strategy, is called the value of the game.

Cont.….

▪ Pure strategy: If a player decides to use only one particular course of action during every play, he is said to use a pure strategy. A pure strategy is usually represented by a number with which the course of action is associated

▪ Mixed strategy: Course of action that are to be selected on a particular occasion with some fixed probability are called mixed strategy. Thus, there is a probabilistic situation and objective is to maximize their expected gains and minimize expected losses.

▪ Two person zero sum game : a gain of one player is loss of other player. In such a game the interest of two players are opposed so that the sum of their net gains are zero.

▪ Also called as rectangular games

We mainly focus on two person zero sum game

• Only 2 players participate

• Each player has finite number of strategies

• Each specific strategy results in a payoff

• Total payoff of each player at the end of each play is zero

• Payoff is a table showing the amounts received by the player named at the left hand side .

1 2 3 . . .

n

1 a11 a12 a13 . . .

a1n

2 a21 a22 a23 . . .

a2n

3 a31 a32 a33 . . .

a3n

.

.

.

.

.

.

.

.

m am1 am2 am3 . . .

amn

Player B

Player A

Decision-Making environments

Uncertainty Risk Certainty

Competitive

SituationMixed

Strategy

2 x 2 order payoff matrix

2 x n or m x 2 order payoff matrix

Graphical method

M x n order payoff matrix

LP method

Pure strategy

Minimax and Maxmini Principal (Saddle

point)

PURE STRATEGIES:GAMES WITH SADDLE POINT

The aim of the game is to how the player must select their respective strategies such that the pay off is optimized. This decision making is referred to as the minimax-maximin principle to obtain the best possible selection of a strategy for the players.

Pure strategies are those strategies where minimax equals maximin .i.e.., where saddle point occurs

Saddle point : It is the payoff value that represents both minimax and maximin value of the game.

Two person sum game with saddle point

P Q

L -3 3

M -2 4

N 2 3

Player B

Player A

Minimum of the row

-3

-2

(2) Maximin

Maximum of column (2) 4 minimax

MIXED STRATEGIES:GAME WITHOUT SADDLE POINT

• In certain cases, there is no pure strategy solution for a game, i.e.no saddle point exists . In all such cases, to solve games both the players must determine an optimal mixture of strategies to find a saddle(equilibrium) point .

• The optimal strategy mixture for each player may be determined by assigning to each strategy it’s probability of being chosen. The strategies so determined, are called mixed strategies.

Mix strategy method can be solved by different solution methods such as:

• Algebraic method

• Matrix method

• Linear programming method

• Analytical or calculus method

• Graphical method

Two person zero sum game without saddle point

A B

P 0 -3

Q -1 0

Using arithmetic method

Game value of Army A

= (0)(1/4)+(-1)(3/4)

= -3/4

Game value of army B

= (-1)(3/4)+(0)(1/4)

= -3/4

Army A: (1/4, 3/4)

Army B: (3/4, 1/4)

Army A

Army B

Oddments of A

1/4

3/4

Oddments of B 3/4 1/4

1

3

3 1

Law of dominance: Law of dominance is used to reduce the size of the payoff matrix

1. For player B who is assumed to be a loser if each element in a column , say Cr is greater than or equal to the corresponding element in another column say Cs in the payoff matrix, then the column Cr is said to be dominated by column Cs and therefore, Cr can be deleted from matrix.

2. For player A , who is assumed to be gainer, if each element in a row ,say Rr is less than or equal to the corresponding element in another row say Rs. Then Rr is said to be dominated by Rs and Rr can be deleted from pay-off matrix.

B1 B2 B3 B4

A1 5 -10 9 0

A2 8 7 15 1

A3 3 4 -1 4

Player B

Player A

B3 B4

A3 15 1

A4 -1 4

In case the law of dominance does not work

B1 B2 B3

A1 8 15 1

A2 3 -1 4

▪ Trial and error method

▪ compare(B1+B2)/2 and B3

(8+15)/2 =11.5 > 1

(3-1)/2 = 1 < 4

Compare (B2+B3)/2 and B1

[(15+1)/2 = 8] <= 8

[(-1+4)/2 = 1.5] <= 3

So delete column B1

Limitations

1. Both the players should be from same industry. In other words business of other fields cannot be compared.

2. Only two of them can be taken at a time.

3. The assumption that players have the knowledge about their own pay-offs and the pay-offs of others is not practical.

4. The technique of solving games involving mixed strategies particularly in case of large pay-offs matrix is very complicated.

5. All the competitive problems cannot be analyzed with the help of game theory.