## Unit VI Markov Chains and Theory of Games for Quantitative Techniques For Business Mcom sem 2 Delhi University

Unit VI Markov Chains and Theory of Games for Quantitative Techniques For Business Mcom sem 2 Delhi University – Quantitative techniques may be defined as those techniques which provide the decision makes a systematic and powerful means of analysis, based on quantitative data. It is a scientific method employed for problem solving and decision making by the management. With the help of quantitative techniques, the decision maker is able to explore policies for attaining the predetermined objectives. In short, quantitative techniques are inevitable in decision-making process.

### Unit VI Markov Chains and Theory of Games for Quantitative Techniques For Business Mcom sem 2 Delhi University

**MARKOV CHAINS: ROOTS, THEORY, AND APPLICATIONS**

Unit VI Markov Chains and Theory of Games for Quantitative Techniques For Business Mcom sem 2 Delhi University – The purpose of this paper is to develop an understanding of the theory underlying Markov chains and the applications that they have.

Markov chain analysis has its roots in probability theory, so we begin with a review of probability. The review will be brief and will focus mainly on the areas of probability theory that are pertinent to Markov chain analysis. As with any discipline, it is important to be familiar with the language of probability before looking at its applications. Therefore, we will begin with a few definitions and a few more will be introduced later as necessary. In probability, the sample space, S, is the set of all possible outcomes for an experiment. Any subset, F, of the sample space S is known as an event. For example, Sheldon Ross explains in his text that if the experiment consists of a coin toss, then

S = {(H)eads,(T)ails}

is the sample space [4]. F = {H} is the event that the outcome of the flip is heads and E = {T} would be the event that the outcome of the toss is tails. Alternatively, if the experiment consists of two successive coin flips, then

S = {(H, H),(H, T),(T, H),(T, T)}

is the sample space (where (H, T) denotes that the first coin came up heads and the second coin came up tails). F = {(H, H)} is the event that both flips came up heads, E = {(H, H),(H, T),(T, H)} is the event that heads shows up on at least one of the coins and so on

The union of two events E and F of a sample space S, denoted E ∪ F, is defined as the set of all outcomes that are in either E or F or both. The intersection of E and F, denoted E ∩ F, is defined as the outcomes that are in both E and F. The complement of an event E is the set of all points in the sample space that are not in E. The complement is written E c . So if we reexamine the experiment of flipping two coins, then every outcome in either E or F or both is F ∪ E = {(H, H),(H, T),(T, H)}. The only outcome simultaneously in both E and F is F ∩ E = {(H, H)}. Lastly, the set of outcomes in S that are not in E is E c = {(T, T)}. These ideas can be represented visually by a Venn diagram. In the Venn diagram we can say that the entire area in the box is the sample space. Then the interior of each of the circles represents an event. The part where the two interiors overlap is the intersection. The area that both the circles enclose together in Figure 1 is the union, and the area that is outside one of the circles is the complement to the event represented by that circle

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**Markov Chains Quick revision notes**

### Unit VI Markov Chains and Theory of Games for Quantitative Techniques For Business Mcom sem 2 Delhi University

### Game Theory: Assumptions, Application and Limitations

Unit VI Markov Chains and Theory of Games for Quantitative Techniques For Business Mcom sem 2 Delhi University – John Von Neumann and Oscar Morgenstern are considered to be the originator of game theory. They mentioned it in the book ‘Theory of Games and Economic Behaviour’.

A game is a situation in which two or more participants take part in pursuit of certain conflicting objectives.

In this case, some players may win by getting positive gain while others may lose. In the same way in competitive market, two or more parties make decisions with conflicting interest and action of one depends on the opponent task. Each and every opponent ‘acts in rational way for resolving the conflicting in own favour.

**Unit VI Markov Chains and Theory of Games for Quantitative Techniques For Business Mcom sem 2 Delhi University **

**Assumptions:**

1. The number of players (competitors) in finite.

2. All players act rationally and intelligently.

3. Each player has a definite course of action.

4. There is conflict of interest between the players.

5. The rules of play are known to all the players.

**The zero sums two person games:**

In a zero sum game the gain of one player is the loss of the other so that the total of gains and losses is always equal to zero. For example, when two firms compete in a duopolistic market the gain of one will be the loss of other. The contest involves in attracting the customers, then the number of customers gained by one firm must be same as number of customers lost by the other. This is two person zero sum game.

In the case of duopoly, assume two firms A and B are competing. Firm A has availability of three strategies A_{1}, A_{2 }and A_{3}. Firm B has four strategies B_{1}, B_{2}, B_{3} and B_{4}. If we consider any pair of strategies, e.g. A opt for strategy A_{2} and B opt for B_{3} it results in 7 per cent gain of market share for A and same loss for B.

In this way the payoff matrix can be achieved in the form of table. The strategies and their corresponding payoff matrix are given in the following table:

If firm A selects the strategy A_{x}, then B will reply by selecting minimum value of B_{1}, B_{2}, B_{3} and B_{4} i.e. B3, for minimum payoff to A. Row minima and column maxima is selected. After this max min and mini max principle is followed and sadle point is decided. In brief, the following procedure is adopted.

(a) Find the minimum value in each row.

(b) The max min value is calculated of the minimum value (that is 16).

(c) Find the maximum value of each column.

(d) The mini max value is calculated of the maximum value (that is 16).

(e) Sadie point is that where the mini max and max min coincides (that is 16).

Thus the optimum solution to the game is:

(i) The best strategy for firm A is A1.

(ii) The best strategy for firm B is B3, and

(iii) The value of the game is 16 for firm A and—16 for firm B.

In some games the max min and mini max value does not coincide. Therefore, saddle point is not existing, There are two methods: graphical and linear programming for solving this case.

**Mixed strategies:**

Unit VI Markov Chains and Theory of Games for Quantitative Techniques For Business Mcom sem 2 Delhi University – If the mini max and max min value does not coincides, it is said the case of mixed strategy. Then the problem of game theory is solved by probability theory.

Solve the following game and determine the value of the game:

The payoff matrix does not possess any saddle point. The players will use mixed strategies. In this case of matrix [ab/ cd] the following formulae is used

R= a + d-b-c

R= 8+1+3+3=15

Value of the game will be

V= (ad-bc)/R

= [8 x 1 – (-3 x -3)]/ 15

= (8-9)/ 15 = – 1/15

Probability for A will be

P_{1}= (d-c)/ R or (1+3)/15 or 4/15

Probability for B will be

P_{2}= (a-b)/ R or (8+3)/15 or 11/15

**Applications:**

Unit VI Markov Chains and Theory of Games for Quantitative Techniques For Business Mcom sem 2 Delhi University – The game theory can be applied to decide the best course in conflicting situations. In business decisions it has wider possibilities. With the help of computer large number of independent variables can be considered with mathematical accuracy. The main advantages of this theory are:

1. Game theory provides a systematic quantitative approach for deciding the best strategy in competitive situations.

2. It provides a framework for competitor’s reactions to the firm actions.

3. It is helpful in handling the situation of independence of firms.

4. Game theory is a management device which helps rational decision-making.

**Limitations:**

1. As the number of players increases in the actual business the game theory becomes more difficult.

2. It simply provides a general rule of logic not the winning strategy.

3. There is much uncertainty in actual field of business which cannot be considered in game theory.

4. Businessmen do not have adequate knowledge for the game theory.

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