GAME THEORY : SOLVING TWO PERSON AND ZERO - SUM GAME PRESENTED BY Ms.K.PARIMALADEVI ASSISTANT PROFESSOR DEPARTMENT OF MATHEMATICS(SF) ERODE ARTS AND SCIENCE COLLEGE. 

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GAME THEORY : SOLVING TWO PERSON AND ZERO - SUM GAME PRESENTED BY Ms.K.PARIMALADEVI ASSISTANT PROFESSOR DEPARTMENT OF  MATHEMATICS(SF) ERODE ARTS AND SCIENCE COLLEGE

Game Theory : Solving Two Person and Zero - Sum Game. 

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Game Theory : Solving Two Person and Zero - Sum Game

Game theory is a type of decision theory in which one’s choice of action is determined after taking into account all possible alternatives available to an opponent playing the same game, rather than just by the possibilities of several outcome results.

PROPERTIES OF A GAME. 1. There are finite numbers of competitors called ‘players’. 

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1. There are finite numbers of competitors called ‘players’

2. Each player has a finite number of possible courses of action called ‘strategies’

3. All the strategies and their effects are known to the players but player does not know which strategy is to be chosen.

4. A game is played when each player chooses one of his strategies. The strategies are assumed to be made simultaneously with an

outcome such that no player knows his opponents strategy until he decides his own strategy.. 

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outcome such that no player knows his opponents strategy until he decides his own strategy.

5. The game is a combination of the strategies and in certain units which determines the gain or loss.

6. The figures shown as the outcomes of strategies in a matrix form are called ‘pay-off matrix’.

7. The player playing the game always tries to choose the best course of action which results in optimal pay off called ‘optimal strategy’.

8. The expected pay off when all the players of the game follow their optimal strategies is known as ‘value of the game’. The main objective of a problem of a game is to find the value of the game.. 

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8. The expected pay off when all the players of the game follow their optimal strategies is known as ‘value of the game’. The main objective of a problem of a game is to find the value of the game.

9. The game is said to be ‘fair’ game if the value of the game is zero otherwise it s known as ‘unfair’.

COMPETITIVE GAME. A competitive situation is called a competitive game if it has the following four properties. 

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A competitive situation is called a competitive game if it has the following four properties

1. There are finite number of competitors such that n ≥ 2. In case n = 2, it is called a two person game and in case n > 2, it is referred as n-person game.

2. Each player has a list of finite number of possible activities.

3. A play is said to occur when each player chooses one of his activities. The choices are

assumed to be made simultaneously i.e. no player knows the choice of the other until he has decided on his own.. 

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assumed to be made simultaneously i.e. no player knows the choice of the other until he has decided on his own.

4. Every combination of activities determines an outcome which results in a gain of payments to each player, provided each player is playing uncompromisingly to get as much as possible. Negative gain implies the loss of same amount.

STRATEGY. The strategy of a player is the predetermined rule by which player decides his course of action from his own list during the game.. 

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The strategy of a player is the predetermined rule by which player decides his course of action from his own list during the game.

Solving Two-Person and Zero-Sum Game. Two-person zero-sum games may be deterministic or probabilistic. The deterministic games will have saddle points and pure strategies exist in such games. In contrast, the probabilistic games will have no saddle points and mixed strategies are taken with the help of probabilities.. 

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Two-person zero-sum games may be deterministic or probabilistic. The deterministic games will have saddle points and pure strategies exist in such games. In contrast, the probabilistic games will have no saddle points and mixed strategies are taken with the help of probabilities.

Definition of saddle point. A saddle point of a matrix is the position of such an element in the payoff matrix, which is minimum in its row and the maximum in its column.. 

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A saddle point of a matrix is the position of such an element in the payoff matrix, which is minimum in its row and the maximum in its column.

Select the minimum element of each row of the payoff matrix and mark them with circles.

Select the maximum element of each column of the payoff matrix and mark them with squares.

If their appears an element in the payoff matrix with a circle and a square together then that position is called saddle point and the element is the value of the game.. 

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If their appears an element in the payoff matrix with a circle and a square together then that position is called saddle point and the element is the value of the game.

To obtain a solution of a game with a saddle point, it is feasible to find out

Example-1 Solve the payoff matrix. Player B. Player A. 

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I II III IV V - 2 0 0 5 3 3 2 1 2 2 - 4 - 3 0 2 6 5 3 4 2 - 6

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ANSWER. Strategy of player A – II. Strategy of player B – III. 

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Example-2 Solve the payoff matrix. Player A. Player B. 

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SOLUTION. Strategy of player A – A2. Strategy of player B – B3. 

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GAME THEORY : SOLVING TWO PERSON AND ZERO - SUM GAME

PRESENTED BY
Ms.K.PARIMALADEVI
ASSISTANT PROFESSOR
DEPARTMENT OF MATHEMATICS(SF)
ERODE ARTS AND SCIENCE COLLEGE
