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Game Theory Basics:
Two-person, zero-sum game
A game with only two players (player A and player B) is called a ‘two-person, zero-sum game’, if the losses of one player are equivalent to the gains of the other so that the sum of their net gains is zero.
Two-person, zero-sum games are also called rectangular games as these are usually represented by a payoff matrix in a rectangular form.
Number of activities
The activities may be finite or infinite.
The quantitative measure of satisfaction a person gets at the end of each play is called a payoff
Suppose the player A has ‘m’ activities and the player B has ‘n’ activities. Then a payoff matrix can be formed by adopting the following rules
Row designations for each matrix are the activities available to player A
Column designations for each matrix are the activities available to player B
Cell entry Vij is the payment to player A in A’s payoff matrix when A chooses the activity i and B chooses the activity j.
With a zero-sum, two-person game, the cell entry in the player B’s payoff matrix will be negative of the corresponding cell entry Vij in the player A’s payoff matrix so that sum of payoff matrices for player A and player B is ultimately zero.
Value of the game
Value of the game is the maximum guaranteed game to player A (maximizing player) if both the players uses their best strategies. It is generally denoted by ‘V’ and it is unique.
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.
Procedure to find the saddle point
Select the minimum element of each row of the payoff matrix. Write them in a new column besides the matrix and mark them with circles wherever they are in the matrix. From the column of the minimum values, find out the maximum value and mark it with circle. This value is known as “Maximin” value.
Select the maximum element of each column of the payoff matrix. Write them in a new row below the matrix and mark them with squares wherever they are in the matrix. From the row of the maximum values, find out the minimum value and mark it with square. This value is known as “Minimax” value.
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. In other words, if the “Minimax” value and the “Maximin” value are the same, then it is the saddle point.
Solution of games with saddle point
To obtain a solution of a game with a saddle point, it is feasible to find out
Best strategy for player A (i.e. the strategy with “Maximin” value)
Best strategy for player B (i.e. the strategy with “Minimax” value)
The value of the game
The best strategies for player A and B will be those which correspond to the row and column respectively through the saddle point.
* If Maximin value = Minimax value = V, then the game is strictly determinable, otherwise not
* If Maximin value = Minimax value = V = 0, then the game is ‘FAIR’, otherwise it is not fair.
Operations Research (OR)
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