PURE STRATEGIES: GAME WITH SADDLE POINT - Quantitative Techniques for management

In a zero-sum game, the pure strategies of two players constitute a saddle point if the corresponding entry of the payoff matrixis simultaneously a maximum of row minima and a minimum of column maxima. This decision-making is referred to as the minimax-maximin principle to obtain the best possible selection of a strategy for the players.

In a pay-off matrix, the minimum value in each row represents the minimum gain for player A. Player A will select the strategy that gives him the maximum gain among the row minimum values. The selection of strategy by player A is based on maximin principle. Similarly, the same pay-off is a loss for player B. The maximum value in each column represents the maximum loss for Player B. Player B will select the strategy that gives him the minimum loss among the column maximum values.

The selection of strategy by player B is based on minimax principle. If the maximin value is equal to minimax value, the game has a saddle point (i.e., equilibrium point). Thus the strategy selected by player A and player B are optimal.

Game Theory Saddle Point Problem

The game theory for saddle point problem is shown below

Example : Consider the example to solve the game whose pay-off matrix is given in the following table as follows:

Game Problem

Game Problem

The game is worked out using minimax procedure. Find the smallest value in each row and select the largest value of these values. Next, find the largest value in each column and select the smallest of these numbers. The procedure is shown in the following table.

Minimax Procedure

Minimax Procedure

If Maximum value in row is equal to the minimum value in column, then saddle point exists.
Max Min = Min Max
1 = 1

Therefore, there is a saddle point.
The strategies are,
Player A plays Strategy A1, (A A1).
Player B plays Strategy B1, (B B1).
Value of game = 1.

Example : Solve the game with the pay-off matrix for player A as given in table.

Game Problem

Game Problem

Solution: Find the smallest element in rows and largest elements in columns as shown in table.

Minimax Procedure

Minimax Procedure

Select the largest element in row and smallest element in column. Check for the minimax criterion,
Max Min = Min Max
1 = 1
Therefore, there is a saddle point and it is a pure strategy.
Optimum Strategy:
Player A A2 Strategy
Player B B1 Strategy
The value of the game is 1.

Example : Check whether the following game is given in Table, determinable and fair.

Game Problem

Game Problem

Solution: The game is solved using maximin criteria as shown in Table.

Maximin Procedure

Maximin Procedure

The game is strictly neither determinable nor fair.

Example : Identify the optimal strategies for player A and player B for the game, given below in Table. Also find if the game is strictly determinable and fair.

Game Problem

Game Problem

The game is strictly determinable and fair. The saddle point exists and the game has a pure strategy. The optimal strategies are given in the following table.

Optimal Strategies

Optimal Strategies

Example : Solve the game with the pay off matrix given in table and determine the best strategies for the companies A and B and find the value of the game for them.

Game Problem

Game Problem

Solution: The matrix is solved using maximin criteria, as shown in table below.

Maximin Procedure

maximin criteria

Therefore, there is a saddle point.
Optimum strategy for company A is A1 and
Optimum strategy for company B is B1 or B3.


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