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The examples above show that not every matrix game has an optimal pure strategy. Investigation of a matrix game begins with finding its saddle point in pure strategies. If a matrix game has a saddle point in pure strategies, the finding of this study point ends the game. If a matrix game has no saddle point in pure strategies, we can find upper and lower net price of this game, which indicate that the first player should not hope to win more than the upper value of the game, and can be confident of receiving a prize is not less the lower value of the game. Such recommendations regarding the conduct of players in a matrix game with no saddle point in pure strategies can not satisfy researchers q practitioners. Improved solutions of matrix games to be found in the use of pure strategies of secrecy and the possibility of repeating the game in the form of parties.

For example, running a series of games of chess, checkers, football, and every time the players apply their strategies in such a way that their opponents are not aware of their contents, and in this way in the middle reach certain win by playing the whole series of parties. These winnings on average greater than the lower value of the game and less of the upper value of the game. The higher the average value, the better strategy applies player. Therefore, the idea to use pure strategies randomly with certain probability.

It provides full privacy of their application. Each player can change the likelihood of their pure strategies in such a way as to maximize his average gain, and in this way to obtain the optimal strategy. This idea led to the concept of a mixed strategy.