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Handbook of mathematics for engineers and scienteists part 152. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 19.2. Mathematical Programming 1025 where aij is the payoff positive or negative of player A against player B if player A uses the pure strategy Ai and player B used the pure strategy Bj. Remark. The sum of payoffs of both players is zero for each move. That is why the game is called a zero-sum game. Let ai minj aij be the minimum possible payoff of player A if he uses the pure strategy Ai. If player A acts reasonably he must choose a strategy Ai for which ai is maximal a max ai maxmin aij . 19.2.1.28 The number a is called the lower price of the game. Let 3j ma a be the maximum possible loss of player B if he uses the pure strategy Bj. If player B acts reasonably he must choose a strategy Bj for which 3j is minimal 3 min 3j min max aij . 19.2.1.29 The number 3 is called the upper price of the game. Remark. The principle for constructing the strategies of player A the first player based on the maximization of minimal payoffs is called the maximin principle. The principle for constructing the strategies of player B the second player based on the minimization of maximal losses is called the minimax principle. The lower price of the game is the guaranteed minimal payoff of player A if he follows the maximin principle. The upper price of the game is the guaranteed maximal loss of player B if he follows the minimax principle. Theorem. In a two-person zero-sum game the lower price a and the upper price 3 satisfy the inequality a 3. 19.2.1.30 If a 3 then such a game is called the game with a saddle point and a pair Aiiopt Bj opt of optimal strategies is called a saddle point of the payoff matrix. The entry v aj corresponding to a saddle point Ai opt Bj opt is called the game value. If a game has a saddle point then one says that the game can be solved in pure strategies. Remark. There can be several saddle points but they all have the same value. If the payoff matrix has no saddle points i.e. the strict inequality a 3 holds then the search of a solution of the game leads
