An n * n matrix is called a Latin square if each row and each column contains the integers 1, 2, ... , n. The follow-ing is an example of a 3 * 3 Latin square. 123231312 Show that any strategy is the minimax strategy for eitherplayer in a game whose payoff matrix is an n * n Latinsquare. What is the value of the game?
An n * n matrix is called a Latin square if each row and
each column contains the integers 1, 2, ... , n. The follow-
ing is an example of a 3 * 3 Latin square.
123
231
312
Show that any strategy is the minimax strategy for either
player in a game whose payoff matrix is an n * n Latin
square. What is the value of the game?
To prove that any strategy is the minimax strategy for either player in a game whose payoff matrix is an n * n Latin square, we need to show that the game is a zero-sum game and that the value of the game is 0.
First, let's consider the fact that the payoff matrix of a Latin square is such that each row and each column contains the integers 1, 2, ..., n exactly once. This means that if player A chooses any row, and player B chooses any column, they will end up with a unique intersection of a row and a column, and the payoff of the game will be the number at that intersection. Since the Latin square is symmetric, this holds true if the players switch roles (player A chooses any column and player B chooses any row).
Now, let's assume that player A chooses a row and player B chooses a column. Without loss of generality, let the number at the intersection of the chosen row and column be x. This means that player A will receive a payoff of x and player B will receive a payoff of -x (since the game is zero-sum). Now, consider any other row chosen by player A and any other column chosen by player B. If the intersection of these choices is y, then player A will receive a payoff of y and player B will receive a payoff of -y. Since each row and column contains the integers 1, 2, ..., n exactly once, it follows that x and y are different numbers. Therefore, player A cannot improve their payoff by choosing a different row, and player B cannot improve their payoff by choosing a different column. This means that any strategy is the minimax strategy for either player.
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