Game Theory: a player's payoff in a zero sum game

Question

I am wondering for a two person zero sum game, is it true that the maximum of a player 1's payoffs is always greater than or equal to zero? Will there be a case that all of the player 1's payoffs are negative and all of the player 2's payoffs are positive, more clearly, will the following payoff matrix for player 1 and player 2 exist? $$\begin{bmatrix}(-1,1)\space(-1,1)\\(-1,1)\space(-1,1) \end{bmatrix}$$

Answer 1

Does the following payoff matrix exist? Well, you wrote it down, didn't you? That looks like a (very boring) zero sum game to me, where one player always loses a buck and the other player always wins a buck.

Answer 2

The definition of zero-sum game is a game which the reward/lost comes only from the players, let's say the reward is money, if all the money of the game comes from the players it is a zero-sum game. So the player which the absolute reward is leaning onto is not important. Because in this situation when creating a matrix of the game the absolute value of the 2 parts in each place if the matrix will be equal than a lot of ppl write only the amount of money player 2 gives player one, and if player one gives money to player two in all possible outcomes it will be looks like this: $$\begin{bmatrix}-1&-1\\-1&-1\end{bmatrix}$$ Ofc the $-1$ can be any value.