Two players, $R$ and $C$, each have two cards. $R$ has one black card with the number 6 written on it and one red card with the number 4 . $C$ has a black card with a 5 written on it and a red card with a 3. They each select one of their cards and simultaneously show the cards. If the cards are the same color, $R$ gets, in dollars, the difference of the two numbers shown. If the cards are different colors, $C$ gets, in dollars, the smaller of the two numbers shown, Write the payoff matrix of this game.

Step 1 ：The game involves two players, R and C, each with two cards. R has a black card with the number 6 and a red card with the number 4. C has a black card with the number 5 and a red card with the number 3. They each select one of their cards and simultaneously show the cards. The payoff depends on the color of the cards and the numbers written on them.

Step 2 ：Each player can choose either their black card or their red card. So there are four possible combinations: R chooses black and C chooses black, R chooses black and C chooses red, R chooses red and C chooses black, R chooses red and C chooses red.

Step 3 ：If both players choose black, then R gets the difference of the two numbers shown, which is \(6 - 5 = 1\). If R chooses black and C chooses red, then C gets the smaller of the two numbers shown, which is \(3\), so R loses 3 dollars. If R chooses red and C chooses black, then C gets the smaller of the two numbers shown, which is \(4\), so R loses 4 dollars. If both players choose red, then R gets the difference of the two numbers shown, which is \(4 - 3 = 1\).

Step 4 ：We can represent these payoffs in a matrix, where the rows represent R's choices and the columns represent C's choices. The entries in the matrix are the payoffs to R.

Step 5 ：The payoff matrix of this game is \[\begin{array}{|c|c|}\hline1 & -3 \\hline-4 & 1 \\hline\end{array}\]