Two-Finger Morra is a game in which two players each hold up one or two fingers. The payoff, in dollars, is the total number of fingers shown. $R$ recelues the payoff if the total is even, and $C$ recelves the payoff if the total is odd. Write the payoff matrix. \[ R_{2} \] 2 $R$

Step 1 ：Two-Finger Morra is a game in which two players each hold up one or two fingers. The payoff, in dollars, is the total number of fingers shown. Player R receives the payoff if the total is even, and Player C receives the payoff if the total is odd.

Step 2 ：We need to create a payoff matrix that represents all possible outcomes of the game. The rows of the matrix will represent the actions of player R and the columns will represent the actions of player C. Each cell in the matrix will contain two values: the payoff for player R and the payoff for player C, respectively.

Step 3 ：The possible actions for each player are showing one or two fingers, so we have actions_R = [1, 2] and actions_C = [1, 2].

Step 4 ：By considering all possible combinations of actions, we can fill in the payoff matrix. If both players show one finger, the total number of fingers is 2, which is even, so player R receives a payoff of 2 and player C receives a payoff of 0. If player R shows one finger and player C shows two fingers, the total number of fingers is 3, which is odd, so player R receives a payoff of 0 and player C receives a payoff of 3. The same logic applies to the other cells in the matrix.

Step 5 ：Thus, the payoff matrix for the Two-Finger Morra game is \[ \begin{bmatrix} (2, 0) & (0, 3) \ (0, 3) & (4, 0) \end{bmatrix} \] where the first value in each pair is the payoff for player R and the second value is the payoff for player C.

Step 6 ：\(\boxed{\text{Final Answer: The payoff matrix for the Two-Finger Morra game is } \begin{bmatrix} (2, 0) & (0, 3) \ (0, 3) & (4, 0) \end{bmatrix}}\)