RockPaperScissors is a zero-sum simultaneous game. There are 3 outcomes: {1,-1} (the first player wins and the second loses), {0,0} (a tie) and {-1,1} (the first player loses, the second wins). Then, its a question of forming the matrix game. Consider the second player plays in order, rock, paper and scissors, consider the outcomes if player 1 chooses rock: {{0, 0}, {-1, 1}, {1, -1}}, paper: {{1, -1}, {0, 0}, {-1, 1}} and scissors: {{-1, 1}, {1, -1}, {0, 0}}. So your payoff array is {{{0, 0}, {-1, 1}, {1, -1}}, {{1, -1}, {0, 0}, {-1, 1}}, {{-1, 1}, {1, -1}, {0, 0}}}. Then, I would set up the GameActionLabels at {{"Rock","Paper","Scissors"},{"Rock","Paper","Scissors"}}, and the GamePlayerLabels, if we were to play together for example, as {"Mitchell","Marc"}.
So in total the call would be
MatrixGame[{{{0, 0}, {-1, 1}, {1, -1}}, {{1, -1}, {0, 0}, {-1, 1}}, {{-1, 1}, {1, -1}, {0, 0}}},
GameActionLabels -> {{"Rock", "Paper", "Scissors"}, {"Rock", "Paper", "Scissors"}},
GamePlayerLabels -> {"Mitchell", "Marc"}]
Best,
Marc Vicuna