What can I do to ensure one of my pins aren't triggered if another pin is triggered?
Consider using a Priority Encoder to encode the player choices. If a player selects "Rock" then the other choices are ignored. If player doesn't select "Rock" but selects "Paper" then they get "Paper" regardless of whether "Scissors" is selected. They only get "Scissors" if they did not choose "Paper" or "Rock".

Truth table for encoding one player's choices looks like this:

Rock    Paper   scissors    encoded as
0       0       0           00 (nothing)
1       x       x           01 (rock)
0       1       x           10 (paper)
0       0       1           11 (scissors)


or fully expanded to show all eight rows:

Rock    Paper   scissors    encoded as
0       0       0           00 (nothing)
1       0       0           01 (rock)
1       0       1           01 (rock)
1       1       0           01 (rock)
1       1       1           01 (rock)
0       1       0           10 (paper)
0       1       1           10 (paper)
0       0       1           11 (scissors)


Then (spoilers!) the game logic would compare these two 2-bit encoded choices to choose the winner:

Player A        Player B        Outcome
00 (nothing)    xx              (pending)
xx              00 (nothing)    (pending)
01 (rock)       01 (rock)       (tie)
01 (rock)       10 (paper)      (B wins: paper covers rock)
01 (rock)       11 (scissors)   (A wins: rock breaks scissors)
10 (paper)      01 (rock)       (A wins: paper covers rock)
10 (paper)      10 (paper)      (tie)
10 (paper)      11 (scissors)   (B wins: scissors cuts paper)
11 (scissors)   01 (rock)       (B wins: rock breaks scissors)
11 (scissors)   10 (paper)      (A wins: scissors cuts paper)
11 (scissors)   11 (scissors)   (tie)


again, X = don't care 0 or 1, so XX is an abbreviation for four rows of the truth table = 00, 01, 10, or 11.


  Further exercise, left for the student: extend this design to play Rock-Paper-Scissors-Spock-Lizard.


I like this question, I may start using this as a job interview question...