Earlier quoted context omitted.
So both players select their cards, then player 1 announces, then player 2, then select, then player 2 announces, then player 1? This seems a bit limiting, as you can't really select Checkmate on the play where you don't reveal first, because you only stand to lose.
I believe the intended turn order is: 1: P1 selects 2: P2 selects 3: P1 reveals 4: P1 selects 5: P2 reveals 6: GOTO 2 I.e. each player always selects immediately before their opponent reveals.
If P2 picks check the first time, then they're done. At any point after if they pick checkmate, since P1 has checkmate selected they will reveal it and P2 will lose.
It seems like a poorly thought through game...