Both players choose a card. Players then in turns reveal their card, and if Check, make another choice. The player first revealing Checkmate wins if their opponent's currently-chosen card is also a Checkmate.
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...
You're assume if someone picks 'checkmate' and the next player picks 'check' the games is over and the checkmate selector loses. I assumed that it means you treat it like 'check' 'check' and continue playing. But neither is actually specified in OPs post.
But let's assume it's your rules. Then winning is easy, just never pick checkmate. Literally never. As soon as your opponent picks it, they lose.
It's a terribly designed game as described.
I think the proposed game has that both of you lose, like tic tac toe. The only way to win is to checkmate as described. Although it is a memoryless game as proposed, so all options (restart, continue, end) are indistinguishable. Maybe if you win, you go again?
Anyways, the game seems to be described to be the equivalent to the political doctrine of mutually assured destruction. Also a terribly designed game.
The player first revealing Checkmate ends the game. They win if their opponent's currently-chosen card is also a Checkmate, otherwise the opponent wins.