But then this just gives the win to the first person to open their card, since in that round they had both selected Checkmate. Or, you have an incentive to rush to open your card when you know you've selected Checkmate, as you want to be the first one to open.

Maybe I should've worded differently for clarity, the game doesn't go forever:

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.

In the proposed game above, there is no rounds, just alternating plays, in which you have to select you play before the other player announces their play, then swap and repeat

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.

Yeah, but what stops P1 from DDos'ing and picking checkmate each time?

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...

Because P1 lost on their first turn if P2 wasn’t about to pick checkmate

That assume a rule that wasn't state.

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.

So is war (the card game), but people still play it

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.

But then you won't know if the other player has selected checkmate when you reveal yours.