Tic Tac Toe
The one everybody already knows the rules to. Best of nothing, settled in a minute.
Four games, played inside the chat you were already having. The invite arrives as an ordinary encrypted message; the match itself is refereed by our server, which means the moves are not private — and that is worth being clear about.
Two phones cannot agree on who won without either trusting each other or trusting something in the middle. Trusting each other means the first modified client wins every match. So our server holds the rules and validates every move — which requires it to read them.
Invites follow the same reachability rules as messages: you can only invite someone you could already message. A game invite is not a way around the PIN.