It's also interesting to analyse a similar game with a bigger deck. Say each player has a number between 0 and 100 instead of an A, K or Q. For a given bet size there's a Nash equilibrium (i.e. neither player can change their strategy to improve their EV) here. For example for a pot sized bet Hero bets the bottom ninth and top two ninths of their hands, and Villain calls with the top four ninths. The more mathematically inclined among you can probably solve this game for a full street of action (i.e. Hero is oop, Villian is in position and can choose to check behind or bet when Hero checks), although it will take an hour or two unless you're cleverer than me.