Even if you don't want the games to be able to refer to the last game you played, it looks like this is greatly convoluted. Why not just consider something simpler like the following?
Game A: If you have an even number of chips, gain one. Otherwise, lose two.
Game B: If you have an odd number or chips, gain one. Otherwise, lose two.
The point being that game A (or B) always leaves you with an odd (or even) number of chips, so that if you keep playing it you lose two every turn, but if you alternate, you win one every turn (after the first, possibly). For simplicity, I'm ignoring the behavior near zero chips, but this still seems to capture the essential properties of the "paradox" in a much simpler fashion.
Exactly - most of the effort was making a game that hid that simple mechanic so that it took some (admittedly interesting) math to figure out where the dependence was.
Game A: If you have an even number of chips, gain one. Otherwise, lose two.
Game B: If you have an odd number or chips, gain one. Otherwise, lose two.
The point being that game A (or B) always leaves you with an odd (or even) number of chips, so that if you keep playing it you lose two every turn, but if you alternate, you win one every turn (after the first, possibly). For simplicity, I'm ignoring the behavior near zero chips, but this still seems to capture the essential properties of the "paradox" in a much simpler fashion.