With this understanding, the paradox resolves itself: The individual games are losing only under a distribution that differs from that which is actually encountered when playing the compound game.
You don't need a modulo-style set up. You just need a dependence of the two games such that a player can predict when to play each game to always be optimal. The ratchet is just one example where the player always avoids the really bad odds in game B that help to make it a losing game.
+ A choice between two "games", each losing on average.
+ At least one game has periods of local payoff great enough to overwhelm the long-run losses in the other game
+ Some flow of information between the games so that playing one game will help you predict the payoff periods of the other game.
The example that flies to mind is investment. Game A is to lose value of money you hold on to via inflation. Game B is the generally losing game of day trading. The net game can be profitable as long as you spend time in A learning to accurately predict game B's upswings.
Of course, the real information flow from game B to game A is already heavily capitalized making the net game even more difficult.
