And, win or lose, you would never know whether your decision had boosted your odds or not.

I'm not sure follow you there, Lighter. By definition the odds are 'if you played the game oodles of times what proportion would you win'. You know that switching improves your odds every time. An individual game on the other hand wins or loses but does not affect the odds: that's true of any game without an 'end-stop' (For example, actual gambling games rarely allow you to continue once you have run out of cash/shirts)