Thank you for your post abramdemski!
I failed to understand why you can’t arrive at a solution for the Single-Shot game via Iterated Play without memory of the previous game. In order to clarify my ideas let me define two concepts first:
Iterated Play with memory: We repeatedly play the game knowing the results of the previous games.
Iterated Play without memory: We repeatedly play the game, while having no memory of the previous play.
The distinction is important: With memory we can at any time search all previous games and act accordingly, allowing for strategies such as Tit-for-Tat and other history dependent strategies. Without memory we can still learn ( for example by applying some sort of Bayesian updates to our probability estimates of each move being played ), whilst not having access to the previous games before each move. That way we can “learn” how to best play the single shot version of the game by iterated play.
Does what I said above need any clarification, and is there any failure in its’ logic?
Best Regards, Miguel
You mention that a Martingale is a betting strategy where the player doubles their bet each time.
A Martingale is a fair game (i.e. the expected outcome is zero). If your outcome is given by a coin toss, and you receive only what you bet, then that is a Martingale game (you win X £ with probability 12 and lose X £ with probability 12 too ).
Then you could say that doubling your bet is a betting strategy on a Martingale game, BUT not that a Martingale game is a betting strategy where the player doubles their bet each time (in the same way that a dog is an animal but an animal is not a dog).
Does that make sense?
Other than that I’m very intrigued by the claim made. Definitely worth reading, but my hopes for something worthwhile are few :P