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
That does make sense, although it is not how they use the term in the book. In the book they explicitly rely on the betting system, for several reasons:
The name of the stochastic process in probability comes from the betting system, and they privilege the historical origination. This does introduce term confusion sometimes, but they apply it consistently in order to capture (what they claim to be) important subtleties.
I utterly failed to communicate this, but they pick up the use of the martingale directly from Jean Ville, who applied it to prove the difference between a frequency which converges via oscillation above and below vs. one which converges only from above—it turns out that if it converges from above, a martingale betting system can make a gambler infinitely rich. This was an important point against von Mises’ collectives, with respect to Kolmogorov’s measure.
They argue that Ville’s (generalized) use of the martingale betting system is a universal test for randmoness. Ville showed (they say) that for any event with probability 1, there is a non-negative martingale which diverges to infinity if that event fails. They extend Ville to say that the probability of an event is the smallest possible initial value for a non-negative martingale that eventually reaches or exceeds 1 if the event happens.
They start from that to build their game-theoretic notion of the martingale, which they then rely on to build the rest of their results.
This still leaves me in the wrong though, because once generalized it no longer is the doubling of bets—it is any system of varying bets. I will make an update to include some of this!