For a fixed, known, negative-expectation game, no betting strategy can change the mean outcome. It CAN change the distribution of outcomes, including the median and modal outcome. Well, depending on how you model the overall game (multiple iterations). One common analysis is “play until you win X or lose Y”, sometimes “or Z iterations”, but that makes generalization much harder, and is irrelevant most of the time.
In this approach, repeated Martingale bets of less than X is simply worse than Martingale starting at X. This is because your “bet handle”, the total money at risk, including re-betting winnings, is higher.
For a fixed, known, negative-expectation game, no betting strategy can change the mean outcome. It CAN change the distribution of outcomes, including the median and modal outcome. Well, depending on how you model the overall game (multiple iterations). One common analysis is “play until you win X or lose Y”, sometimes “or Z iterations”, but that makes generalization much harder, and is irrelevant most of the time.
In this approach, repeated Martingale bets of less than X is simply worse than Martingale starting at X. This is because your “bet handle”, the total money at risk, including re-betting winnings, is higher.
Much more interesting is postive-value bets, where you get to use logarithms. There’s a fair bit of LW discussion under the tag https://www.lesswrong.com/tag/kelly-criterion .