Hmm. It seems martingaling is better only if your aiming to less than double your initial balance.
I'm willing to bet you're wrong on that too if you like?
The optimal strategy for going from X btc to Y btc is the one that minimizes the amount wagered. In the previous example of 3BTC -> 4BTC, the martingale of 1BTC -> 2BTC is better because 49.5% of the time only 1BTC is being wagered. We can do even better by betting less at a smaller winrate -- consider for example the following martingale of 3 bets:
0.58740084 BTC at 2.70241438x (36.6339%) -> if we lose 0.93244059BTC at the same rate -> if we lose 1.48015697BTC at this rate.
The expected values of each strategy:
1 bet -- (3btc @ 1.3333333x) -- 0.7425 success rate, EV = 2.97 BTC (-0.03)
2 bets -- (1btc, 2btc) --0.744975 success rate, EV = 2.9799 BTC (-0.0201)
3 bets -- (pattern above) -- 7455684 success rate, EV = 2.9823 BTC (-0.0177)
In theory, if BTC were infinitely divisible, one could start at an infinitesimal amount at "near-infinite" odds and apply this martingale to eliminate the house edge and obtain an EV of 3BTC.
In practice, small martingale style bets are better because they introduce scenarios where less BTC is bet and reduce the loss from house edge. The starting and target BTC amounts are irrelevant.
Thanks, and you're exactly right.
But now who's going to bet against me?!?