E[X(n+1) | history] = X(n). A fair game: no drift in either direction.
Optional stopping is the powerful consequence: under suitable conditions the expected value at a stopping time equals the starting value. That solves gambler's ruin in one line - if wealth is a martingale and you start at k with barriers at 0 and N, then k = P times N, so P = k/N.
The trap. Optional stopping needs conditions - a bounded stopping time or bounded increments. The martingale doubling strategy appears to produce guaranteed profit precisely because it violates them, requiring unbounded capital.