There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is “bold play”: in each round of play bet the smaller of either your entire wealth or an amount that would ensure that you would achieve your target amount if you win in that round. See for example Section 1.7 of Billingsley's Probability and Measure.
In trying to understand this intuitively I came up with this argument:
If the game is unfair then the expected earnings have a negative trend which is linear over time. On the other hand like any random walk the standard deviation of the earnings is proportionate to the square root of time. If we let enough time to pass the linear will dominate the square root and we will certainly be ruined. So our only hope is to compress the time for which the game is played and the way to do so is to play large stakes.
My question is whether this intuitive argument comes close to what is happening?
Also, is there is a way to turn this argument into a rigorous proof? The idea came to me from knowing that random walks can be embedded into Brownian motion but unfortunately I don't know enough about such embeddings to judge if the reasoning above is sound.