But this is so slow! I’m a busy grad student! I can’t spend all this time gambling, so instead, let’s blow an afternoon proving what the optimal strategy is using martingales. On the other hand, if we bet just one star each time, we would win an average of 1.5 stars for each star we bet, and if we played long enough we’d be sure to make money. Obviously we don’t want to bet all our stars on green, since there’s some decent probability we’d lose everything. The only thing standing between us and the rare scarecrow of our dreams is finding the optimal way to bet on this wheel. In this post, I’ll show that the optimal strategy is to bet half your stars on green in each round.
Winning gives you star tokens, and there’s big incentive to get lots of stars: you can win a rare scarecrow! Betting on a roulette wheel biased in our favor seems like a great way to get them. At some point you go to the Stardew Valley Fair, where there’s a roulette wheel you can play which comes up green with probability $3/4$ and orange with probability $1/4$. I’ve been playing a bit of Stardew Valley recently.
