Interesting gambling theory question/puzzle
I saw a discussion the other day that reminded me of a problem I came up with and I posed it to a bunch of friends. None of them were able to get the right answer and reasoning, so I figured it would make a good blog post.
I originally thought of this in the old days when online casino bonuses were prevalent. If you’re not familiar with them, you would deposit some amount, get a bonus, and have to meet a wagering requirement to withdraw the bonus. So you’d do something like deposit $100, they’d give you $20 on top of that for free, and you’d need to wager $1000 at blackjack to withdraw. Blackjack pays back around 99.5% if you play perfectly, so you’d end up losing $5, thus profiting $15 after the bonus. This is actually how I started my bankroll many years ago.
So, here’s the problem:
You deposit $400 in an online casino and are given a $100 bonus immediately, so you have $500 to bet with. You can withdraw only after betting a total of $2500. Let’s say you play a game where you flip coins and if you win you get 1.99 times your bet and nothing if you lose. This has a 99.5% return like blackjack, but I’m abstracting it because in blackjack you can run into bad EV spots where you make a bet and then don’t have enough to split or double down. The table limits for this game are minimum bet $1 and maximum bet $100. How much should we bet to maximize EV, and why? Does it even matter? If so, why? If not, why not?
Time and variance are not factors in this problem. We don’t care that making lots of small bets takes more time than making a few large bets and we only care about maximizing EV.
Post answers or discussion in the comments. I’ll reveal the answer in less than a week and may drop some hints if nobody gets it.