The Kelly criterion is a way of choosing a number to bet from the set of rational numbers between 0 and your bankroll (inclusive). However, this isn't the set y

The Kelly criterion is a way of choosing a number to bet from the set of rational numbers between 0 and your bankroll (inclusive). However, this isn't the set you're choosing from when playing the Powerball. You're rightly accounting for this. However, you're accounting for this incorrectly by adjusting the size of the bankroll ("Oh well, if you are a centimillionare you can buy a ticket and be rational."). Your bankroll is fixed. The correct adjustment is to change the set of values from which the fraction of the bankroll can be chosen. The optimal bet for a +EV bet is always positive when the set of possible bets includes positive numbers, because a zero bet has effectively a zero EV. My intuition is that you want the number in the set of possible bets that is closest to the Kelly Criterion number when weighted along some curve (I'm too lazy to calculate which curve on my mobile device). If this is correct (I haven't proved my "closest on a curve" idea) then the optimal bet is to buy a $2 Powerball ticket (the smallest positive bet). That said both these ideas assume an infinite number of future bets following the same strategy, with no other games to play. If you play the Powerball every time the EV is positive, this is like an investment that will yield a return, on average, long after you are dead. And there are alternate games (the stock market, education, job searching) which yield more consistently and possibly with a higher EV. Finally, the goal isn't to maximize your bankroll. I just spent the cost of a Powerball ticket on a coffee, which has a -EV, and I couldn't be happier with my decision.