What is the optimal strategy for this made-up casino game? Part 1: Let's say I walk into a casino with 100 dollars in my wallet. I sit down to play a game where my payoff or loss each round is aX dollars, where X is a continuous random variable uniformly distributed between [-2, 2]. However, because I'm friends with the manager, he gives me a special deal. If X < -1, I only lose a dollars instead of losing anywhere between a and 2a dollars, as I rightfully would. (Note that the random variable is deliberately [-2, 2] instead of [-1, 2], in order that the initial determination of the random variable will have an EV of 0.) The random variable X is redetermined every round.
The one choice I get to make is the value of the constant a, which is fixed forever once I choose it. If I want to maximize the amount of cash in my wallet at the end of 50 rounds (and implicit in this question is the need to avoid going bust), what value should I pick for a?
Part 2: Everything is the same, but instead of the payout or loss each round being aX dollars, it is now a(X+b). Let's say b = 0.1, so my payout or loss every round will be a * [-1.9, 2.1]. The special deal with my buddy the manager is still on for any X < -1, so I only lose a dollars instead of anywhere between a and 1.9a dollars. How does my selection of a change?
 A: This is my best guess so far. Comments/thoughts/suggestions, please?
Kelly Criterion states that f, the optimal fraction of your bankroll to bet, is (bp-q)/b, where b is the net odds you receive on the bet, p is the probability of winning, and q is the probability of losing.
Since the random variable is uniformly distributed across [-2, 2], you have a 50/50 shot at winning and losing, which means that p = q = 50%. 
Now to find b, the odds of your bet. The expected value of a winning round is 1, given that the RV must be somewhere between (0, 2]. The expected value of a losing round is -0.75, since a losing round happens when the RV is between [-2, 0), and in half of the losing rounds the RV will be between [-1, 0), in which case you lose -0.5 on average, while in the other half of the losing rounds the RV will be between [-2, -1) - in which case you only lose -1 due to your special deal with the manager. So EV(losing round) = (50% * -0.5) + (50% * -1) = -0.75. Therefore, you are risking 0.75 on average to win 1 on average, hence the odds of your bet, b = 1 / 0.75 = 4 / 3 = 1.33
Plug into the formula and you get f = (4 / 3 * 50% - 50%) / (4 / 3) = 12.5%
Therefore you should bet 12.5% of your bankroll every time, or 12.50 on a bankroll of 100. Since every bet implicitly costs you 0.75 to play, you should select the constant a such that a = 12.5 / 0.75 = 16 2/3
