Thank you very much for the explanation. I ran my numerical solution, and it fully agrees with yours. I am convinced now. Below are the top 20 strategies. The concept of the "liferoll" does indeed add another dimension to this game. I am moving on to crafting the "progressive risk" solution now. Code: LifeRoll: $1,000,000 BankRoll: $1,000 R16 R14 RC RN BC BN Log(BankRoll + LifeRoll) 100.0 0.0 0.0 0.0 0.0 0.0 13.8193340330 99.9 0.1 0.0 0.0 0.0 0.0 13.8193331951 99.8 0.2 0.0 0.0 0.0 0.0 13.8193323570 99.7 0.3 0.0 0.0 0.0 0.0 13.8193315187 99.9 0.0 0.1 0.0 0.0 0.0 13.8193315120 99.9 0.0 0.0 0.0 0.0 0.1 13.8193312492 99.6 0.4 0.0 0.0 0.0 0.0 13.8193306802 99.8 0.1 0.1 0.0 0.0 0.0 13.8193306740 99.8 0.1 0.0 0.0 0.0 0.1 13.8193304112 99.9 0.0 0.0 0.0 0.0 0.0 13.8193302763 99.5 0.5 0.0 0.0 0.0 0.0 13.8193298414 99.7 0.2 0.1 0.0 0.0 0.0 13.8193298358 99.7 0.2 0.0 0.0 0.0 0.1 13.8193295730 99.8 0.1 0.0 0.0 0.0 0.0 13.8193294383 99.4 0.6 0.0 0.0 0.0 0.0 13.8193290023 99.6 0.3 0.1 0.0 0.0 0.0 13.8193289973 99.8 0.0 0.2 0.0 0.0 0.0 13.8193289909 99.6 0.3 0.0 0.0 0.0 0.1 13.8193287346 99.8 0.0 0.1 0.0 0.0 0.1 13.8193287281 99.7 0.2 0.0 0.0 0.0 0.0 13.8193286001
I see nothing extraordinary here. It's just fractional Kelly. If your net worth is say $50K but the most you're willing to gamble (aka lose) is $1K, then figure out the strategy that maximizes your potential gain. I doubt there is a general formula for this. Everything is dependent on both the net worth and the betting bankroll. Every time your Kelly fraction exceeds (bankroll)/(net worth), you're going to end up with fractional Kelly betting. Nothing new here. Of course explaining all this to the interviewers will probably risk blowing the interview.
Because it's cancelled by the bankroll. This: (4/37)*log(1,000,000 + 36,000) + (33/37)*log(1,000,000) = 13.819 Is equivalent to this: (4/37)*log(1,000,000 + 1,000 + 35,000) + (33/37)*log(1,000,000 + 1,000 - 1,000) = 13.819
It is extraordinary, because what we calculated before as "constant and optimal" {R16: 8.1%, R14: 5.4%, RedColor: 48.6%, BlackNumbers: 37.8%} is no longer constant or optimal. Specifically, this allocation would change, depending on your starting life roll, and then, even more interestingly, if the life roll is large enough, the allocation would change on every spin, becoming progressively riskier with every spin, and ending with the last bet of 100% of the bankroll on R16.
None of what has anything to do with the interview question. They want to know how will you grow that $1000. If you start going off the rails with this "life roll" shit, you can kiss that job goodbye. Try to remember: this is not a real casino game. It exists only because the interviewers made it up. I doubt any real-life casino anywhere has offered anything like this game Ever. So good luck explaining to the interviewers how you blew your bankroll on the last spin because you were trying to maximize your at-home net worth instead of maximizing your bankroll.
I really don't care if my answer pleases the interviewer, or if I get a job. My interest here is to apply what we collectively leaned in this discussion to trading strategy selection and portfolio optimization. If it were an interview, yeah, I'd figure out the Kelly fractions, and it would probably please the interviewer immensely. But a good answer turned out to be a lot more interesting than that.
So is your trading account entirely dispensable because the rest of your assets are "sufficiently large"?