What is your strategy?

[nonlinear5]

That's correct. If W is large (say $1mn), I believe betting $1,000 on R16 maximizes the expected log of liferoll in your one-spin example.

Goal: Max E[log(W + B_1)]

Candidate solution:
Bet $1000 on R16.

B_1 = $36,000 with probability 4/37
B_1 = $0 with probability 33/37

E[log(W + B_1) =

(4/37)*log(1,000,000 + 36,000) + (33/37)*log(1,000,000) = (4/37)*13.851 + (33/37)*13.816 = 13.819

You can check other allocations to see if they offer a higher expected return than this. My intuition is that they will not, for large enough W. I expect $1mn is large enough, but could be wrong.

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.

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

 

[tradingjournals]

That's correct. If W is large (say $1mn), I believe betting $1,000 on R16 maximizes the expected log of liferoll in your one-spin example.

Goal: Max E[log(W + B_1)]

Candidate solution:
Bet $1000 on R16.

B_1 = $36,000 with probability 4/37
B_1 = $0 with probability 33/37

E[log(W + B_1) =

(4/37)*log(1,000,000 + 36,000) + (33/37)*log(1,000,000) = (4/37)*13.851 + (33/37)*13.816 = 13.819

.

Why you did not subtract $1000 from the million in the second log?

 

[kut2k2]

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.

 

[nonlinear5]

Why you did not subtract $1000 from the million in the second log?

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

 

[nonlinear5]

I see nothing extraordinary here.

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.

 

[kut2k2]

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.

 

[nonlinear5]

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.

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.

 

[SplawnDarts]

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.

Any interviewer that DOESN'T understand that is an indication the firm is too stupid to work for.

 

[kut2k2]

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"?

 

[SplawnDarts]

So is your trading account entirely dispensable because the rest if your assets are "sufficiently large"?

Potentially, sure.