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I thought about this again, and I am still not clear about this algo. To maximize E[log(W+B_10)], I need to try different allocations of B_9. But as we have established, the optimal allocation would depend on B_9. That is, the optimal 10th allocation is a function of B_9. I was hoping to solve...

Okay, makes sense now, thanks much.

Hey markettimer, I have a question. What you have advocated is maximizing the log of the sum of the liferoll and the bankroll: Utility = Max[E(Log(L+B))] where L is the liferoll, B is the bankroll But the notion of geometric growth is meaningful in terms of getting from the starting wealth...

The original question didn't ask "What's the optimal strategy to grow the bankroll?". It's more open-ended. As markettimer pointed out, no utility function was specified, either. It could be quadratic, logarithmic, or it might as well be risk-neutral (i.e. maximization of absolute bankroll, or...

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...

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...

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

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...

Wait, there is something wrong here. I initially subscribed to this idea of "progressive risk increase as we get closer to the final spin", but now I am questioning its validity. Let's change the rules with respect to the number of spins. Instead of 10 spins, there is only 1, i.e. you can play...

Good point. I'll see if I can set up the numeric solution to this.

Oh, I think I understand. B_9 can be figured out from the recursive call, E[log(W+B_8)]. I'd take a shot at computing it, but in this particular instance of the game, we know the answer already: the bet would be sized at 100% of the bankroll, on every spin, because of the built-in arbitrage.

To maximize E[log(W+B_10)], we need to know what B_9 is. What do we use in place of B_9?

This feels right in spirit (base the bets progressively more on the liferoll and progressively less on the bankroll as we get closer to the end of the game), but the proportions need refinement.

Right, but that would only be the case if you are not allowed to use your "life roll" from the start, such as with this game that sets the starting amount. If the rules allowed you to use any amount to start with, you'd use your entire life roll, and the bet sizes would always be against that...

markettimer makes a good point that this strategy would be sub-optimal, even if the utility function is the geometric rate of growth. If the odds are heavily weighted in your favor, you should deploy your entire wealth, and use the Kelly fraction of your entire wealth, rather than deploy a...

Well, presumably it takes less than 10 minutes to play 10 spins. I think it would be irrational not to play. As illustrated in the latest results, there is an optimal point in between these "either" and "or". This is a valid case, and it was discussed here, as well. If the rules allowed you...

I'll take the credit for calculating the bet allocations, but the idea of arbitrage is that of SplawnDarts. Yes, if the utility function that we seek is the terminal wealth (rather than the log of terminal wealth), then everything changes. To maximize terminal wealth with a fixed proportional...

Yeah, it's been fun. kut2k2, feel free to start a new thread to see if k*E is a good performance measure.

Okay, here are the top 20, with all bets accounted for correctly: R16 R14 RC RN BC BN F(R16,R14,RC,RN,BC,BN) 8.2 5.5 47.1 1.4 0.0 37.8 0.2115470448 8.3 5.6 45.3 3.0 0.0 37.8 0.2115470448 8.4 5.7 43.5 4.6 0.0 37.8 0.2115470448 8.5 5.8 41.7 6.2 0.0 37.8 0.2115470448 8.2 5.5 47.0 1.5 0.0...

I was wrong. A bug in my code led me to believe in the red numbers "superiority". I am re-running the sim, and will post the results shortly.