What's the formula from which this relationship is derived? And how does the growth rate scales when Kelly is reduced? As Kelly is reduced, the growth rate decreases slower than the variance, which implies that the risk-adjusted growth (growth divided by variance) goes up. I'd like to know at which fraction of Kelly this risk-adjusted growth reaches its peak. I am guessing it would occur at somewhere around 1/4 of full Kelly.
You didn't prove anything, you did not even try. I can't blame you, proving the Kelly fraction is the best bet for a single spin is impossible task. But I am going to show you that even on 10-spins, it is not the optimum bet, at least from a statistical point of view. I run a 10-spins MC simulation, comparing your 8%-bankroll bet strategy (Red16) vs a flat $50 bet strategy (same Red16). In 70% of the 10-spins runs, $50-fix bet comes ahead of your Kelly bet. In the scenario presented, 10-spins run, Kelly is NOT optimal.
Now, if you change the "Kelly" bet strategy, and use 8% of the max Bankroll instead of 8% of the current Bankroll, and compare with a fixed $80 bet, the stats change as follow: $80 fixed-bet: better in 35.4% of the 10-spin runs 8% of max Bankroll: better in 28.7% of the 10-spin runs Tie: 35.9% Fixed-bet still comes ahead.
I think the interviewers wanted to see which candidates did think with their brain, and which candidates were just using cookbooks without analysis of the problem at hand.
But in the 30% of the outcomes, the payout on the Kelly bet is many times larger than the payout on the fixed $50 bet, isn't it? Why are you ignoring these 30% of the outcomes?
I am not ignoring that the 30% has a much larger outcome, I am just highlighting that as an individual who has only 1 go at the 10-spin run, I want to maximize my probability of having the best outcome, not the potential total gain in the unlikely event of 10 times 16.
This 10-spin game is much different from a infinite process, it seems this is difficult to grasp for some. My utility function is to maximize my chances of winning some. It seems most people on this thread are focused on maximizing the potential gain, at the cost of having a low probability such high gain. This isn't optimal asset allocation IMO - it is gambling.
Ok, so you are uncomfortable with the high variance of the results when using full Kelly. Me too. But it doesn't disprove the hard mathematical fact that Kelly maximizes the compound rate of return, either over 1 trial, or over 1 million trials. Mine is to maximize the return-to-risk ratio. The full Kelly is, indeed, the fine edge between the "aggressive" and the "insane" capital allocation.
All those Kelly graph apply to infinite processes. It is ridiculous to say they apply to a unique 1-spin game. As for the 10-spin game, you prefer a higher probability (than fixed-bet) of winning less or losing, in return for higher gains in lower probability scenarios. That is your choice, but can't be called optimal asset allocation, optimal gamble for sure.