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 to borrow and/or sell shares to investors, and then bet arbitrarily large amounts, then you should borrow as much you can, and take advantage of the arbitrage (i.e. risk-free profit) offered by the rules of this game.
I think you are confusing net worth and bankroll btw. Net worth could be a million pounds. I may however only take £100k of that as being my bankroll or account size. The Kelly fraction is then applied only to that bankroll of £100k. So say you have a trading strategy that has a positive edge, you calculate the kelly fraction, say it is 10%, you then bet 10% of your bankroll per trade you make,, recalculating the 10% after each trade. In a sense you then only are risking £100k of the million net worth.
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 portion of your wealth and use the Kelly fraction of that portion of your wealth. Let's say the game is the flip of the coin, and the "heads" pays 100:1, while in the case of "tails" you lose your bet. It would be irrational to not use your entire wealth while using Kelly to size the bets. In fact, the case of "heavily weighted odds" is just for the sake of illustrating the effect. More generally, if the odds/edge are even slightly in our favor, you should use the Kelly of the entire wealth.
This goes to what I was saying earlier - clearly the last bet of a sequence like this should be against your "life roll", not your immediate cash in hand. But the first bet of a long sequence should be on cash in hand since it alters the value of the subsequent bets. For bets close to the end but not the last, the answer is ???. Of course in this case, due to the arbitrage, we're always going to use all the money every time. So it's something of an academic debate. But with a non-arbitrage situation, it would be very relevant.
Maybe this was discussed, but was a confidence intervals approach considered in the optimization? For instance, if the question were that I only need 40% confidence in my maximization objective, then strategies other than kelly (or in conjunction with kelly) could be used.
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 life roll, whether it is the first bet, or the last one. Yeah, maybe something of this form: Utility(i) = Log[BankRoll^(1 / i) + LifeRoll^(i / N)] where BankRoll is the starting amount allowed ($1000 for this game) LifeRoll is the player's net worth N is the total number of spins (10 for this game), i is the spin number, [1..N]
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.
For a utility function, you could still use expected log of terminal wealth. This is a constrained maximization problem, where the key constraint is the initial bet cannot exceed $1000, and no bet can exceed what the $1000 bankroll has grown to. You could work backwards to solve the problem. In other words, on spin ten, you have bankroll B_9 (B_9 being the bankroll after the ninth spin) and outside wealth W (total wealth W+B_9). You want to maximize E[log(W+B_10)]. B_10 is a distribution that depends on how B_9 is allocated. This is a straightforward maximization problem where you simply choose allocation fractions xj_10, how to allocate B_9 across the j possible bets on the wheel for the 10th spin. Now, working recursively, you would solve for the allocation fractions on the ninth spin, starting with bankroll B_8. And so onâ¦