What is your strategy?

[nonlinear5]

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

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.

 

[markettimer]

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

Treat B_9 as a variable and solve for the optimal xj_10 as a function of B_9. It gets cumbersome quickly. That's why I was suggesting maximizing the risk neutral expectation, not expected logarithm. Not only are there some assumptions where this is optimal (i.e., ability to diversify by selling shares in the bankroll, or large W relative to B), but it considerably improves tractability.

 

[markettimer]

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.

Agreed that 100% would be wagered, but the allocation across bets would vary. The arbitrage has a lower expected return than the riskier R-16. Presumably, the allocation to R-16 would increase as you get closer to the final spin.

 

[nonlinear5]

Agreed that 100% would be wagered, but the allocation across bets would vary. The arbitrage has a lower expected return than the riskier R-16. Presumably, the allocation to R-16 would increase as you get closer to the final spin.

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

 

[SplawnDarts]

Agreed that 100% would be wagered, but the allocation across bets would vary. The arbitrage has a lower expected return than the riskier R-16. Presumably, the allocation to R-16 would increase as you get closer to the final spin.

This sounds right.

 

[kut2k2]

Suppose you are worth $1,000,000. Suppose you can bet up to $1,000 on just one spin, with the same odds given in the OP. How much would you bet? If your goal is to maximize geometric mean returns, it should be for the $1,000,000 net worth, not the $1,000 you are allowed to wager. When people are discussing Kelly here, it is for the $1,000 limit, which shows a misunderstanding of what the Kelly criterion says. The objective is to maximize the expected logarithm of net worth over the set of outcomes. Maximizing the expected logarithm of the initial bankroll does not make sense as a goal.

No, I've read plenty about Kelly and this is the first I ever heard of this. This is just your own arbitrary spin. That's your choice but don't act like this is how it's "supposed to be" for everybody. Most people don't want to risk their entire net worth when they gamble, even when the odds are on their side. Case in point: the casino owners never put their entire worth on the line with their clientele, and the odds are always on the owners' side.

 

[kut2k2]

Either you do the fractional betting as discussed in the thread or you go for the "arbitrage " where you allocate the entire bankroll across the wheel such that you make money on each spin.

No if you allocate the entire bankroll across the wheel, you lose on 30 out of every 37 spins. The only way to make money on every spin is to bet on Red and the individual nonred numbers in an appropriate division of the bankroll.

 

[nonlinear5]

Agreed that 100% would be wagered, but the allocation across bets would vary. The arbitrage has a lower expected return than the riskier R-16. Presumably, the allocation to R-16 would increase as you get closer to the final spin.

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 this game only once. Then with accordance to the above the line of thinking, your bet should be 100% of the bankroll on R16, given that your "liferoll" is significantly larger than your "bankroll". Correct? If so, I don't see how this maximizes the log of the life roll. If you are maximizing the life roll itself, only then this bet would be optimal.

 

[markettimer]

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 this game only once. Then with accordance to the above the line of thinking, your bet should be 100% of the bankroll on R16, given that your "liferoll" is significantly larger than your "bankroll". Correct? If so, I don't see how this maximizes the log of the life roll. If you are maximizing the life roll itself, only then this bet would be optimal.

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.

 

[markettimer]

No, I've read plenty about Kelly and this is the first I ever heard of this. This is just your own arbitrary spin. That's your choice but don't act like this is how it's "supposed to be" for everybody. Most people don't want to risk their entire net worth when they gamble, even when the odds are on their side. Case in point: the casino owners never put their entire worth on the line with their clientele, and the odds are always on the owners' side.

Well, I'm not suggesting to bet your entire worth, just to take it into account deciding the optimal wager. There is no "correct" answer here because the OP does not specify an objective function. If you take the objective to be "maximize the expected log of the initial $1000 bankroll after 10 spins," then that is one interpretation. Note that this does not maximize the risk neutral expected value after 10 spins. My interpretation is, here is a great deal where you are limited to a $1000 wager. In deciding how to bet, I'd like to know how much I'm worth. If $1000 isn't much to me, I'm more willing to lose it all in the hopes of ending up with more. Therefore, my objective is to maximize the risk neutral expectation (or, similar in this case, but much more analytically cumbersome, maximize the expected log of total wealth when total wealth far exceeds $1000).