What is your strategy?

[kut2k2]

Only had time to browse a few pages, but what struck me was that everyone seemed to assume the optimal strategy does not change with spin number.

If we assume the goal is to maximize the expected value of terminal wealth after 10 spins, the optimal bet changes with each spin. In particular, on the final spin, the optimal strategy is to bet everything on R-16. You have to work backwards to identify the optimal strategy at each spin. In other words, having a positive expectation on the 10th spin creates something like risk aversion on the 9th spin. Note that risk aversion doesn't exist on the terminal spin if the goal is to maximize expected terminal wealth.

I agree that the problem takes the character of optimal portfolio allocation, particularly on the early spins.

It is a very challenging, interesting problem.

There is an article called "The Good and Bad Properties of the Kelly Criterion". It is available as a free pdf and is easily found via Google. One of the good properties is this:

The Kelly bettor is never behind any other bettor on average in 1, 2, ... trials.
Source: M. Finkelstein and R. Whitley (1981) Optimal strategies for repeated games. Advanced Applied Probability 13: 415 - 428.

The rules of this game are fixed for all ten spins, so it's safe to assume that the optimal strategies are fixed as well.

 

[markettimer]

There is an article called "The Good and Bad Properties of the Kelly Criterion".

Thanks for the link.

I am familiar with the Kelly criterion (have an economics phd, studied dynamic programming problems like this ad nauseam in my first year macro sequence). The reason I think we differ is due to what you are considering the maximization problem. You are assuming we would like to maximize geometric mean return, I believe. Whereas I assume the goal is to maximize the expected terminal wealth. This actually does not imply maximizing geometric mean return for the $100 bankroll. The simple one-spin game is an example of why that is true.

There are good reasons to think the expected return (not geometric mean return on initial $100 bankroll) should be maximized:

(1) The bankroll is initially limited to $100. In other words, any bet is likely to be a small fraction of your total wealth. While some recommend 8% (or $8 initially) be applied to a given bet, for example, a true Kelly adherent, who maximizes geometric mean of net worth, would recommend 8% of total net worth. Therefore, unless you started this game penniless (and technically with no human capital either) we cannot reach the bet size that would be recommended by the Kelly criterion were total net worth considered.

(2) There is no rule against selling shares in your bankroll to outside investors. If you are able to diversify the risk of this game in such a way, then even if you are poor, you should try to maximize the mathematical expectation across the 10 spins.

In other words, either we should apply Kelly to total wealth (not $100), or maximize expected terminal wealth. In the former case, for large starting wealth, I believe the problem converges to maximize the mathematical expectation (i.e., maximizing the geometric mean of total wealth converges to the same problem as maximizing expected terminal wealth as $100 becomes small relative to one's net worth).

 

[Visaria]

The bankroll is $1000, not $100, for starters. You only have a thousand dollars, nothing more. You are only allowed to lose yur initial thousand dollars. There is no need for outside investors lol.

 

[Visaria]

Kelly fraction applied per spin maximises the expected return over 1 spin, 5 spins, 10 spins, 1 million spins. The optimal fractions to use (because we have more than 1 possible +ev bet and one of them overlaps) have already been discussed.

 

[Visaria]

Congrats to whoever came up with betting on all outcomes to create an "arbitrage" situation. Didn't think of that!

 

[markettimer]

Kelly fraction applied per spin maximises the expected return over 1 spin, 5 spins, 10 spins, 1 million spins. The optimal fractions to use (because we have more than 1 possible +ev bet and one of them overlaps) have already been discussed.

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. If you understand this, you realize the $1,000 limit is a binding constraint--you'd like to wager much more than that. As your wealth grows, the problem gets much closer to maximizing the expected terminal wealth (risk-neutral), even for someone trying to maximize geometric mean returns (over net worth). The solution to the risk-neutral maximization problem is to bet as much as you are allowed on R-16 when you only have one spin left.

The clearest example I can give you is the following: Suppose you can bet up to $100 on a weighted coin toss, where heads is 99% likely to happen. I'll give you even odds, so if you bet $100 on heads and win, I'll give you $200 (a $100 profit). Maximizing geometric mean return for the $100, which is what is being advocated by apparently everyone here, would have you risk less than $100 on this coin toss. However, clearly, you should put $100 on heads--in fact, you'd like to risk almost everything you have, but are constrained. Do you see why maximizing Kelly only makes sense across your entire net worth, not the arbitrary $100 limit? Kelly would suggest you put almost your entire net worth on heads, for argument's sake, $1mn. However, you can only bet $100. Therefore, you should do the next best thing and bet $100.

 

[kut2k2]

Congrats to whoever came up with betting on all outcomes to create an "arbitrage" situation. Didn't think of that!

Visaria, thanks for your contributions to the thread. The person who came up with the strategy you speak of is nonlinear5.

 

[nonlinear5]

Visaria, thanks for your contributions to the thread. The person who came up with the strategy you speak of is nonlinear5.

I'll take the credit for calculating the bet allocations, but the idea of arbitrage is that of SplawnDarts.

If we assume the goal is to maximize the expected value of terminal wealth after 10 spins, the optimal bet changes with each spin. In particular, on the final spin, the optimal strategy is to bet everything on R-16.

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 bet, you'd bet 100% on R16, on every spin. And, as you suggested, you can do even better with a variable bet which changes depending on the result of the previous spin.

However, I disagree with your assertion that maximization of terminal wealth makes more sense than maximization of the log of the terminal wealth. The former utility function has all the properties of irrational gambling, while the latter one approximates the rational, risk-averse behavior.

I thought about this problem of "terminal wealth maximization" vs "log of terminal wealth maximization", and here is what I figured out. Let's say you know for sure that you have only one day left to live, and that you can play the originally proposed game with your entire wealth. Furthermore, assume that you don't care what happens after you die (i.e. you don't care about who inherits your debt or wealth). What would then be your strategy with this original game? The answer, it seems to me, would be to seek the "terminal wealth maximization", i.e. bet your entire wealth on R16, on every spin of the wheel. If you lose everything, you don't care. If you win, you can potentially win an astronomical amount of money (35^10 times your initial wealth). This mental experiment does support your advocacy of the "terminal wealth maximization", because, effectively, every spin is the last spin that you can ever bet on. But this is different from the case when you expect to live a number of years, because in these coming years you'd have any chances to play similar games of weighted odds. Just because the original game ends after 10 spins, it should not be treated as the last game you can play.

 

[Visaria]

If i knew i had only 1 more day to live, i wouldn't be wasting my time playing a roulette game regardless of how much ev there was!

The value of the time i had left would outweigh any possible monetary gain.

 

[Visaria]

- You have a starting bankroll of $1000, and are allowed to play for ten spins.

markettimer, you only have $1000. That is it. No million, no outside investors, nothing mate apart from a grand.

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.