What is your strategy?

[markettimer]

Thanks again, markettimer. I'll see if I can figure it how to code that in.

If you do, please let me know how you do it. I've actually never used a computer to solve one of these problems, so have been limited to textbook problems / proofs that can be solved with a few notebook pages of calculus.

 

[nonlinear5]

If you do, please let me know how you do it. I've actually never used a computer to solve one of these problems, so have been limited to textbook problems / proofs that can be solved with a few notebook pages of calculus.

Sure. The article that you linked specifically says that Bellman equation can be solved by backwards induction numerically on a computer. So, I need to understand it, and the rest (programming it) should be straightforward.

 

[kut2k2]

Only had time to browse a few pages....

I should have paid more attention to this.

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.

Here's a shocker: it does not matter what your freakin' net worth is. Because of the betting constraint, all you can maximize is your bankroll ($1000 initially).

You're hung up on the expected return per spin. There's a reason why expectation alone is a poor performance metric. If you bet the full $1000 on R-16, there is a 89+% chance that you walk home penniless. There is an almost nine in ten chance that you go home broke. Period. Doesn't matter how much wealth you have at home, you leave that casino with nada. With the wheel arbitrage bet, there is a 100% chance that I go home with at least $973.

My expectation is much less than yours but I like my chances much better than yours.

To drive the point home, unbeknownst to you, the casino has offered an even-money side bet to onlookers on whether you win or lose your spin. Since your probability of losing your bet on R-16 is 33/37, the Kelly fraction of the side bet is 33/37 - 4/37 = 29/37 = 78%. That's huge. A Kelly fraction of 78% is absolutely huge and the bet is against you, not with you. This is why expectation by itself is meaningless.

 

[nonlinear5]

Here's a shocker: it does not matter what your freakin' net worth is.

If your utility function is the log of the bankroll, then the net worth does not matter. If your utility function is the log of the net worth, then the net worth does matter.

Because of the betting constraint, all you can maximize is your bankroll ($1000 initially).

That's incorrect. The utility function of the log of the net worth is totally legitimate, even in the presence of the bankroll constraint.

 

[kut2k2]

If your utility function is the log of the bankroll, then the net worth does not matter. If your utility function is the log of the net worth, then the net worth does matter.




That's incorrect. The utility function of the log of the net worth is totally legitimate, even in the presence of the bankroll constraint.

You've ignored a large majority of my post. Did any of that register with you?

Also, the thing to be maximized is not log($), it is log(rate of change of $). If your $ goes to zero (always a possibility), then log($) goes to negative infinity, which is meaningless.

 

[nonlinear5]

Also, the thing to be maximized is not log($), it is log(rate of change of $).

That's incorrect, as well.

 

[kut2k2]

That's incorrect, as well.

Prove it.

 

[nonlinear5]

Prove it.

Look it up. I will not argue with you until you get the fundamentals right.

 

[kut2k2]

Look it up. I will not argue with you until you get the fundamentals right.

Are you kidding me? I'm done indulging your pathetic ignorance of the math behind optimal geometric growth.

http://www.elitetrader.com/vb/showpost.php?p=1577681&postcount=1

 

[markettimer]

Also, the thing to be maximized is not log($), it is log(rate of change of $). If your $ goes to zero (always a possibility), then log($) goes to negative infinity, which is meaningless.

These problems are equivalent.

A few posts earlier, I showed how max E[log(W(N)/W(0)] is the same problem as max E[log(W(N)]. Here I'll show that this is the same as maximizing the expected compound growth rate.

Note that W(N)/W(0) can be rewritten as (1+CGR)^n, where CGR = compound growth rate.

Therefore, by maximizing E[log(W(N))], we are also maximizing E[log((1+CGR)^n)] = nE[log(1+CGR)], which is the same as maximizing E[log(1+CGR)]. You are likely more familiar with this version of Kelly, but it is mathematically the same as maximizing the expected log of wealth.