Question: Math Boy's Cumulative ROR Article

Link to article: http://www.bj21.com/bj_reference/pages/riskofruin.shtml

MB wrote:

"While it may be very liberal, say he is playing via half the Kelly criterion without resizing. Then the probability of doubling a bank before going bust (RoR) is about 1.8%."

If a player is using 0.5 kelly, then wouldn't the probability of doubling the bank before going broke be 98.2%?

MB wrote:

"1) CRoR=1-(1-RoR)^n.

Here, CRoR stands for cumulative risk of ruin. Unfortunately, this means that that the probability of not surviving 5 years in the above scenario is (1-13%)^10=17%. The probability of not surviving 10 years is (1-13%)^20=31.1%".

First, where does MB come up with 13% for ROR? Second, (1-13%)^10=25% , not 17% as MB wrote. Same goes for (1-13%)^20, it equals 6%, not 31%. Somebody please clarify, this is an interesting article.

MJ

is your risk of ruin betting full kelly without resizing after losses.

You wrote: "13% ROR is your risk of ruin betting full Kelly without resizing after losses."

What you meant to write, at the end, is "without resizing after losses OR WINS."

Don

MB stipulated 0.5 kelly and also made it clear that the counter could double a 20k bank twice in a year. Unfortunately, the 40k winnings are used to pay expenses. If expenses = earnings, then ROR becomes 100% in the long run.

The only way it could be full kelly is if expenses were 1/2 earnings and the counter did NOT stop prematurely when he doubled the bank. In other words, he would play until he either went broke or won all the money in the world. But these are not the parameters we are given!

MJ

Full Kelly ROR is 13.5%, as to the rest of the post, You'll have to hope Mathboy ventures to the free pages.

You may have found them. The big ideas are what counts. When you are dealing with ruin, the exact number is not really important. Say a plan has a 1.5% RoR versus a 1.4% RoR. It doesn't make much difference. What is important is that you are consistent with your measurements and know the order of magnitude of risk you are taking.

Math Boy

Math Boy,

Please reread what I wrote in my 2 posts for this thread. I am not disputing minor discrepancies, but what appear to be major errors in your figures. You claim 13% ROR when it is more like 100% ROR. Then, your calculations yield incorrect figures at the beginning of your article. I explained all this.

Reread what I wrote in both my posts and then let us know if these are MAJOR mathematical errors on your part.

MJ

MB wrote:
"1) CRoR=1-(1-RoR)^n.
Here, CRoR stands for cumulative risk of ruin. Unfortunately, this means that that the probability of not surviving 5 years in the above scenario is (1-13%)^10=17%. The probability of not surviving 10 years is (1-13%)^20=31.1%".

First, where does MB come up with 13% for ROR? Second, (1-13%)^10=25% , not 17% as MB wrote. Same goes for (1-13%)^20, it equals 6%, not 31%


Regarding the 13% RoR, you're correct. For half-Kelly without resizing, 1.83% should be used for the RoR instead of 13%. The statements (1-13%)^10=17% and (1-13%)^20=31.1% are certainly untrue and don't follow the CRoR=1-(1-RoR)^n equation given at the beginning of the post. However, the results of 17% and 31.1% are actually correct; MB simply wrote the equations incorrectly in his post.

Just replace those two equations with the following and all is well:

10 years: CRoR=1-(1-RoR)^n = 1-(1-.0183)^10 = 17%

20 years: CRoR=1-(1-RoR)^n = 1-(1-.0183)^20 = 31%

By the way, though (1-13%)^10 = 25% and (1-13%)^20 = 6% are true statements, they're inconsistent with common sense; they suggest that the CRoR decreases as n increases, which is clearly illogical.

David

By the way, though (1-13%)^10 = 25% and (1-13%)^20 = 6% are true statements, they're inconsistent with common sense; they suggest that the CRoR decreases as n increases, which is clearly illogical.

If you add the 1- to the beginning of each those equations, you get the correct CRoRs for full-Kelly of 75% after 10 years and 94% after 20 years. My point in saying that your "corrected" figures were illogical is that, since MB's original answers seemed at least reasonable, and the figures of 25% and 6% are clearly unreasonable, consider that the problem is more likely with the left side of the equations than with the right :-)

MB wrote:

"While it may be very liberal, say he is playing via half the Kelly criterion without resizing. Then the probability of doubling a bank before going bust (RoR) is about 1.8%."

If a player is using 0.5 kelly, then wouldn't the probability of doubling the bank before going broke be 98.2%?

Your numbers look reasonable. I probably screwed up here and wrote 1.8% instead of 1-1.8%.

"1) CRoR=1-(1-RoR)^n.

Here, CRoR stands for cumulative risk of ruin. Unfortunately, this means that that the probability of not surviving 5 years in the above scenario is (1-13%)^10=17%. The probability of not surviving 10 years is (1-13%)^20=31.1%".

First, where does MB come up with 13% for ROR? Second, (1-13%)^10=25% , not 17% as MB wrote. Same goes for (1-13%)^20, it equals 6%, not 31%. Somebody please clarify, this is an interesting article.

MJ

The 13% is an estimate of what your RoR is if you start out betting kelly on your bankroll and never resize. On the numbers, you are right. I probably edited the example and did not change everything. As some other posted has noted, sometimes I wrote the (1-r)^n number instead of the correct 1-(1-r)^n number.

MB stipulated 0.5 kelly and also made it clear that the counter could double a 20k bank twice in a year. Unfortunately, the 40k winnings are used to pay expenses. If expenses = earnings, then ROR becomes 100% in the long run.

The only way it could be full kelly is if expenses were 1/2 earnings and the counter did NOT stop prematurely when he doubled the bank. In other words, he would play until he either went broke or won all the money in the world. But these are not the parameters we are given!

Your writings above seem to be correct. I probably screwed up somewhere in the editing. I do remember that I was assuming expenses and profit taking were done continuously as hours were put in. People usually take them in chunks, but that is harder to model.

Math Boy

Thanks for your response. Everything you wrote makes sense. However, the one area of potential disagreement would be with your use of 0.5 kelly.

As I wrote above:
"MB stipulated 0.5 kelly and also made it clear that the counter could double a 20k bank twice in a year. Unfortunately, the 40k winnings are used to pay expenses. If expenses = earnings, then ROR becomes 100% in the long run".

Assuming the simple ROR formula, ROR should be 100% due to expenses. Our only saving grace is the fact that there is a Goal constraint (winning 20k). This constraint makes ROR < 100% as the experiment will end prematurely on those RARE occasions when we reach our Goal. I'm not really sure what the ROR would be given these parameters, but it is a helluva lot higher than 1.8%; perhaps even as high as 90%!

On another note, what did you think of the rest of this article? It seems like MB has found a way to double the bank indefinitely without increasing C-ROR. He has tamed the long run using mathematics. I will have to study this article much more when I get some time.

MJ

Thank you for the clarification. Interesting article! On a side note, you might want to go back and fix some of those typos to avoid confusion for future readers.

MJ

Frankly, I'm not sure what RoR is "correct"--I felt like I was speculating enough just trying to resolve some of the inconsistencies in the article :-)

As far as the article goes in general, I found it interesting, at least from a theoretical standpoint. Practically speaking, it confirms what we already know: don't spend too much relative to your income; the more you spend, the greater your risk; it's better to make a lot of money over a short period than to make that same amount over a long period :-)

David