Not never go broke. Your chance of going broken will always stay at 13.5%.Originally Posted by jimmybond007
Your RoR figure assumes you will never resize bets. If you will resize bets the RoR figure is still true but it is based on a false assumption so it doesn't apply once you resize you have a new RoR. For that matter with each change in BR a new RoR figure would apply to your current situation. The old figure would still apply to the original starting point until you resize bets.
RoR is a snapshot for any given time. It is true from that specific time perspective given the assumption that you always bet the same. If you lose half your BR your current time frame RoR while go up by a higher factor than double.
Playing full Kelly assumes resizing so your chance of going bust is almost 0 but your RoR at any given time frame is 13.5%. This is not a discrepancy because the assumption RoR is based on is constantly violated. Your chance of busting out if you don't resize your bets to maintain a 13.5% RoR is 13.5% (RoR by definition). If you resize your bets to maintain a constant current time frame RoR of 13.5% your chances of busting out is 0. When you can't make a bet due to table minimums you will still have some money left.
Simply, risk of ruin should not apply to a serious AP intending to grow his/her bankroll.
Any system, if able to freely bet a fraction of one's bankroll, will never go broke. This includes full Kelly, fractional Kelly, bet 50% every time, etc. Blackjack is an exception because (a) minimum bets prohibit the ability to freely bet fractionally, and (b) bets may be (frequently) made with a -EG, but are part of a larger +EG system.
The AP needs to realize when the system ("game") in which they are playing becomes -EG, they need to stop that system and develop another system which yields a +EG. This will happen before any zero bankroll situation. (Actually, AP should give up a +EG game when a better EG opportunity arises)
The most important calculation for the AP to make is not % risk of ruin, but what is the EG for the system he/she is playing for a given set of assumptions, and maximizing EG thru game choices, reduction of costs to participate, etc.
Another important calculation is the probability that his/her system may reach a point of -EG due to expenses not part of +EG play, such as frequency of cover bets, costs to participate, minimum bet, etc.
How is that CE?
From what I gather, CE allows one to rate their risk allowance that they are comfortable with, having CE at 0.5 means one is willing to take on risk that is close to half of their expectation.
Lets say there is a coin toss. The outcome is a provably fair 50:50 odds. The punter is given two choices: 1.) You can take 5 dollars no questions asked, or 2.) Call heads/tails and if the called face is up, you will get 10 dollars; otherwise you get 0 dollars.
If I take the 5 dollars, I have a CE of 0.5. Correct? For me, if I was a high risk punter, I would take the toss. If I was a low risk punter, I would take the cash.
CE changes based on ones perceived risk of a given outcome based on the available value and variance of any expectation(s).
So, how does this fall into bankroll growth? I must assume at this point that taking on a greater risk means a greater reward, but it also means smaller betting events leading to a higher N0, no?
First I said like CE, EG measures the "expected rate of bankroll growth over a given number of iterations". CE is Certainty equivalent which measures the certainty of BR growth.
No.
CE = EV - Var*(B/2), where B is your bet size.
So if either variance or bet is zero CE = EV which is 5. Both variance and bet are 0 in this case.
You should always take the cash in this instance. Why risk being below expectation when you can get the same EV at no risk? The certainty of BR growth is higher by taking the guaranteed amount since you know exactly where your results will be after a known number of trials. If you take the $0 or $10 option BR growth is less certain.
Because higher CE means smaller swings away from expectation which has you closer to your expected growth. In other words BR growth is more certain.
If you look at the CE formula you will notice that CE is maximized when the product of variance and bet size is minimized as compared to EV. Higher bets have higher EV in a plus EV situation but but if variance is high enough CE decreases.
Perhaps the way to put this in perspective for betting is to substitute EV with its equal B*Adv:
CE = B*(Adv - Var/2)
Last edited by Three; 04-21-2017 at 06:09 AM.
Like CE was more of a question. From what he wrote EG could be exactly EV which measures the expected rate of BR growth. Or a measure of certainty of BR growth.
It is useful to imagine expectation as the man walking state to his destination EV on a graph and results to be where a dog is that he has on a leash. Variance is a measure of how long that leash is. If you just take the certain $5 you are carrying the dog straight to your destination since variance is 0. If there is high variance the dog is on a long leash and could be anywhere within the range of the leash.
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