Southpaw
Well-Known Member
The Replenishable Bankroll
What is a replenishable bankroll? To answer this question, it may be easiest to first describe what isn't a replenishable bankroll. In this case, I will describe Anthony Parker (AP) who does not have a replenishable bankroll. AP has saved up 250k through working as a physician, but has now quit his job, so as to pursue the more vicarious lifestyle of an advantage player. He is not eligible for social security, and his wife that worked as an accountant divorced him after being told of AP's aspirations. Therefore, AP has 250k, but no income other than what he will make as an advantage player. AP's bankroll is not replenishble--he has no outside income to continue to build his bankroll while riding the ups and downs of advantage play.
To the contrary, I will describe Ben Campbell (BC) (not the one from 21 of course) who has a very replenishable bankroll. BC graduated from pharmacy school three years ago and is currently employed as a pharmacist where he makes 104k per year. He receives a check for 2k every single week of the year. BC is young and has not yet settled down yet--he does not live a luxurious lifestyle, is unmarried, has no kids, and lives in a relatively cheap apartement. BC has calculated that he can live quite comfortably with only $500 of the 2k check that he receives every week. Over the past three years working as a pharmacist, BC was able to save up a 10k bankroll that he intends to use for advantage play. (He would have been able to save up a lot more had he not been paying off his school loans).
So, BC's bankroll situation is clearly a bit different than AP's; whereas AP has a fixed 250k to work with, BC has a bankroll of 10k that can be replenished by 1.5k every single week--BC's bank is an example of a highly replenishable one.
Traditional Kelly Criterion would have BC only betting in proportion to the 10k that he has saved up, but the fact of the matter is that he can afford to play significantly higher stakes. (Let us assume that BC is well versed in advantage play, and his stomach has already become tolerant of gut-wrenching losses). So much of the rhetoric out there regarding bankroll management is better suited for the case of AP where the bankroll is not replenishable. The fact of the matter, however, is that very few players today are playing with a non-replenishable bankroll. Were BC only to play against his saved up bank of 10k, he would hardly be maximizing his utility.
We can say that BC is playing against a "virtual bankroll" if you will, and to maximize his utility, he should be playing against this virtual bankroll, rather than his smaller saved up bankroll of 10k. The next appropriate question is how do we calculate BC's virtual bankroll?
The following statements describe the nature of a virtual bankroll with respect to a few variables:
*The size of the virtual bankroll is proportional to the actual or saved up bankroll.
*The size of the virtual bankroll is proportional to the EV to be had over time period "t". (The more you are expected to win, the less likely your actual bankroll is to tap out out when playing your virtual bank at a given RoR).
*The size of the virtual bankroll is inversely proportional to the SD from the EV over time period "t".
*The size of the virtual bankroll is proportional to the replenishability of the bankroll over time period "t."
The most interesting conclusion to be had from the following statements is that your virtual bankroll is a function of how frequently you play. In the case of BC, the replenishability rate is a fixed 1.5k per week. However, what is not fixed is how often that BC plays. The more frequently that BC plays, the larger his SD per week (per time "t") will become, while the replenishability rate per week (per time "t") will remain at 1.5k per week. The consequence of this is that if BC plays less frequently his virtual bankroll will be higher, subsequently allowing him to play for higher stakes.
The next question to be asked is how much should you bet when you are playing on a replenishable bank. Let us take the case of Ben Campbell who has a bankroll of 10k that is replenishable by 1.5k per week. Let us further assume that Ben Campbell plays the game that under ideal conditions yields 3.48% advantage. However, the game he is playing is not ideal. The pay-table is 541 and is dealt at 40 hands per hour, but he gets a perfect-read only 80% of the time, ace/face/other 5% of the time, face/other 5% of the time, and no read the remaining 10% of the time. The EV/hour under these conditions is 0.98 units, whereas the standard deviation per hour is 10.96 units. Assume that BC will play this game for 5 hours per week, and is willing to accept a risk of full kelly (RoR = 13.5%). (It is hard to say what an acceptable RoR is for a replenishable bank. Obviously, it should be lower than for a fixed bankroll that can be replaced, but how much lower? For instance, even if BC taps out his replenishable bank, he will be out of action only until he gets his next paycheck. But then again, you have to consider the stability of one's income source).
Again, the question becomes, "How much should BC bet assuming the above parameters?
I will derive a formula that will indicate the proper betting-level, given the information above.
(I deeply apologize for how hard these formulas are going to be to read. I first did the derivation using equation editor in MS Word only to find that Google Docs will not show equations made in equation editor. Needless to say, the equations were much easier to read when using equation editor. I suggest following along using a pad of paper.)
I will begin by working with a formula provided by Don Schlesinger on page 140 of BJA II that is used to find the appropriate bankroll when RoR, EV/time and Sd/time are known:
B= - (σ^2/2E) lnr
Where B equals Bankroll, σ equals standard-deviation per hour, E equals EV per and r equals RoR.
The first operation to be done is to incorporate the bankroll’s replenishability into the equation. It is perfectly acceptable to consider the replenishablility as EV. Take the case of BC as an example. If he plays 5 hours per week, and his bankroll replenishes 1.5k per week, then we can say his bank replenishes by $300/hour of table time. Replacing hourly EV (E) with the sum of the hourly win-rate (R) and the bank’s hourly replenishability (P) yields:
B= - (σ^2/(2(R)+P)) lnr
This equation does not include the wager-size (W), which is what we are looking for. However, hourly standard-deviation (σ) can be rewritten as the product of the wager-size (W) times the standard-deviation in units (σ_units), and the hourly win-rate (R) can be rewritten as the product of the wager-size (W) and the win-rate in units (R_units) . Making these changes yields the following:
B= - (〖((W)(σ_units ))〗^2/(2(W)(R_units )+P)) lnr
The square of the product is equal to the product of the squares, so the numerator becomes:
B= - (〖(W)^2 (σ_units )〗^2/(2(W)(R_units )+P)) lnr
Multiplying both sides of the equation by 2(W)(R_units )+P yields:
2B(W)(R_units )+BP= -(〖(W)^2 (σ_units )〗^2 )lnr
Moving all items to the left side of the equation yields:
(〖(W)^2 (σ_units )〗^2) lnr+2B(W)(R_units )+BP=0
Wager-size (W) is what we are ultimately looking for. The formula at hand is a quadratic. Wager-size can be solved for by using the quadratic equation. (Note that I removed the plus/minus sign typically found in the quadratic equation and replaced it with a minus sign because using plus yields a negative monetary sum, which is irrelevant for our purposes):
W= (-2B(R_units )-√((〖((2B)(R_units ))〗^2)-(4) ((σ_units )^2) (lnr)(BP)))/((2(σ_units )^2) lnr)
Again, to make the final product a little prettier to look at, the square of the product is equal to the product of the squares:
W= (-2B(R_units )-√((4B^2 )((R_units )^2)-(4) ((σ_units )^2) (lnr)(BP)))/((2(σ_units )^2) lnr)
Where W equals wager-size, B equals actual bankroll, R_units equals win-rate per hour in units, σ_units equals the standard deviation per hour in units, r equals the accepted RoR (if accepted RoR was 13.53%, r would equal 0.1353), and P equals the bankroll's replenishability per hour played (See Bold Text Below).
Those of you that have been skimming through this document and did not read the derivation may be a bit confused as to what P, the bankroll's replenishability per hour played is. To help explain this concept, let us take the case of BC. If he plays 5 hours per week, and his bankroll replenishes 1.5k per week, then we can say his bank replenishes by $300/hour of table time--Thus P = $300.
Well, now that we have derived this equation, let us put it to work. Let us take the case of BC one last time. Recall that Ben Campbell has a bankroll of 10k that is replenishable by 1.5k per week. Let us further assume that Ben Campbell plays the game that under ideal conditions yields 3.48% advantage. However, the game he is playing is not ideal. The pay-table is 541 and is dealt at 40 hands per hour, but he gets a perfect-read only 80% of the time, ace/face/other 5% of the time, face/other 5% of the time, and no read the remaining 10% of the time. The EV/hour under these conditions is 0.98 units (R_units), whereas the standard deviation per hour is 10.96 units (σ_units). Assume that BC will play this game for 5 hours per week, and is willing to accept a risk of full-kelly (RoR = 13.5%, thus r = 0.135). We calculate that P the bankroll's hourly replenishability is equal to $300 because BC's bank is replenished by 1.5k per week and he gets 5 hours of table time per week.
Plugging into the equation (do it on your own to double-check your's and my math!), you find that BC should, ideallly, place $159.62 as his ante.
Now, I know there will be some gawking regarding playing a virtual bankroll at an RoR of a whopping 13.5%. Thus, I decided to do the calculation for when the RoR is only a measly 0.1%. It turns out that while playing at 1/135 the risk, BC is still able to place an ante wager that is virtually half (~$73) what he would have been placing whilst accepting 135x the risk! If that isn't interesting, then I don't know what is!
Anyways, I hope you enjoyed the above discussion of bankroll replenishability. Furthermore, I hope that some may find use of the equation derived here. Lastly, I'd be forever grateful if some of the other math guys here double-checked my work; I'd hate to be the one that led rookies to overbetting their replenishable banks.
Spaw
