The Kelly Criterion Revisited

[heavy]

Yeah, we've talked about it before but it deserves another mention. The Kelly Criterion. Those of you familiar with the Kelly Criterion probably first discovered it while betting the ponies. The Kelly Criterion sprung from the work of John Kelly at AT&T's Bell Labs in 1956. His original formulas dealt with long-distance telephone transmission signal to noise ratios. Exciting stuff, right? But the gambling community quickly grasped that Kelly's approach could be used to help calculate the optimal amount to bet on a horse race.

Essentially, the Kelly is an "up as you win" wagering system designed to maximize the expected growth of the player's bankroll. This is accomplished by adjusting the size of the bet to an optimum level based on the player's advantage or positive expectation. It assumes that the player only places action when he has a positive expectation.

In horseracing, this is based on the player's own handicapping of the race. Of course, the Kelly has also been adapted for other "bets," including blackjack, sports betting, and playing the stock market. Pro blackjack players often talk about wagering Kelly-based units. They speak of betting a full Kelly, although most of the guys I know go with a partial Kelly wager. Does the Kelly have any application in casino craps? Well, the answer to that is - depends. It depends on your advantage - and how consistent it is.

Let's walk through a simple example first, then I'll show you how to determine just what fraction of your bankroll should be wagered based on the Kelly.

Let's say you have $100 bankroll and are skilled enough at tossing the dice to win on the point of ten fifty percent of the time. For the sake of this example we'll bet $20 - or 20% of your bankroll on the ten. Winning bets will be paid at 2-to-1. We'll ignore the commission for now just to keep it simple. If you win this first $20 bet, you win $40, increasing your bankroll to $140. How much is your next bet?

Based on Kelly's principles, you would again bet 20% of your bankroll. Remember, that percentage is dictated by your advantage on the ten - not by your bankroll. But since your bankroll increased to $140, you're next bet is larger than your initial bet. The next bet is $28. Now imagine you win again employing the same strategy. Your wager of $28 would win $56, increasing your total bankroll to $196. That's a very nice two-win increase over your initial $100 bankroll - almost doubling your buy-in.

But what do you do on a loss? Simply put - the same thing. Let's say you continue this strategy and bet 20% of $196 - or $39 on your next hand. If you lose that bet your total winnings slip back to $157. That's still $17 more than you had after your first win. If you continue to play, your next wager will still be 20% of your bankroll. But since your bankroll is now $157, your bet size is reduced to $31. Now let's assume you lose that wager as well. You still have $126 in the rack - a net $26 win for the series. Note that you had two wins and two losses for the series, but are still dollars ahead. That is the power of the Kelly.

Most people who play the Kelly base their progressions on the saying, "Bet the fraction of your bankroll that equals your percentage advantage." However, this saying assumes you are playing at an advantage on even-money wagers only. Like my blackjack pro pals, I think a better choice would be to bet a fraction of your bankroll equal your percentage advantage divided by the "to" odds on that wager.

For example, if you have a 20% advantage on the nine and you are getting 7-to-5 odds, then the fraction of your bankroll to bet is (0.20)/1.4 = 0.14 =14%. Of course, if you toss out 14% of your bankroll on a single number, lose, then post about it on the craps forum - a certain fat guy we all know and love will probably say you over-bet your bankroll. But that's a topic for another day.

The Kelly Criterion is a recognized money management system used extensively in horse and sports betting, and in the stock market. Whenever the question of optimal betting size pops up in handicapping or money management books, you usually see Kelly formula mentioned. It's a great "up as you win" way to play - provided you put the tracking time in to know where your advantage is. Therein lies the rub.

[Mad Professor]

There were a couple of issues that Heavy graciously left the door open on, so I'll mention them.

~The first being Risk-of-Ruin. Without getting into the sordid details of that (because it contains the much-dreaded "M" word {math}, R-o-R takes into account not only the pos-ex advantage of your bets, but also the frequency at which they hit, and of course a conservative-but-profit-tractive bet-size-to-bankroll ratio...with an eye to avoiding total bankroll ruin while at the same time, optimally growing it.

~The second being that there are common-sense limits that have to be placed on your bet-size-to-bankroll ratio because of the aforementioned VOLATILITY. For example, just because you have, let's say, a 14% advantage over a given wager, doesn't mean that your positive edge will gloriously manifest itself 'on-time and on-schedule' on every bet that you make...just as not every horse that you bet on, nor every stock that you buy, will always turn into an instant winner.

To think otherwise would be preposterous.

So when you have an advantage...even a sizeable one...you are STILL going to have to put up with a fair amount of toss-to-toss, hand-to-hand, and session-to-session volatility.

The question is, do you have a strong enough constitution to maturely handle it, and is your bankroll large enough to endure it?

If not, prepare for little to no bankroll-growth, but lots of tractionless "I-know-I've-got-the-skill-but-I-can't-seem-to-get-ahead-and-stay-ahead" frustration.

MP

[Mad Professor]

Your EDGE Determines Your Bankroll Requirements

…and conversely; your bankroll determines how well you’ll be able to exploit any D-I edge you have developed.

~If you try to use too large of a percentage of your bankroll trying to maximize your profit on a bet where you have too little edge; then there is very little likelihood of emerging with a net-profit even though you’ve meticulously documented and confirmed your advantage.

Clearly, your edge and your bankroll not only go hand in hand, but they are also so closely tied together, that you have to structure your bets in symbiotic lock-step with your current bankroll.

Ø Failure to tie them together (in both the breadth and the depth of the bets that you make); pretty much guarantees your failure as an advantage-player even if you shoot the dice with an unbelievably large edge.

Let’s face it, most players are usually under-capitalized when they go to the casino, and still more have unrealistic profit expectations on the money that they do bring.

It is in those unrealistic expectations that modest profits are needlessly squandered and often times ignored in the temporarily blind pursuit of impractical win-objectives.

Ø Your edge over the casino determines how much of a chance you have in terms of coming out on top, and how much money you will likely need to reasonably do so.

Ø Bankroll size is a critical factor in determining which betting-methods (Flat, Regress, Press, etc.) and specific bet-types (PL w/Odds, Come, Place, Field, Props, etc.) are best suited to your profit objectives and risk-tolerance.

Most players let greed rule their decisions instead of common sense, so they ignore that ratio...and as a result, most of them have trouble surviving the casino battles.

If you have ever wondered why so many players come to these boards and then fade away (even though they get good enough at the physical part of dice-influencing); it's largely due to over-betting their bankroll and not being disciplined enough to stick to the bets where they have a confirmed advantage…and to ignore all of the ones where they don’t.

You may want to re-read that last paragraph again even though I’m not specifically talking about you.

HOW MUCH of a bankroll you have available to fund your advantage-play wagers should determine the breadth, depth and size of all the bets that you make.

If you don't have the money to comfortably make an Initial Steep Regression, then your gaming approach has to be modified down to a point where it is comfortable and you have just as good of a chance of winning; or better still...

Ø A wise advantage-player will wait until he accumulates a properly sized bankroll (in order to make the bets that his skill-based talent indicates he has the strongest advantage over); and he’ll stay away from the casino until he does.

I’m going to tell you something that most players don’t want to hear…


Insufficient Bankrolls Produce Insufficient Results

If you play without a sufficient amount of money; then there is an incredibly good chance that you won’t emerge from your sessions with a net-profit that is reflective of your current abilities.

Worse still, an insufficient bankroll often means that your skill won’t even get a fighting chance to produce a net-profit before it is battered and shattered despite the fact that you have a substantial edge over the house.

In other words…

If you short-change your bankroll, you short-change your chances of winning.

So what constitutes a “sufficient” bankroll?

By using the Kelly Criterion to properly size our bets, it allows us to use what I would characterize as a "balanced approach". It's one which recognizes our desire to profitably exploit our edge over the house, and blends it with the need to preserve and grow our bankroll while avoiding risk-of-ruin.

As I explained in my book, "The Mad Professor's Crapsshooting Bible", your edge-per-ROLL on most bets will be about one-quarter to one-third of your edge-per-HAND; or expressed the opposite way, your edge-per-HAND will be three to four times higher than your edge-per-ROLL (except on one-roll prop-bets where the two figures will be the same because those types of bets get resolved one way another with each new outcome).

As such, your Kelly-based bet-sizing is based on your edge per-ROLL, however as you know, there are limits as to how much of your total bankroll you should reasonably expose to any given wager, as well as limits on how much of your total bankroll you can reasonably apply across multiple wagers at the same time.

If you have, let's say, a 5% edge-per-HAND; then it probably equates to an edge-per-ROLL of between 1.25% and 1.66%, so that lower per-ROLL percentage is the amount of your total bankroll that you can reasonably wager on that bet.

However, if your advantage over the house reaches above the 2.5% edge-per-ROLL threshold; then you should restrict your 7-exposure on that one bet to 2.5% of your bankroll.

Likewise, if you have strong advantage over multiple wagers which each exceed 2.5% edge-per-roll; then the maximum cumulative amount that you should reasonably expose to a 7-Out would be capped at about 3.2% of your total gaming bankroll.

In other words, when you have a positive edge over multiple wagers (even if they all individually exceed 2.5% edge-per-roll); then you should still restrict the cumulative multi-wager 7-exposure amount to about 3.2% of your total bankroll in order to maintain that balance between profitably exploiting your edge, against the need to preserve and grow your bankroll...all the while avoiding risk-of-ruin.


Your EDGE Determines Your Bankroll Requirements…and conversely; your bankroll determines how well you’ll be able to exploit any D-I edge you have developed.

If you short-change your bankroll, you short-change your chances of winning.

MP

[heavy]

One thing missing from the above is how to determine how much bankroll you have. The true AP considers his bankroll to be the total amount of money he is willing to LOSE over the course of his lifetime of play in order to have a fair shot at a win. If a player is willing to lose - say $300 a month and plans to play for the next 30 years then his "bankroll" is $108,000. That doesn't mean you HAVE $108K in the bank today - it simply means you're willing to risk that much over the long run. Remember, you're willing to lose $300 a month on this scenario. I don't have time to run through a complete example this morning, although I have done it in the past (I think on the old forum). I probably have an example on the home computer, but I'm not at home at the moment. More later.

[Maddog]

Another good post Heavy!
In your example you use a 20% "advantage" derived by tracking. Could you direct us to where this "advantage" number is displayed in BoneTracker?

“Advantage” in this context applies to the notion of betting. Advantage means that given an initial bet of “Z”, with a probability “P” of winning an amount “X” and a probability “Q” of losing an amount “Y”, there is a positive expectation of winning at a rate of “A” over time. The most common equation is written as:

Advantage = ((probWin*amountWon) + (probLoss*amountLost))/amount bet

For example, the bet “Place 8”

-1.515% = ((.4545)×$7) + (.5454×(-$6)))/$6

What makes this formula difficult is that “ProbWin” and “ProbLoss”, because those values represent the probability of rolling an 8 before rolling a 7, and the probability of rolling a 7 before rolling an 8, respectively. Many people incorrectly plug in the 5 ways to roll an 8 vs 36 possible results (5/36) and use that as the probability for the 8. That is correct for one roll, but the Place 8 is not decided on a single roll.

And of course this gets more complicated as one combines multiple bets.

The point I’m trying to make, is that if you intend to determine your “Advantage” you must do so in the context of the BET that you intend to make. If you will make many bets and/or many varying bets as the situation warrants, then you must determine your advantage for each situation. Or as an alternative, establish an estimated advantage based on the amalgamation of all bets that will be played.

With this background in mind we can get back to the original question of “where is the advantage number displayed in BoneTracker”.

The simple answer is that BoneTracker does not show a specific “Advantage” number, as that would require BT to have a notion of specific BETS on which to base the advantage formula. Instead the focus of BoneTracker is to provide the DI with a tool that tracks your practice results in order to establish a baseline and a benchmark of one’s ability to execute a controlled toss and impart influence on the dice roll outcome.

The more complex answer is that on the Stats page BoneTracker is keeping track of the probability of rolling each number and with this information; one can then perform the calculation for the specific bet in question.

The DICETOOL integration does some of this calculation for certain common bets and can be seen on the DiceTool tab. Keep in mind however that DiceTool performs a specific approximation based on Stanford Wong/MadProfessor’s Foundation Frequency Model. This has been documented several times in several threads. Also keep in mind the importance of multiple reasonable sample sizes when one is determining current tossing skill based advantage. In other words you cannot to a few hundred throws and believe that those results represent your life-time advantage.

[Mad Professor]

Bankroll: It's About You and Your Money

It is important to determine just how much money you are willing to put into your dice-influencing pursuit.

Traditionally, craps players have defined their "bankroll" as the amount of money that they use to buy in when they walk up to a crap table. Unfortunately that isn't the true measure of your bankroll, nor is it the true measure of how much money you are willing to venture on dice-influencing; so let's clear up some definitions.

~The money that you use to buy in at the table is called your "session stake" or "session buy-in.”

~Your "total bankroll" is a much larger amount, and is directly tied to both your overall financial capability and your casino-gaming intentions.



Defining Bankroll, Defining Your Intentions

If your bankroll isn't your typical session buy-in, then what is it?

If you want use logic to determine the proper amount of money to wager at craps (or any other investment where you accept a chance of losing in order to have a chance of winning); then you need to determine the total overall size of your bankroll.

Therefore, recommendations for bet sizing should rightly be expressed as a percentage of your bankroll as well as being tied directly to the actual positive edge that you have over certain wagers.

Suppose your bankroll is twice the size of mine. If we are playing dice together, then your percentage edge over the house is the same as mine for any given bet. We should both be betting on the same things; however, your bets should be twice the size of mine.

Whatever I win, you will win twice as much; whatever I lose, you will lose twice as much.

In its simplest form, then, the totality of your bankroll should be defined as the amount of money that, if you lose it, you will stop betting on craps.

To determine your bankroll, you have to look at your dice-influencing intentions.

~Your bankroll for the purpose of playing craps, is only loosely tied to your personal wealth. Being a millionaire does not mean your bankroll is a million dollars.

For example, if your net worth is a million dollars, but losing $10,000 at craps would cause you to pull away and not risk any of the other $990,000 at the tables, then your bankroll for the purpose of properly sizing your bets at craps is $10,000.

Though this at first seems far removed from the traditional gambler's definition of bankroll, it is important to understand that the objective behind dice-influencing is to remove as much of the gamble as possible from the process and replace it with positive-expectation wagers.

The casinos themselves are in the gambling business, but they don't gamble. They know that their edge over the player will redeem itself. Although they won't win every wager that they book, they'll still show weekly, if not daily, net earnings from each game. Their win-rate will vary from day to day, but it will almost always be on the positive side of the ledger.

There are some rare exceptions, but generally this holds truw more than 99% of the time.

Advantage players are in the gambling business too; but like the casinos in which they ply their skilled-trade, their wagers aren't gambling in the broad definition of the word.

Rather, they are in the advantage-play business; they have a validated edge over certain bets, and that is where the majority of their betting-weight is (or should be) placed.

They know that if they stick mostly to bets where they have a verified edge over the house, their skill will redeem itself. Although it won't manifest itself with a win on every day that they play, they'll show a long-term profit from their geared-to-skill-and-bankroll wagering.

Likewise, their earnings rates will vary from month to month, but generally if advantage players stick to properly sized positive-expectation wagers, earnings are going to be on the positive side of the ledger.

In forcing yourself to take a brutally frank look at the total amount of money that you'd reasonably be willing to venture on dice-influencing, you have to answer some tough questions.

You have to be honest with yourself. You might think you are wealthy enough to invest $20,000 in craps, but you have to ask yourself if you would stick with crapshooting if your first few trips go badly and you fall $5000 in the hole. If the answer is no, then your bankroll is nowhere near $20,000, and is probably somewhat less than $5000 too.



Determining Bankroll Size

Here's an exercise that will help you determine an appropriate bankroll size: pick an acceptable loss. That is, pick an amount of money that if you lost it, you know for sure you'd still be willing to make another dice trip.

~Say, for example, you pick $1000. That means if you lose $1000 on one trip you would not be happy, but you'd still put together another $1000 to make another casino visit to play craps.

~Now try to imagine having a series of losing trips where you drop your acceptable-loss amount on each trip. Suppose, for example, you stick with that $1000 as an acceptable loss per trip. If you visited casino dice tables three times and lost $1000 on each trip, putting you a total of $3000 in the hole, would you go back for a fourth trip?

~If yes, and it happens again, would you go back for a fifth and sixth and seventh trip?

~Continue that procedure until you have settled on a total-loss amount where you are no longer certain you'd make another trip to toss dice.

That amount is your total bankroll.

You have to be very definite about your game-quitting amount. Obviously, too, you have to keep accurate track of your wins and losses, to know where your bankroll stands at any given point in time.

Of course, it is up to you to decide what part of any increase in your wealth will be used to augment your bankroll.

~Money you win at craps is generally assumed to be added to your bankroll, but that does not have to be the case. It's more important that your bankroll at any given time be the maximum amount of money you could lose without giving up craps. That maximum acceptable loss does not have to be equal to your starting bankroll plus the amount you have won at craps.



What Bankroll Is and What It Is Not

So, if your total bankroll is the amount of money that would cause you to give up craps if you lost it, let me tell you what it is not:

~It is not the amount of cash you have in your wallet when you visit a casino.

~It is not the amount of money you have in a bank account. Your bank account can be more than or less than your bankroll. However, some players do set up a separate bank account for their gambling needs. If the amount in that account is the amount that if you lost it, you would give up craps; then, yes, you can define that as your bankroll.

~Your bankroll is not a fixed percentage of your income, except by coincidence.

~Your bankroll is not the amount of your personal wealth, though it probably is smaller than your wealth.

~If your current personal wealth is modest but you have a large income, and you are willing to finance future casino trips out of future earnings if need be, then your bankroll for the purpose of sizing your bets could well be larger than your current personal wealth.

By the way, if you aren't willing to define just how much you'd be willing to lose before abandoning your dice-influencing efforts; then there's a possibility that a gambling problem is hiding just below your "I'm an advantage play dice influencer" facade.

Many crapshooters who make the transition into the advantage-play world of dice-influencing find they have difficulty in treating money with the respect that a pro does. They have a hard time shedding the disrespect that they treated money with in their random-gambling days. When you get serious about your dice-influencing skills, you also have to get serious about your money.



The Bigger Your Bankroll, the More You Can Win

While determining the size of your bankroll requires focusing on the possibility of losing, keep in mind that there is a benefit to having a larger bankroll:

~The larger your bankroll, the bigger you can bet.

~If your dice-tossing skill gives you an edge, then the bigger you bet, the more money you will win.

Those two last items may seem like obvious and timeless verities, but you would be surprised at the number of dice setters who can't or won't look at the game of craps in quite such a business-like manner.

That's okay if you use your casino adventures for entertainment rather than making money. It is important, however, to understand that the longer you continue to use the game for its entertainment value, the longer you delay and retard your dice-influencing profitability.

If you call your bankroll $500 when in fact it would take a loss of $5000 to knock you out of craps, then your future dice winnings will be only a fraction of what they could be, or should be.

Of course, your future dice losses will also be a fraction of what they could have been, so that is a valid reason and incentive to choose a small number as your bankroll size. Your bankroll-determining decisions come down to your own comfort level and personal financial situation.

Having an edge means your future dice wins are expected to overwhelm your future dice losses. Bet sizes that properly balance future wins and future losses therefore are the keystone to doubling and re-doubling your current total gaming bankroll and not just doubling a buy-in or two..

The validity of all this advice for you depends on the quality of the number that you select for your bankroll.

~If you select too large a number for your bankroll, then my proportioned-to-advantage betting advice will put too much weight on the possibility of future wins and give too little consideration to future losses.

~If you select too small a number for your bankroll, then my proportioned-to- advantage betting advice will put too much weight on future losses and too little on future wins.

~Under-estimating your bankroll amounts to giving too much importance to the losing side of the equation without giving the winning side a fair enough chance to succeed and prosper.



Growing Your Bankroll

You can recalculate your bankroll any time. As your dice-influencing skill grows, you might be willing to risk more money on craps, and therefore be willing to increase the maximum loss you are willing to accept. From that point forward, that new amount will be your bankroll. Ultimately, what growing your bankroll means is being willing to accept a larger loss on your way to making larger and larger gains.

Some players dedicate a high percentage of their profit at craps to the overall amount that they'd be willing to lose. As a result, they are able to increase the base amount of their proportioned-to-advantage bet-levels, so that their dice-influencing edge is able to increase their earnings-rate at a faster clip.

This bankroll-growth and loss-tolerance re-assessment cycle allows you to build steadily expanding dice-influencing revenue for yourself.



Your Profit Is Your Money

The profit that you have struggled to make can disappear quickly. That's because the edge you have over this game as a dice-influencer is often not that large.

~A small edge means that it can take time to build up a profit, but which can also be wiped out in a minutes.

The way to handle that situation is to select your bankroll-size properly.

If you are not willing to lose it, then don't call it part of your bankroll.

Once you win any money, it is yours, and it is no longer the casino's. It is yours to pay taxes on, and yours to treat the same way you would if had been yours in the first place. You earned it, and you are worthy of it. You are not just a temporary custodian of it.

Whether the money you just won increases the size of your bankroll is up to you.

~Increasing the size of your bankroll will justify betting bigger, but that's not required. Betting bigger means winning more when you win, but it also means losing more when you lose. You can devote only part of your winnings to increasing the size of your bankroll if you want. Any winnings that you do not make part of your bankroll, augments that portion of your wealth that is not bankroll for craps.


MP

[Jonah]

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[Bankerdude80]

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