By Blackjack Rebel.

In my last article, High-Low (HL) count with 7m9c, I had suggested that Knock-Out (KO) is a better count for the shoe game than HL and I said I would not cover why in that article.  So I am writing this article to explain why the KO is a better primary count for the shoe game than HL is. For the two deck game still use HL as the primary count but for the shoe game, KO is much better.¹⁾EDITOR NOTE: This article was originally published on Oct 22, 2024.  It has been republished here to simplify the presentation and make it easier to understand and use.  The author is hopeful this new KO with TCRC article will draw interest and appreciates any comments or feedback.

This article will cover just the primary KO count along with a Table of Critical Running Counts (TCRC) for eight and six decks as an alternative to the High-Low (HL) for a shoe game. Some symbols used in this article are tc (or simply “t”) = true count, dp = decks played, dr = decks remaining, n = number of decks and crc = critical running count. 

The KO is the HL with the sevens counted as +1 instead of 0 as in the HL. The HL is balanced at zero and so counting the sevens as +1 creates the unbalanced KO with an unbalance of four per deck. Similar to the balanced HL, start the KO at zero at the beginning of each shoe.

A major problem with switching from HL to KO was how to get accurate true counts with the unbalanced KO. This is solved with the Table of Critical Running Counts (TCRC). Use the following TCRC to get the KO true count from the KO running counts and decks played for playing strategy variations and also determine how many units to bet based using the KO running count and decks played. There is no need to do any true count calculations. Just memorized the TCRC and the betting table included in this article for the six and eight deck games which has a simple pattern. For example for a KO true count of one the crc increases by three for each deck played and for a KO true count of six the crc decreases by 2 for each deck played and they all converge to the pivot of 24 for six decks and 32 for eight decks as decks are played. The formula for six decks is crc(t) = 24 + (t – 4)*dr and for eight decks is crc(t) = 32 + (t – 4)*dr. If t = 0 then crc(t) = 4*n + (t – 4)*dr gives crc(0) = 4*n – 4*dr = 4*(n – dr) = 4*dp.

The KO indices for strategy variations are basically the HL indices. Three notable exceptions (all with a dealer up card of Ten) where the KO index is less than the HL index (see KO Risk Averse Indices chart) are: stand on hard 14 v T if KO ≥ crc(7), surrender hard 14 v T if KO ≥ crc(2) and surrender hard 13 v T if KO ≥ crc(6).

In addition to the ease of use of the KO count system where true counts can simply be looked up in a TCRC (instead of a true count calculation that is required with the HL), the KO true count at true counts of three or more where large bets are made and accuracy in true count calculations for betting and playing strategy variations is very important, are more accurate than the corresponding HL true counts.

HL has a pivot of a true count of zero and KO has a pivot of a true count of four. At a true count of six, for example, HL is six true count points away from its pivot and KO is two true count points away from its pivot and so KO true count of six using TCRC is three times more accurate than a HL true count calculation of six. In this situation, if HL true count where to obtain the same accuracy as the KO true count, decks would need to be estimated to the nearest third of a deck and HL true count calculations would need to be accurately divided by decks to the nearest third of a deck.

Consider the following example which demonstrates the accuracy of KO true count estimates as compared to HL true count estimates when the true count is close to the KO pivot of a true count of four. In the case of splitting T,T v 6 the expected value index is five but the risk averse index is six. So for six decks and estimated dr = 3 then using the risk averse index, split T,T v 6 if HL ≥ 6*dr = 6*3 = 18 and split T,T v 6 if KO ≥ crc(6) = 24 + 2*dr = 24 + 2*3 = 30. Now suppose the actual decks remaining were four and not three.

• Splitting T,T v 6 with HL = 18 is splitting T,T v 6 not a HL true count of six but at a HL true count of HL / actual decks remaining = 18 / 4 = 4.5 so T,T v 6 is actually being split below T,T v 6 expected value index of five even though the risk averse index of six was used.
   
• Splitting T,T v 6 with KO = 30 is splitting T,T v 6 not at a KO true count of six but at a KO true count, using tc(KO) = 4 + (KO – 4*n) / dr, of KO = 4 + (30 – 4*6) / 4 = 4 + 6 / 4 = 5.5. Thus even though the estimated decks remaining was off by a full deck to the actual decks remaining, the KO true count was at 5.5 and so was still above the KO expected value index of five for splitting T,T v 6.

Accuracy of tc(KO) to tc(HL) at tc = 3, 4, 5, 6 for n = 6 decks:

1. KO = crc(3) = 24 – dr    three times as accurate as HL = 3*dr
    
2. KO = crc(4) = 24    exact as compared to HL = 4*dr
    
3. KO = crc(5) = 24 + dr   five times as accurate as HL = 5*dr
    
4. KO = crc(6) = 24 + 2*dr three times as accurate as HL = 6*dr

Thus KO is superior to HL for the shoe game because:

1. Power: KO is the theoretically most powerful level one count.
    
2. Accuracy: KO is more accurate than the HL at true counts of three or more. KO estimation to the nearest deck is sufficient.
    
3. Ease of use: KO with its TCRC or the simple formula crc(t) = 4*n + (t – 4)*dr is easier to use than HL true count calculations.

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By Karel Janeček.

I wrote the following article as a paper at Bradley University in May/June 1997.  I believe that it can be of an interest to readers of Blackjack Review magazine since it offers a solution to the problem of optimal betting for a given maximum spread.

The practical calculation was done by the spread sheet-based software SBA Calculator 4.0.  SBA Calculator worked with the output of the Statistical Blackjack Analyzer (SBA) software.  With the help of SBA and SBA Calculator one could analyze practically any available rules and conditions in blackjack and, for given maximum spread and tipping strategy, determine the optimal betting strategy for each count.          

I would also like to give credit to many of discussion participants on the BJ21 web page which did a lot of great work in this field — namely Brett Harris and Winston Yamashita to name two at least.¹⁾EDITOR NOTE: This article was originally published in Blackjack Review magazine in 1998.  A few minor edits were performed to reflect this.

Summary

In this paper I will analyze how to optimally wager one’s bets in a positive expectation game.  I will assume that a person has fixed wealth known with certainty.  The wealth does not necessarily have to be currently available cash, rather the present value of future cash flows which are assumed to be known with certainty.

An extension of this work could consider the optimal wagering in the context of a portfolio income.  The topic becomes more complicated in this case. 

I will assume a logarithmic utility of a fully rational person.  However, similar results could be obtained for any other risk averse utility function.  It is understood that a rational person is risk averse, and not risk neutral, nor risk seeking.  For example, as we show further, a risk neutral person would lose all his/her wealth in finite time with probability one.²⁾The term “probability one” means 100% probability.

In the last part of my paper I will analyze a practical application of positive expectation wagering — the casino game of blackjack.  In this game it is possible to get an advantage for a sophisticated player by a method of tracking cards called card counting.  In practical application I will also consider constraints for application of optimal wagering in practice, namely the constraint of maximum bet spread.

“The “Optimal Wagering with Bet
Constraints” analysis is of
major importance especially
to people who are interested in
playing positive expectation games.”

Player’s Utility

A common misunderstanding is that the ultimate goal of a player is to maximize his/her expected win.  This very false idea is equivalent to risk neutrality of the player.  If one were to maximize expectation, one would bet his/her entire wealth on any (arbitrarily small) advantageous situation.  Obviously, with this method one would eventually lose the entire wealth providing that the game has a non-zero probability of a loss.

The proper player’s utility maximization (rather than expectation maximization) betting for a risk averse player will always be a certain proportion on one’s wealth.  The optimal proportion will depend on:

1) the underlying probability distribution of  the possible bet outcomes (the expectation and variance being usually the most important).

2) the utility function (degree of risk aversion) used by the player.

The utility function should, among the general characteristics of a utility function, reach an important objective: the limit inferior of player’s wealth as a random process is positive infinity (for time approaching to infinity).  An intuitive explanation of this condition would roughly be that the player’s bankroll will grow indefinitely, while the player is experiencing a zero risk of ruin (he/she will never “get broke”), nor will the player’s wealth be approaching zero.

“…the precise algorithm for
determining the optimal
betting strategy with spread
constraints has not been
discovered until now.”

It is important to realize that the requirement of limit inferior approaching infinity is the proper sufficient and necessary condition.  For example, a weaker requirement of limit superior approaching infinity is not sufficient.  In this case, it is possible that player’s wealth would aimlessly fluctuate between infinity and zero.  An example of a continuous random process fluctuating this way would be the geometric Brownian motion. 

As mentioned above, a risk neutral linear utility function (leading to expectation maximization) would generally cause the player to “go broke” in a finite time with probability one.  On the other hand, it is possible to reach the objective of infinite limit inferior of player’s wealth.  The case in which the limit superior is infinity and limit inferior is zero will exist somewhere between these two cases. 

Appropriate Utility Function

I chose to provide the analysis for a logarithmic utility function.  Under the logarithmic utility function, the player is maximizing the expected logarithm of wealth at each time. The logarithmic utility function, maximizing the rate of wealth growth, possesses several desirable properties that make it superior to other utility functions: decreasing marginal utility of wealth, decreasing absolute risk aversion, and constant relative risk aversion.  Furthermore, some empirical evidence suggests that wealthy investors, at least, appear to posses the logarithmic utility.

Let us assume that we are dealing with a wager opportunity with a fixed probability distribution of possible outcomes.  It is obvious that the proper bet will always be a certain constant fraction f of player’s wealth, regardless of the wealth current level.  We assume that f >= 0, since the player cannot place a “negative” bet.

It is good to realize that we are dealing with a special case of portfolio problem where there are only two possible investment opportunities: the wager on the game (positive expectation and risk), versus no investment (no return and risk-free).  A rational investor will chose a certain combination of these two options.

Player’s Objective

The player’s objective under the logarithmic utility can be expressed as:

max E(log(W(n))), where W(n) is player’s wealth at time n (after n rounds).

The player’s wealth after playing n rounds, always betting fraction f of one’s current bankroll, will be the product:

(1) W(n) = PI (1+Xi*f), where Xi is the outcome of round i (either positive or negative), Xi are i.i.d. (independent identically distributed) bounded random variables.

Xi is measured in the terms how many initial bets were won or lost. Xi will be positive if the player won round i, negative if lost.  For example, if our wealth is $100,000, f is 1 percent, our bet is $1,000.  If we lose this bet, Xi is -1 (one bet lost).  If we win $2,000 on this bet, Xi is +2 (two bets won). 

Let us assume, without loss of generality, that the “lowest possible value” of Xi is -1 (which roughly means that we cannot lose more than what we bet, and that it is possible to lose the whole bet).  This assumption is only for technical reasons.  The “lowest possible value” shall be defined as inf(x, F(x) > 0), where F(x) is the distribution function of Xi.  Even without the technical assumption, the infimum is greater than -infinity since the random variable Xi are bounded, and since we assume that the probability of Xi < 0 is positive (otherwise the player would always bet his whole wealth and never lose), the “lowest possible value” is thus negative.  Thus, if the “lowest possible value” was not equal to -1 for some special games, we can simply say that our initial wager is by definition Xi / inf(x, F(x)>0) instead of Xi.

Note that while Xi has to be always greater than or equal to -1 according to our definition of initial wager (we cannot lose more than the initial wager), Xi can be greater than +1 (we can win more than the initial wager).

Expression (1) can be rewritten as:

(2) W(n) = exp(SUM(log(1+Xi*f))

The player’s objectives can then be written as:

(3) maximize E(log(W(n))) =  E(SUM(…)) = SUM (E[log(1+Xi*f)])

Since Xi are i.i.d. random variables, our objective reduces to:  maximizing {over all f >= 0} E[log(1+X*f)], where f is the proper proportion of player’s wealth to be wagered, X is a random variable with the same probability distribution as Xi.

An interesting observation here is that f has to be always lower than 1.  We will never bet and risk the entire bankroll, since in the case of a loss, log(1+X) approaches minus infinity for X approaching -1, and the expectation thus becomes minus infinity also.  We would be better off betting zero, in which case the expected rate of growth (3) is also zero.

Another very important property is that f > 0 if and only if E(X) > 0.  If player’s expectation is negative, optimal wagering will call for negative bet which is not possible given our assumption f >= 0, and the optimal bet must thus be zero.  Similarly, if the expectation is zero, the optimal bet will be zero.  Only in the case when the expectation is positive, the player will place a non-zero wager.  The proof of this theorem follows from Jensen’s inequality:

Since log is a concave function on the interval [0, infinity), Jensen’s inequality gives E[log(1+X*f)] <= log(E(1+X*f)) = log (1+EX*f).  The last term is positive for a positive value of f if and only if EX > 0.  For EX <= 0 the last term is always lower or equal to zero, equal to zero for f = 0 (and f = 0 is thus the optimal bet for EX <= 0).

Determining the Optimal Wager Using Taylor Series

Since the random variable X >= -1 everywhere and 0<= f < 1, it follows that X*f > -1. I need to make another technical assumption: X*f < 1 almost sure (a.s.).  In other words, the optimally betting player cannot win his whole wealth or more in one round.  This assumption will exclude some extreme games from the analysis, however, it is not any constraint for practically available games. 

Since -1 < X*f < 1 a.s., we can use the Taylor series for log(1+X*f) around X = 0 on (3).  We obtain:

(4) log(1+X*f) = X*f – (X*f)^2 / 2 + (X*f)^3 / 3 – (X*f)^4 / 4 + …

The objective (3) then becomes (after dropping the index i):

(5) maximize (over 0 <= f < 1) E(X) * f – E(X^2) * f^2 / 2 + E(X^3) * f^3 / 3 – ….

After taking first derivative of the absolutely convergent series (5) with respect to f we obtain final equation for f:

(6) E(X) – E(X^2) * f  + E(X^3) * f^2  – E(X^4) * f^3  + …  =  0,

and since the second derivative is lower than 0, the solution to (6) gives us the optimal betting fraction f.

The solution to (6) can be generally provided in a numerical form while an explicit solution for f will generally not exist.  Note also that since the solution f will be always lower than one, the terms farther to the right will have lower impact on the solution of equation (6).  In practically available games, f will not be greater than 10% (mostly around 1%), so the farther terms can probably be neglected for practical purposes.  And this is indeed a commonly used methodology in practice, as we shall see further.

Example

Let us consider a special game, in which the player loses his wager with probability p, and wins A with probability 1-p.  In this simple case, we can calculate the moments of the alternative distribution as (an analysis of this game has first been provided by Thorp, [5]):

Mi = A^i*(1-p) – p, for i = 2*k+1

Mi = A^i*(1-p) + p, for i = 2*k

Substituting the moments into equation (6) and simplifying, we obtain:

(7) A*(1-p) * (1 – Af + (Af)^2 – (Af)^3 + …) – p * (1 + f + f^2 + f^3 + …) = 0, equivalently (0 < p < 1):

(8) A*(1-p) / (1+Af) = p / (1-f)

Solving equation (8) for f gives:

(9) f = [A*(1-p) – p ] / A

Note, that the nominator A*(1-p) – p of the expression (9) is the expectation of the game.  The denominator A simply says how many times the absolute value of win is greater than the loss (loss is always -1 according to our technical assumption). 

Again, the expectation of the game has to be positive so that the optimal wager was positive.  The solution to (9) is positive if and only if A*(1-p) – p > 0, or in other words if the player has an advantage.  If the player is playing at a disadvantage or an even game, the optimal bet is zero.

The solution for optimal f is valid for A*f < 1 (given our technical assumption X*f < 1), which is equivalent to A*(1-p) – p < 1.  It is true, however, that (9) is valid even for A*(1-p) – p >=1 (see below).

As a special case, let’s take a game where one can either win or lose 1 unit.  In this case, the optimal bet f is simply player’s expectation (substitute A=1 in (9) ).  This a commonly known form and commonly used strategy of proportional betting: bet proportionally to player’s advantage.

A note:

Formula (9) can be derived using a simpler methodology directly from general form (3), without using the Taylor’s series.  However, my objection was to illustrate a general approach to this problem.  The simpler methodology would work in the following way:

“A common misunderstanding
is that the ultimate goal of a
player is to maximize his/her
expected win.”

After N rounds, players’ bankroll will be (1-f)^L * (1+A*f)^W, where L is the number of losses, W is the number of winnings.  Expected logarithm of player’s bankroll will than be:

E(L) * log(1-f) + E(W) * log(1+A*f).  Since the probability of a loss is p, and probability of a win is 1-p, we will maximize

(10)  p*log(1-f) + (1-p)*log(1+A*f)

Setting the first derivative of (10) to zero gives optimal f exactly the same as in (9).  We did not need the technical assumption X*f < 1.

Commonly Used Approximation

It is not practical to solve a complex equation like (6).  In practice, player’s advantage will be very small (around one percent, very rarely more than five percent).  The optimal fraction of bankroll to be bet will thus be relatively small (in the order of several percent).  It follows than that the higher members of Taylor series (6) will become very insignificant.  For example, if the optimal  f for a specific game is one percent, f^2 will be 0.0001, and the third member of (6) is negligible.  Similarly, the other members (moments) in (6) will be even more negligible.

A common practical approach is to take only the first two members of equation (6).  In this case we approximate the logarithmic utility function with a mean-variance approximation and find the optimal solution as a function of mean and variance only. 

It has been shown in [1], [2], [3] that while the mean-variance framework is precise only for a quadratic utility function (which has rather undesirable properties), the mean-variance approximation is very good in most cases for other utility functions as well.

After dropping the other members of the series, the equation then reduces greatly to E(X) – E(X^2) * f = 0, which yields sub-optimal:

(11) f = E(X) / E(X^2)

f then becomes a simple ratio of player’s expectation (advantage) and the second moment.  Instead of (11), commonly used is

(12) f = E(X) / var(X), where var stays for variance

Since var(X) = E(X^2) – [E(X)]^2, and E(X) (player’s expectation) is usually small in order of no more than several percent, [E(X)]^2 is in order of hundreds of percent, and (12) gives practically the same numerical results as (11).

Optimal Wagering if the Underlying Probability Distribution Changes

In some games the underlying probability distribution of game outcomes differs for different rounds.  The advantage and also variance of different rounds can be different.  An example would be the casino game of blackjack, where the player’s advantage changes depending on which cards have already been played.  In this specific game, the player will actually face a negative expectation situation most of the time.

A common practical constraint in the blackjack game is that the player needs to place a certain wager in all game situations, including the prevailing negative expectation situations, in order to be allowed to place the positive expectation wagers.  The reason for placing also negative expectation wagers can be for example cover purposes where the player does not want to be eventually barred from the game by the game provider.  Still, the player will be allowed to place bigger bets in the positive expectation situations, and thus overall reach a positive expectation.

The requirement of placing a bet in all situations in the game should be understood in the sense of saying that the ratio between player’s maximum bet and minimum bet (further referred to as spread) is a given constant.  If the player’s spread was not constrained, the player would bet arbitrarily small amount (practically zero) in negative expectation situation, while the optimal bets in positive expectation situations would be the same as in the case without any constraints.

Problem Formulation

Our problem can be formulated in the following way:

A player is playing a game in which he is required to place bets in each situation -km, …, -1, 0, 1, 2, …, kp, which occur with probabilities p(-k),…, p(-1), p0, p1, p2, …, p(kp), where km (k minus) and kp (k plus) are some arbitrarily constant.  At least one situation has positive expectation.  Given a maximum spread of 1 to N, what is the player’s optimal fraction of his/her wealth to be placed in each situation? (Note: I consider situations -k to k rather than 1 to k so that the notation was consistent with practical application to the casino game of blackjack, example of which will be done in detail in the next section.)

The solution to this problem is similar to the simpler one.  Instead of maximizing E[log(1+X*f] over all f, we will now maximize  (13) E[log (PI(1+Xj*fj)^pj)] , maximized over all f1, f2, …, fk with the constraint that fj/fi <= N for all i,j running from 1 to k (constraint on the spread).  Fj are the proper fractions to be wagered in situation j, Xj is the random variable of results describing the probability distribution of situation j, and pj is the percentages of occurrence of situation j.  To see why (13) is the proper expression to maximize, it is enough to realize that the player’s wealth W(n) can be expressed as: W(n) = PI (1+Xj,i*fj)^nj (over all j), where nj is the number of occurrences of situation j, E(nj) = pj * N, Xj,i is the i-th random choice from random variable Xj (i runs from 1 to nj).  We can directly substitute pj instead of for example nj/n in (13), since E(nj/n) = pj (by definition), and E[log(…)^(nj/n)] = E[(nj/n)*log(1+Xj*fj)] = E(nj/n) * E[log(1+Xj*fj)] (due to obvious independence of random variables nj and Xj) = pj * E[log(1+Xj*fj)].

The expression (13) gives a sum of Taylor series similar to (4): (14) SUM pj*( Xj*fj – (Xj*fj)^2 / 2 + (Xj*fj)^3 / 3 – (Xj*fj)^4 / 4 + … ), where the summation runs from j=1 to k.  The objection is now to maximize the expected value of (14) with constraints fj <= N*fi for all i, j.

A Solution for the Mean-Variance Approximation

Similarly as in the case of no spread constraint, we can use a mean-variance approximation to the logarithmic utility by dropping all moments higher than the second moment.  (13) is greatly reduced to: (15) maximize E[ SUM pj * (Xj*fj – (Xj*fj)^2 / 2) ] over fj, subject to fj <= N*fi for all i, j.

Without loss of generality, let us make the assumption that the situations -km to kp are sorted by their Expectation/Second Moment ratio, where -km has the lowest ratio, while k has the highest ratio.  For example, in the game of blackjack, situations -km to around 0 have a negative expectation and equivalently negative Expectation / Second Moment, while situations 1 to kp have a positive one.  Also, I will occasionally say “count” j instead of “situation” j, which directly corresponds to the terminology commonly used in the game of blackjack.

Obviously, we will be betting minimum (further referred to as unit) in some situations (on some counts).  These situations will include negative expectation situations (which are count -km to around 0 for the game of blackjack), and possibly also some positive expectation situations for which the Expectation / Second Moment ratio is not high enough to justify an increased wager.

We will be betting some amount between one unit and the maximum of N units in some positive expectation situations.  These situations have a specific importance.  I will call them intermediate situations, or intermediate counts.  Lastly, we will be betting the maximum of N units in at least one situation.  These situations will have the highest Expectation / Second Moment ratios, as will be obvious later.

Let us denote the one unit as f.  We will wager some multiple bj of f, 1 <= bj <= N, on each count j.  A minimum bet is such a bet where bj = 1, while a maximum bet is for bj = N.  For intermediate counts, 1 < bj < N.  For simplification, I will also call bj a “bet”, although a precise terminology is multiple of minimal unit f. 

We can rewrite (15) as:

(16) max E [ SUM pj * (Xj*bj*f – (Xj*bj*f)^2 / 2 ] over f, bj, subject to 1 <= bj <= N for all j, equivalently

(17) max f * SUM [pj*bj*E(Xj)] – f^2 / 2 * SUM [pj*bj^2*E(Xj^2)] over f, bj, 1 <= bj <= N.

It follows easily by differentiating (17) with respect to f, and setting the result to zero, that f maximizing (17) is:

(18) f  = SUM [pj*bj*E(Xj)] / SUM [pj*bj^2*E(Xj^2)] = P / V,

where P is Expectation of the game, and V is Second Moment of the game (further referred to total expectation and total second moment respectively).  (Yamashita obtained independently practically the same result in [7].)

The second derivative of (17) with respect to f  is always negative, which means that  f as given by (18) is indeed the optimal f maximizing (17).  Unfortunately, we still do not know bj, except that we know that bj is one for all j such that E(Xj) <= 0 (we bet one unit in a negative or zero expectation situation).

By substituting for optimal f from (18) to (17) and a simple rewriting, our objection simplifies to:

(19) max 1/2 * P^2 / V over all bj, 1 <= bj <= N

Note, that expression (19) is the ratio of total Expectation squared and total Second Moment.  The second moment is almost equal to variance for all practically available games since expectation is small in order of percent, and the square of expectation is thus negligible.  Substituting variance for the second moment is a common practice.  I will continue to use the second moment since there is probably no advantage in using variance.

As a byproduct, it has been shown that the proper objective of a game is to maximize so called Sharp ratio, which is equal to Expectation / Second Moment (again, more commonly used approximation is Expectation / Standard Deviation).  Furthermore, it follows that the Sharp ratio is also the proper comparison of lucrativeness of different games in a sense that the higher Sharp ratio, the superior game.  Note, however, that the square of Sharp ratio is more appropriate absolute comparison.  Game X is twice better (provides twice higher utility to the player) than game Y if and only if the square of the Sharp ratio of game X is twice the square of the Sharp ratio of game Y.

To find an equation for optimal bj, we can differentiate (19) with respect to bj, and set the first derivative to zero.  This way we will get an expression for all intermediate bets bj, 1 < bj < N.

The first derivative of (19) with respect to bj equals (prime ‘ denotes the first derivative with respect to bj):

(20) (2*P’*P * V – V’*P^2) / V^2

Setting (20) to zero and substituting pj*E(Xj) and 2*pj*bj*E(Xj^2) for P’ and V’, respectively, immediately yields:

(21) bj = E(Xj) / E(Xj^2) * V / P, valid for all intermediate bets bj, 1 < bj < N

The proof that (21) is the proper expression for bj is not complete yet since we have not shown that we indeed found the maximum of (19).  The technical and argument is left for Appendix 1.

The important result (21) appeared probably for the first time in [8], although Harris’ proof was not complete.  Harris and others seemed to have missed another very important result, though, which follows.

Theorem:

For all intermediate situations j for which 1 < bj < N, the proper optimal wager is equal to fraction E(Xj) / E(Xj^2) of player’s wealth.  The fraction is the same regardless of what the allowed spread is (if there is any spread constraint at all), and regardless of what the optimal betting unit f is, as soon as situation j remains as intermediate situation.

Proof:

The proof follows directly by substituting from (18) and (21) for f and bj, respectively:

(22) fj = f * bj = P / V * E(Xj) / E(Xj^2) * V / P = E(Xj) / (E(Xj^2)

where fj is the proper fraction of player’s wealth to be bet on intermediate count j, equation valid for 1 < bj < N.

This result directly contradicts the generally accepted view that the player would “scale down” his/her bets if he/she is required to place also negative expectation bets.  On the other hand, it is obvious that the player’s betting strategy should be different since a situation with constrained spread is vastly different from a situation where the player has no spread constraint and does not need to place negative expectation bets.  The solution to this problem stems from the value of the optimal betting unit f.  Relation (22) is valid only for intermediate count bets, not for maximum, nor minimum bets.  Especially maximum bets are the ones which strongly depend on the spread.  The higher the spread, the higher the optimal maximum bet.  I will provide several numerical examples in the section “Numerical Examples.”

How to Find a Numerical Solution

It is still not clear how to numerically find a solution to the optimum betting problem.  While we do have equation (21) for bj for intermediate counts j, the right side of the equation still depends on all bj, including the intermediate ones.  While it is possible to converge to an optimal solution using for example a method of trial and error on a spread sheet, this approach is very lengthy.  It is possible to get the optimal values almost directly.  For this, we will need to improve equation (21) first.

(21) is not the only expression we can obtain for bj.  By setting (20) to zero we can also get: 0 = (2*P’*P * V – V’*P^2), equivalently 0 = 2*P’ * V – V’ * P =

2*pj*E(Xj) * [V – pj*bj^2*E(Xj^2)] + 2*pj*E(Xj) * [pj*bj^2*E(Xj^2] – 2*pj*bj*E(Xj^2) * [P – pj*bj*E(Xj)] – 2*pj*bj*E(Xj^2) * [pj*bj*E(Xj)], equivalently 0 = E(Xj) * (V – pj*bj^2*E(Xj^2)] – bj*E(Xj^2) * [P – pj*bj*E(Xj), which immediately yields:

(23) bj = E(Xj) / E(Xj^2) * [V – pj*bj^2*E(Xj^2)] / [P – pj*bj*E(Xj)] , which is valid again for all intermediate counts j, 1 < bj < N.

Note, that Pj = P – pj*bj*E(Xj) is total expectation without the contribution of situation j, and Vj = V – pj*bj^2*E(Xj^2) is total second moment without the contribution of situation j.  By comparing (23) to (21) it follows immediately that V / P = Vj / Pj for all intermediate counts j.  According to an algebraic rule (see Appendix 2), it follows then that we can “remove” the contributions to V and P of all intermediate counts j, not just one at a time.  In other words, we get:

(24) V / P = Vx / Px , where Px is is the expectation and Vx is the second moment, each without the contribution of all intermediate counts j, 1 < bj < N:

Px = P – SUM (over intermediate counts j) pj*bj*E(Xj) = SUM (over minimum bets) pi*E(Xi)  + SUM (over maximum bets) pi*N*E(Xi),

Vx = V – SUM (over intermediate counts j) pj*bj^2*E(Xj^2) = SUM (over minimum bets) pi*E(Xi^2) + SUM (over maximum bets) pi*N^2*E(Xi^2)

Combining (24) and (21) yields:

(25) bj = E(Xj) / E(Xj^2) * Vx / Px, for all intermediate counts j, 1 < bj < N

The advantage of (25) relative to (21) lies in the fact that (25) is a direct formula for bj which does not contain any other intermediate bets bi, 1 < bi < N.  One problem still remains, though.  It is yet to be determined which counts are intermediate.  This problem is equivalent to finding the lowest and the highest intermediate count — the first count j at which bj is greater than 1, and the last count i at which bi is lower than N.  The solution can be quite simply provided by a spread sheet program.

Numerical Examples

As a practical application of the Optimal Wagering problem I used the casino game of blackjack.  I got the necessary data from a Monte Carlo simulation.  The software used was a commercial blackjack simulation and analysis software Statistical Blackjack Analyzer, version 3.1.  The simulator played approximately 540 million blackjack rounds which decreased the standard error of the observed data sufficiently to be negligible.  The specific blackjack rules used were 2 decks, doubling allowed after splitting, re-splitting up to four hands except aces, one card to split aces, face up game, penetration 78 cards.  The counting system used was Hi-Lo with Illustrious 18 strategy indices for playing deviations from so-called blackjack basic strategy.  I analyzed three different spreads: 1) spread 1 to 2 (a minimum spread required so that the player could reach positive expectation), spread 2) 1 to 6 (probably the most common spread used in practice under these conditions), and finally spread 3) 1 to 12 (used in ideal conditions).  See Appendices S1, S2, and S3, respectively, for data and spread sheet calculations for the three different spreads.  Note, that the player is playing with a disadvantage on counts 0 and below, and he/she is playing with an advantage on counts +1and higher.  The results are:

1) For the 1 to 2 spread (Appendix S1), the player bets 1 unit on counts 0 and below, and immediately jumps to the maximum of 2 units on positive counts.  There are no intermediate bets.  The expectation is 0.40% of a unit per round, with second moment (variance) 2,67 units squared.  The optimal fraction f (the minimum bet) is 0.150% of player’s wealth, the Sharp ratio is 0.2447%.  Assuming 100 round per hour and bankroll (wealth) of $10000, the player will be winning 100*0.40% * $15.0 = $6 per hour in average.

2) For the 1 to 6 spread (Appendix S2), we already have intermediate bets on counts 1, 2, and 3.  The maximum of 6 units is reached at count 4.  The expectation is 2.48% of a unit per round with variance 9,19 units squared, the optimal minimum bet is 0.270% of player’s wealth, the Sharp ratio is 0,8172%.  Under the same assumptions as above, the player wins roughly $67 per hour which means that spreading 1 to 6 is over 11 times better than spreading 1 to 2.  Another way to put it is that a player spreading 1 to 2 needs more than 11 times higher wealth to reach the same results as a player spreading 1 to 12.

3) For the 1 to 12 spread (Appendix S3), we reach maximum bet at count +7 (while +6 is very close to maximum).  The expectation is 4.91% of a unit per round with variance 22.81 units squared.  The optimum unit bet is 0.215% of player’s wealth.  Note, that this f is already lower than for the 1 to 6 spread.  The player with $10000 bankroll wins roughly 4.91 *$21.5 = $106 per hour.  Spreading 1 to 12 is approximately 60% better than spreading 1 to 6.

Conclusion

The “Optimal Wagering with Bet Constraints” analysis is of major importance especially to people who are interested in playing positive expectation games.  Probably the best example of such a widely spread and very favorite game is the casino game of blackjack.  According to my knowledge, the precise algorithm for determining the optimal betting strategy with spread constraints has not been discovered until now, although the new independent work of Harris [8], Yamashita [7], and several other participants of the Internet discussion group, URL http://www.bj21.com, have become very close.

There is still one imperfection left.  This imperfection lies in the mean-variance approximation of logarithmic utility.  It is almost surely not feasible to obtain a reasonable closed-end solution for the precise logarithmic utility, even for the simplest case of no spread constraint.  Fortunately, some evidence suggests that the mean-variance approximation is reasonably precise for all practically available situations. 

Appendix 1 (maximality proof)

Since [1,N]^(2k+1), a subspace in R^(2k+1), is a compact in R^(2k+1), the expression (19) must have a maximum (b(-k),…, b(k) on this compact.  Let 1 < bj < N for any j (inner point). Then, the partial derivative of (19) with respect to bj must be zero since (19) must have local maximum in bj for fixed bi, i<>j.  However, there is only one point where the partial derivative is zero. It follows than that for any bj, 1 < bj < N, equation (21) in the paper must hold.

Note: It is possible that the set of all {bj, 1 < bj < N} is empty (no bj exists).  This can happen for example if the player can never reach a positive advantage with the given betting spread  in which case he should bet zero in all situations.

Appendix 2 (algebraic rule)

Theorem:

If (x+y+z) / (a+b+c) = (x+y) / (a+b) = (x+z) / (a+c), then (x+y+z) / (a+b+c) = x / a, for any x, y, z, a, b, c real numbers, abc <> 0.

Proof:

If (u+v) / (r+s) = u / r, it follows immediately that u / r = v / s, for rs <> 0.  In other words, we get y / b = (x+z) / (a+c) = (x+y) / (a+b) = z / c, or y / b = z / c.  Since y / b = z / c, it is obvious that (y+z) / (b+c) = y / b = z / c = (x+y+z) / (a+b+c).  One more application of the rule gives (x+y+z) / (a+b+c) = (y+z) / (b+c) = x / a, which finishes the proof.

Bibliography

1. 1. Lawrence B. Pulley, “Mean-Variance Approximations to Expected Logarithmic Utility”, Operation Research, No. 4, July-August 1983
   2. Yoram Kroll, Haim Levy, Harry M. Markowitz, Mean-Variance Versus Direct Utility Maximization, The Journal of Finance, No. 1, March 1984
   3. H. Levy and H. M. Markowitz, “Approximating Expected Utility by a Function of Mean and Variance,” American Economic Review 69 (1979)
   4. David P. Baron, “On the Utility Theoretic Foundations of Mean-Variance Analysis”, The Journal of Finance, No. 5, December 1977
   5. Edward O. Thorp, “Beat the Dealer”, 1966
   6. Horwath, Scott, “A Demand for Lotto Tickets by Risk Averse Individuals”, working paper, Bradley University
   7. Winston Yamashita, “Kelly Generalization”, Internet discussion group http://www.bj21.com/messages/948.html, May 1997
   8. Brett Harris, “There is a Formula!”, Internet discussion group http://www.bj21.com/messages/1027.html, May 1997
   9. Winston Yamashita, “Kelly Generalization”, Internet discussion group, March 1997

ABOUT THE AUTHOR: Karel Janecek is (was) the author of the IBM compatible software package Statistical Blackjack Analyzer (SBA) which is no longer available.

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