Setting Table Maximum Betting Limits

Risk and volume dependence in relation to determining maximum betting limits for independent trial, negative expectation table games.
By Andrew MacDonald
Senior Executive Casino Operations, Adelaide Casino, 1995

Casino Analyser
Reference
Maximum Bet
Volatility

Secondly, let us consider how to take into consideration the effect of various bet distributions on what table maximum bet limit can be set with comfort. Clearly, the simple method would be to assume that all bets occurred at the maximum bet limit and thus adopt whatever maximum limit had been calculated based on the maximum loss which the shareholders would accept. This, however, would be a highly conservative approach as it would be extraordinarily rare for all bets to occur at the maximum limit. Indeed, such an approach, while simple, could lead to the adoption of a table maximum bet limit that sub-optimised the overall casino’s performance by inhibiting turnover. To address this the following is provided:

Consider how a variation in bet distribution will affect the variance of the expected return to player. Assume that the average bet size is $100,000 with the following distribution.

Bet size= x	Frequency= f		x.f	(x)^2.f
150,000	15%		22,500	3,375,000,000
125,000	20%		25,000	3,125,000,000
100,000	35%		35,000	3,500,000,000
75,000	15%		11,250	843,750,000
50,000	10%		5,000	250,000,000
25,000	5%		1,250	31,250,000
SUM	100%		100,000	11,125,000,000
When the bet size is represented as a relative frequency as illuatrated above the standard deviation follows:
Standard deviation		= sqrt ( sum (x^2.f) – (sum(x.f)) sqrt
		= sqrt ( 11,125,000,000 – (100,000^2)
		= sqrt ( 11,125,000,000 – 10,000,000,000 )
		= 33,541

One can assume that the variance of the expected return when all bets are fixed at some value B is approximately equal to V.N , where V is a constant close to 1 and relates to the variance of the game in question (ref 4) and N is the number of hands dealt. Then the standard deviation of the expected return when there is a bet distribution with a mean bet size of B is approximately equal to:

= B. sqrt ( V.N.( 1 + ( SB/B)^2))
where SB is the standard deviation of the variable bet distribution.

Consider the win W net of tax and variable expenses assuming N hands of equal bet size B

W = [ ( NE -1.645sqrt(NV) )(1-T) – NC ] B
where N = Number of Hands
E = Expected Win Percentage
V = Variance of Baccarat Hand with Constant Bet Size
T = Tax Rate
C = Commission Rate

Then the win for a variable bet distribution with a mean bet size Bav and a variance of VN( 1 + (SB/Bav)^2 is
W = [ ( NE -1.645sqrt(NV(1 + (SB/Bav)^2) )(1-T) – NC ] Bav

If Bmax is the maximum fixed bet size which will produce an acceptable risk profile then by equating the win net of tax and variable expenses of N hands at bet size Bmax to N variable bets with a mean Bav it is possible to determine a limiting factor on the variance of the variable distribution such that the maximum allowable risk is not exceeded.

Let F be the factor which accounts for the increased variance of the variable bet distribution. Then solving for F in the following equation

[( NE -1.645sqrt (NVF))(1-T) – NC ] Bav = [( NE -1.645sqrt(NV) )(1-T) – NC] Bmax
-1.645sqrt NVF) = [ NE -1.645sqrt(NV) – NC/(1-T) ] Bmax/Bav – NE + NC/(1-T)
F = {1/(1.645sqrt(NV))(NE – NC/(1-T))(1- Bmax/Bav) + Bmax/Bav}^2

Therefore provided 1 + (SB/Bmax)2 = F is less than the right hand side of the above equation, the overall bet distribution will satisfy the risk constraints.

Thus in relation to the maximum bet size that may be offered, based on the shareholders acceptance of a loss point, the bet limit represents a bet distribution with mean, x.f (Bav) , and variance, SB , where the mean bet multiplied by one plus the square of the standard deviation of the bet distribution divided by the mean bet is equal to what may be referred to as the effective bet limit.

Effective Bet Limit = Bav ( 1 + ( SB/Bmax )^2 ).

The effective bet limit used then to calculate a maximum loss point would be greater than the mean but less than the maximum bet on offer, if bet sizes had some range. This is an appropriate methodology if bet sizes and frequencies can be predicted with some degree of accuracy, however, it is vulnerable to a situation where the bet distribution actually achieved is skewed towards much higher bet levels than anticipated. Then the shareholders may experience greater fluctuations in profits than desired or losses which they were not prepared for. The other potential use for this is to determine whether or not based on known bet distributions the actual results fall within an acceptable range.

Example

As an example of how this information may be used, let us consider the following:

Maximum bet = $85,695 (refer to page 7 example) to incur a maximum potential loss of $10 million after all costs.

An average bet of $80,000 with the following range and distribution could be applied and still produce a lesser maximum loss point than the shareholders accepted.

The limiting value F when the average bet size Bav is $80,000 and the maximum bet size Bmax is $85,695 is as follows

F = (1/(1.645sqrt(NV))(NE – NC/(1-T))(1- Bmax/Bav) + Bmax/Bav}^2 ={1/(1.645sqrt(34,500.0.97))(34,500.0.0125-34,500.0.007/(.08625)).
(1-85,695/80,000)+85,695/80,000}^2
=1.07207

Bet size= x	Frequency= f		x.f	(x)2.f
120,000	2.5%		3,000	360,000,000
100,000	35%		35,000	3,500,000,000
80,000	32.5%		26,000	2,080,000,000
60,000	22.5%		13,500	810,000,000
40,000	5%		2,000	80,000,000
20,000	2.5%		500	10,000,000
SUM	100%		80,000	6,840,000,000
Standard deviation		= sqrt ( sum (x^2.f) – (sum(x.f))^2 )
		= sqrt ( 6,840,000,000 – 80,0002^2 )
		= 20,976.
Effective Bet Limit		= Bav. ( 1 + ( SB/Bav )^2).
		= 80,000. ( 1 + ( 20,976/80,000)^2 )
		= 80,000. ( 1.06875 )
		= 85,500.

Note that the actual value of 1 + ( SB/Bav )^2 equals 1.06875 for the above distribution which is less than the allowable maximum calculated at 1.07207.

Therefore the game will behave as if there were 0.933N hands of size $85,500 rather than N hands with equal bet size $80,000. Based on 0.933N hands of fixed bet size $85,500 there is a one in twenty chance that the company will lose $9.98M which falls within the acceptable risk profile.

The effective bet limit at $85,500 is less than $85,695 and therefore implies that it should be appropriate to allow a table maximum bet limit of $120,000 on the assumption that the distribution of bets actually experienced will reflect the above. Clearly, a reasonably accurate assessment of how the market will respond to this range of limits is required, however, if this sort of distribution is achieved then there is only a one in twenty chance that the company will lose $10 million or more based on the assumptions regarding fixed and variable expenses also being accurate.

It is important to recognise that this must be market driven or historical and cannot be derived a priori. Of interest is that if an a priori determination were attempted, a higher table maximum bet limit could be provided, but at the cost inevitably of deriving a lower average bet and thus reduced profitability, if the shareholders’ maximum loss point is not otherwise increased. This would be a direct result of possibly falsely attempting to increase the bet range and therefore, implicitly, increasing the variance of the bet distribution.

To finally evaluate how all the above may be applied practically in a casino operation that does not market to high-end players, the following is provided.

Setting Table Maximum Betting Limits

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