Non Negotiables

Introduction | Walkthrough | Mathematical Ratios | Revenue Calculation Comparisons for Various Games After Tax | Non Negotiables on Roulette | Other Methods of Explanation | Conclusion |

The commission payment is therefore calculated essentially on the loss of non negotiable chips over the period. An often used statement which holds some validity is “once through the cage equals twice over the tables”.

What this means is that actual turnover may be calculated using a ratio of two to one if non negotiable turnover is known.

For example: Given that Baccarat is a 50/50 game (exclude ties) and assuming that the bet distribution is $50K Banker and $50K Player, then after a singe hand, the Casino will be in possession of a single $50K chip and turnover will be equal to $100K.

Hence, once through the Cage, twice across the table.

This helps some people understand why on a game like Baccarat where the house advantage is around 1.25%, casino operators are prepared to pay commissions of around 1.5% to 1.8% or even higher on non negotiable chips. Essentially the house advantage can be looked at as being twice that of “normal” due to the ratio of actual versus non-negotiable turnover being 2:1.

Thus Baccarat could be looked upon as having a house advantage of 2.5% on non negotiable play or conversely the non-negotiable commission percentages could be halved to bring them back in line with actual turnover. Either way the same result is achieved.

To consider this more mathematically and find a formula which more aptly defines the ratio the following is provided.

To entirely convert non-negotiable chips to zero in a theoretical sense one would have to continue to play until exhaustion.

After each bet if the theoretical return were obtained we would have an equation as follows:-

RESIDUAL AMOUNT FOLLOWING WAGERING

WAGER

1

p

p

p^2

p^2

p^3

p^3

p^4

p^4

p^5

etc

where “p” is the probability of the wager winning. The total amount wagered in this series would equal:- 1 + p + p^2+ p^3+ p^4, + ….., p^n This would provide the ratio of actual amount wagered to non negotiable losses. This is in fact a simple geometric progression with a solution that may be found in numerous mathematical text books. That solution is:-

R =

___1___

1-p

Where R = Ratio What then of non negotiable chips and Baccarat.In this case, the probability of the wager winning (p) is equal to 0.4932 for “Player” and 0.5068 for “Bank” based on relative probabilities (which is correct given that ties have no effect on non negotiables).

Thus R

= 1 / (1-p)

= 1 / (1 – 0.4932) for “Player”

= 1.97

or

R

= 1 / (1-p)

= 1 / (1-0.5068) for “Bank”

= 2.03

If equal amounts were bet on both “Player” and “Bank” then “p” would equal 0.5. “R” would then equal 1 / (1-p) = 1 / 0.5 = 2. What about other non even payoff games such as Roulette? Using single zero Roulette for our example “p” would equal the following:-

Straight ups “p”

= 0.027

R

= 1.028

Dozens and columns “p”

= 0.324

R

= 1.480

Even chances “p”

= 0.486

R

= 1.947

Thus as can be seen the ratio of actual turnover to non negotiable losses is not fixed but rather is a function of the probability associated with the individual game and the particular wager placed. Thus our truism of “once through the cage equals twice over the tables” is roughly valid only for even chance games with a low house margin. If non negotiables are played solely on straight ups on roulette (for example) the ratio is much closer to one.

How is this additional information of use?

When constructing commission programs using non negotiable chips it is essential to understand the cost of the program. Let us look at several simple examples and assess this in table form.

Example:-

Baccarat : (equal play on “Player” and “Bank”)
Edge : 1.26%
Theoretical turnover = 2 x Cage turnover
Casino tax = 20% win (example only)
Theoretical win = edge x turnover
Theoretical win = 1.26% x (2 x Cage turnover)
Theoretical win = 2.52% x Cage turnover
Theoretical win (after tax) = 2.52% x Cage turnover – 20% (win)
Win after tax = 2.016% x Cage turnover
Win after tax = 2.016% per turn