Player Loss

Introduction | What is the cost of rebating? | How large a value? | Unit Normal Linear Loss Integral | Baccarat (Player/ Bank 50% Equivalent) | Roulette (single number play on single zero Roulette) | Conclusion |

The above table for Baccarat is interesting in that it depicts the percentages of loss which can be paid for various numbers of hands to maintain a 50% rebate on theoretical loss equivalence. From this it can be seen that quite attractive rebates may be paid. A central question which arises, however, is what is the best or simplest manner by which to calculate the number of hands. While a simple method may be to take time played and employ standard decision rates, that is inappropriate due to potentially widely divergent bet levels.

This is important because when dealing with actual loss some bets may be statistically insignificant. To demonstrate using extremes if we had 1000 hands with bets of $1000 and one hand with a bet of $1,000,000 clearly the player’s final result will be primarily determined by whether they win or lose the $1,000,000 hand. It can be said therefore that the 1000 hands are insignificant. Thus a reasonable method of calculating the number of hands played for the purposes of determining a rebate on loss is to divide the total turnover by the player’s maximum bet. This criteria may be particularly useful when the Casino permits a table differential to be employed which potentially allows an unlimited maximum bet to be placed.

Determining the maximum bet placed is generally a simple proposition if dealing with an individual player. In a junket group situation where members of the same group may for example bet against each other on Baccarat, the bet could be considered to be the difference between the opposing bets, even though the turnover is the sum of the opposing bets.

For other non-even pay off games such as roulette, the mathematics remains similar, however, because the player wins more when they do win but this occurs less often, the percentage of actual loss which may be rebated is relatively less.

To incorporate this factor into the previously shown formula requires the calculation of the variance for a specific game for one result. In an even money game such as Baccarat (when playing Bank or Player) the variance may be approximated as one and therefore the previously shown formula was valid.

In any game the calculation of variance is accomplished by summing the square of the player wins multiplied by the probability of the returns. The standard deviation then becomes the square root of the number of hands multiplied by the average squared result.

In a game with multiple betting options at the same game with varying payoffs but the same house advantage (eg Roulette) the variance figure utilised when calculating a rebate on loss would necessarily be the maximum figure.

The appropriate numbers for various games are:-

In games with multiple betting options at varying payoffs and house edges it would be appropriate to fully calculate the rebate payable on every option and utilise the variance from the result which returns the least to the player as otherwise any requirements on data collection by staff may be prohibitive.

When performing this calculation the following formula may be used:-

1. Calculate the variance for one play.

2. Find expected loss for the player.

3. Find standard deviation of player result = square root of (hands multiplied by variance (refer point 1))

4. Calculate z = expected loss / standard deviation.

5. Look up UNLLI table corresponding to z.

6. Multiply UNLLI value by standard deviation.

7. Add number calculated from point 5 to number calculated from point 2.

8. Take whatever percentage of point 2 is to be returned and divide by the result of point 7.

This then produces the following example of a table of rebate percentages applicable to be paid for the game of roulette (when playing single numbers on a single zero game) and which maintains a 50% equivalence on theoretical loss.