In Part III of this series, we discussed an approximate formula for calculating an equivalent rebate on actual losses in baccarat. (See Casino Enterprise Management Volume 8, Issue 1.) [Note: Using this formula on as few as 100 hands may not be advisable, and using this formula for games other than baccarat could be disastrous.] Now we will use a method of repeated random sampling called a “Monte Carlo simulation” to assess the accuracy of the results of this formula. We will also propose an alternative method for calculating rebates on actual loss. This alternative method is more accurate and profitable to the operator than the approximate formula—if followed correctly.
This article, while very mathematical, deals with a simple question: How do we calculate a rebate on player losses? Although our previous article used a well-established approximation, our computer simulation reveals that using the formula has many inaccuracies—some quite startling. For example, one player rebate that the approximation formula calculates at 20 percent should be only 10 percent, and the simulation method puts it closer to 8 percent. The mathematics are not for the faint of heart, but considering the potential revenue lost by “over-rebating,” studying this effect is worth the effort for any operator with a rebate on loss scheme.
Rebate on Player’s Actual Loss
As discussed in Part I of this series, in a rebate on player loss program, premium baccarat players are informed in advance that on each trip, losing players will be given back a fixed percentage of their trip loss. For example, suppose baccarat Player A has won $4,000 and baccarat Player B has lost $10,000. If the casino offers a 15 percent rebate on loss, then Player A will go home with her $4,000 in winnings and Player B will get a $1,500 rebate on his losses (.15 x $10,000 = $1,500).
At first glance, there does not seem to be anything wrong with this—Player B still lost $8,500 on this trip. The problem lies in the fact that casinos can return a portion of their long-term profit or theoretical win (which is related to the house edge of a game) without being negative, but when an x percent rebate, for example, is given on a player’s actual loss, x has to be calculated so that the casino win from the player remains positive.
The standard approach to calculating a rebate on actual loss that will be equivalent to an x percent rebate on theoretical loss is by using a formula given in Practical Casino Math by Robert C. Hannum and Anthony N. Cabot (2nd Edition, 2005) (see below). However, it is generally not known that this formula is based on a well-known statistics theorem called the Central Limit Theorem (CLT), which states that the probability distribution of the mean of a sample from any distribution with finite variance can be approximated by a normal distribution, provided that the sample size is large enough. In other words, in the context of a rebate on loss program for baccarat, the number of hands played by a player must be sufficiently large for the formula to give accurate results.
Now let’s use a Monte Carlo simulation to assess the accuracy of this equivalent rebate on loss formula.
The Monte Carlo Simulation
A computer code was developed using a statistical language called “R” in order simulate n hands of baccarat where a player is betting $1 per hand. The process of playing n baccarat hands is repeated 1,000 times, thus obtaining 1,000 values of player win amounts when n baccarat hands are played. The following pseudo-algorithm was used:
P(Player Win) = P(X = 1) = .4462466
P(Player Loss) = P( X = –1 ) = .4585974
P(Tie) = P(X = 0) = 1 – .904844 = .095156
where X represents the $win amount of the player.
(1) Generate results of n baccarat hands ( x1, x2, …, xn) by generating n values from the following probability distribution:
Values of X +1 0 –1
Probability .4462466 .4585974 .095156
(2) Calculate the $win amount after n baccarat hands by adding the $win for each hand:
Player win from n baccarat hands = x1+ x2+…+ xn = y. The player’s expected value (EV) is estimated by y/n, and the house edge is –y/n.
If the player $win amount from n hands is negative (i.e., if y is a negative number), calculate player $win after giving a rebate of r percent as follows:
If y < 0, then w = (1– r/100)xy, where w = player win after an r percent rebate is applied. The player’s EV after rebate is w/n, and the house edge after rebate is –w/n. Steps (1) and (2) were repeated 1,000 times for each choice of n (number of baccarat hands), yielding 1,000 values of player win (y) and player win after an r percent rebate (w). The simulation was run for the following 16 values of n: n = 100, 200, 300, 400, 500, 600, 700, 800, 900; 1,000; 2,000; 3,000; 5,000; 10,000; 15,000; 20,000 For each combination of input parameters, the output of the simulation is 1,000 values of y and w: y1, y2, …, y1,000 = player $win amounts when the number of baccarat hands played is n w1, w2, …, w1,000 = player $win amounts after rebate when the number of baccarat hands played is n The player EV values and house edge with and without rebate are calculated as shown above, and then histograms of observed EV and house edge are plotted. The Results Part of the results of this simulation were presented in an earlier paper, where it was shown that the percent of win returned to the player under a 20 percent rebate program is in fact closer to 95 percent when the number of baccarat hands is 100 and is approximately 30 percent when 1,000 baccarat hands are played. In other words, the actual rebate percent is much higher than intended. Figures 1 to 4 show the histograms of 1,000 observed casino win amounts (–y values) and casino win amounts after a rebate is given to a losing player (–w values). Figure 1 shows the house edge histograms for 100–400 baccarat hands. It is easy to see from the histograms in Figure 1 that the casino is in the red a significant part of the time and that the volatility in casino win is relatively high. But when the number of baccarat hands is increased to 20,000 (Figure 4), the casino is in the black almost all of the time. Figures 1–4 also show that, as expected, the house edge is reduced when a rebate on loss is given. Figure 1: Observed House Edge with a 20% Rebate ( n = 100, 200, 300, 400) Figure 2: Observed House Edge with a 20% Rebate (n = 500, 600, 700, 800) Figure 3: Observed House Edge with a 20% Rebate (n = 900; 1,000; 2,000; 3,000) Figure 4: Observed House Edge with a 20% Rebate (n = 5,000; 10,000; 15,000; 20,000) Assessment of Accuracy Suppose a casino is giving a 20 percent rebate on actual loss to a player who has played 100 baccarat hands, betting on the player hand each time. We calculated the house edge and standard deviation of this wager in the January 2010 issue of CEM, with the following results: House Edge = .01235 sd = .95115 The approximate formula for calculating an equivalent rebate is: Using this formula, we can calculate the following values: z = (100 x .0124)/[.95115 x √100] = .1307 UNLLI(.1307) = .3373 (determined using the UNLLI table provided in the January 2010 issue of CEM) Equivalent Rebate = [100 x .0124 x .2]/[100 x .0124 + .3373 x .95115 x √100] = .056, or 5.6% These numbers mean that if the casino intends to give back 20 percent of its theoretical win from this baccarat player, then the actual rebate on loss given to the player should be 5.6 percent. Now let’s suppose a casino is giving a 20 percent rebate on actual loss to a player who has played 300 baccarat hands, again betting on the player hand each time. Using the approximate formula, we can calculate the following values: z = (300 x .0124)/[.95115 x √300] = .2258 UNLLI(.1844) = .2944 (from the UNLLI table) Equivalent Rebate = [300 x .0124 x .2]/[300 x .0124 + .3154 x .95115 x √300] = .1, or 10% Now let’s see how the equivalent rebates calculated in these two examples compare to those calculated by Monte Carlo simulation, which we can also use to estimate an equivalent rebate on loss corresponding to a specified percentage of theoretical loss. This is easily calculated in Step 2 of the simulation experiment described earlier; we can either use a trial-and-error approach or perform a bisection search on r to obtain the value r0 that yields the specified percentage of theoretical loss. (The bisection search method can be used to calculate an equivalent rebate using the simulation code, as described below.) Running the simulation code 1,000 times gives us a 1,000 x 32 matrix W, the first 16 columns of which contain the observed casino win amounts without rebate, and the last 16 columns of which contain the corresponding values when an r percent rebate is applied: W(i,j) = observed casino win in iteration i, i = 1, 2, … ; 1,000 Columns j = 1, 2, …, 16 correspond to casino win without rebate for number of hands n(j), and columns j = 17, 18, …, 32 correspond to casino win with rebate for number of hands n(j). W(i,j+16) = (1–r/100) x W(i,j) if W(i,j) > 0
M(j) = the mean of 1,000 W(i,j) values, j = 1, 2, …, 32, with first 16 columns representing the mean casino win without rebate, and the next 16 columns representing the mean casino win with rebate.
100 x (M(j) – M(16+j))/M(j) = the estimated percent of casino win returned to the player.
Here the j-th column corresponds to n(j) baccarat hands given in the description of the simulation experiment.
Using the above formulas for calculating equivalent rebate by simulation, a computer code in the R language was developed to calculate the percent returned to the player based on the simulation results.
Now let’s recalculate our rebate for the example player above who played 100 hands at the casino with a 20 percent rebate on loss program. Let H(r) be the actual percent returned to the player when the casino gives r percent back to every losing player. The code can be used to calculate H(r) for r = 4 and r = 7: H(4) = 16.87, and H(7) = 29.52. This implies that the value of r corresponding to 20 percent falls in the interval (4, 7).
In a bisection search, the value of the function H(r) is calculated at the midpoint of the interval: r = (4 + 7)/2 = 5.5, and H(5.5) = 23.20 > 20, implying that the desired r is inside the interval (4, 5.5), the midpoint of which is (4 + 5.5)/2 = 4.75. H(4.75) = 20.03, which is close to 20 percent.
Therefore, the casino should be giving r0 = 4.75 percent to each losing player (instead of 20 percent) if the intent is to give 20 percent of the casino’s theoretical casino win to losing players.
It is important to note that the approximate formula based on CLT had given r0 = 5.6 percent. In other words, the UNLLI-based approximate formula is a bit generous. When we used the simulation code to calculate the actual percent returned to the player with r0 = 5.6 percent, it resulted in an actual percent returned of 23.6 percent—not 20 percent.
Now let’s recalculate for our losing player who played 300 hands. Using the same method as above, we start with the (8, 9) as initial interval for r, with H(8) = 19.39 and H(9) = 21.82, and go through the following sequence of bisection searches to find r0 = 8.25:
Interval Midpoint Actual Percent Returned
(8, 9) 8.5 20.61
(8, 8.5) 8.25 20.00
The UNLLI-based approximate formula had resulted in r0 = 10 percent, which is again higher than the percent obtained from simulation. When we used the simulation code to calculate the actual percent returned to the player with r0 = 10 percent, it resulted in an actual return of 24.25 percent—again, not 20 percent.
The UNLLI-based approximate formula tends to return a higher actual percent of win than the casino intends to provide. This could result in large overpayments of rebates to losing baccarat players in the long term. We therefore recommend that casinos use the Monte Carlo simulation instead of the UNLLI-based formula to calculate a rebate that is equivalent to a specified percent of theoretical casino win. We also recommend running the simulation for at least 1 million baccarat hands (instead of the 1,000 used in this article). Mathematics can be critically applied to show the difference between expected win and actual game play, and if this difference is not carefully understood, it will result in rebates that are higher than anticipated or intended. In this case, we needed to resort to computer-driven simulations to illustrate the difference, but in today’s profit-focused world, blindly following incorrect assumptions may be illustrated on the casino’s balance sheet—and may no longer be hidden by the fundamental profitability of the business.