Authors’ Note: This article introduces the difference between game play and expected value, following from our CEM Blog series on the difference between actual and theoretical win. In this article, we show how mathematics and statistics play a crucial role in many of the marketing decisions routinely taken by the casinos to increase their base of loyal customers. Specifically, this article discusses the commonly used “rebates on player loss” program and demonstrates why careful analysis is required in using this marketing tool.
The history of rebates or discounts on a player’s actual losses goes back to the late 1970s, when high rollers would gamble for 30 to 40 hours on one trip, taking breaks for eating and sleeping, and generating gambling debts exceeding a million dollars, which was a large amount back then. These gambling debts remained unpaid for large periods of time, and the idea of “rebates on player loss” was born.1 This program has now become an expensive marketing tool and is offered by the majority of large U.S. and international casinos, sometimes without doing a proper mathematical and statistical analysis. Researchers have considered the problem of assessing the profitability of premium players and have shown that high rollers don’t necessarily translate to high profits.2 A formula to calculate a rebate on actual loss that is equivalent to a specified percent of player theoretical loss has been developed3 based on the well-known normal distribution approximation for total casino win in a sequence of N trials. The issue of how large N must be before this formula can be used, however, has been ignored in gaming and statistical literature. In this article we will address this issue. We will also show via examples why, if not properly used, this marketing tool may actually be generating huge losses for the casinos. In other words, some casinos may be paying their high rollers to fly in on a chartered plane, stay in luxury suites and take home gambling winnings as well.
The Rebate on Loss Program
A rebate on loss program works like this: Premium players are told in advance that they will receive a discount of X percent on any gambling loss incurred on a particular visit. With a 10 percent rebate on loss, a premium player losing $50,000 would, for example, get back a $5,000 rebate, whereas a player winning $50,000 would keep the entire win amount. Large Strip casinos in Las Vegas have been rumored to offer loss rebates of as high as 25 percent to their premium players.4 The question is, does a 25 percent rebate on a player’s actual loss on a trip equal 25 percent of the player’s theoretical loss?
Let’s discuss the rebate on a loss program for baccarat, since this is the game for which rebates on loss are most common. In baccarat, a player can bet on either the player hand (which pays even money) or the banker hand (which pays even money minus a 5 percent commission). The probabilities of the events of interest are:
probability that player hand wins = 0.4462466
probability that banker hand wins = 0.4585974
probability of a tie = 0.0951560
The calculations in this article will involve two parameters of the game: the expected value (EV) of a single unit wager, and its standard deviation (sd). The EV of a $1 wager is the dollar amount a player can expect to win in a very long sequence of a particular wager. The sd measures the long-term volatility in player win amount. The general formulas for EV and sd of a wager are given as:
EV = Event_Probability x Payout
sd = Σ Event_Probability x (Payout – EV)2
where Σ denotes the process of adding over all possible outcomes in the particular bet.
For a game with three outcomes (like baccarat), if a wager pays $A-to-1, the EV and sd for the wager are calculated as:
EV = Win Probability x A + Tie Probability x (0) + Loss Probability x (-1)
Since a bet on the player hand in baccarat pays even money, EV($1 wager on player hand) = 0.4462466 x (+1) + 0.0951560 x (0) + 0.4585974 x (-1) = – 0.01235.
The house advantage of this wager is 100*(-EV) = 100 x ( +.01235) or 1.235%.
The sd of a $1 bet on player hand is 0.4462466 x (+1 + .01235) + 0.0951560 x (0 + .01235) + 0.4585974 x (-1.01235) = 0.95115.
Suppose a casino is offering a rebate of 25 percent on actual loss in baccarat. A premium player places one wager of $10,000 on the banker hand, and the casino charges a 5 percent commission on winning bank hands. What is the EV of this wager?
EV($10,000 wager on the banker hand with 5 percent commission and a 25 percent rebate on actual loss) = 0.4585974 x (.95 x $10,000) + 0.0951560 x (0) + 0.4462466 x (- 0.75 x $10,000) = +$1,009.80.
Here, .95 in the first term comes from the 5 percent commission on banker hands, and .25 in the third term is the 25 percent rebate on actual loss. Note that the player’s EV is +$1,009.80, which translates to a casino loss of $1,009.80 on a $10,000 wager, or a negative house advantage of -10.09 percent.
This simple example shows that a 25 percent rebate on actual loss makes the banker bet a losing proposition for the casino. In other words, giving back 25 percent on actual loss is not the same as giving back 25 percent of theoretical loss.
In this example, we present the results of 100 simulated baccarat hands of $1 wagered on the player hand, for each of 10 players. Suppose the casino is giving a 20 percent rebate on actual loss.
Table 1 shows the simulation results. Player 1 ties on the first hand, loses the second hand and loses the 100th hand. Player 1’s total loss in 100 hands played is found by adding these 100 results (not all results are shown). Since Player 1 has lost $12, $2.40 will be returned, resulting in a loss of $9.60 for Player 1. Player 2 has won $9, so the rebate does not come into play here.
The 10 players’ combined loss from 100 hands each is obtained by adding the Player Win columns; the sum of the Player Win – No Rebate column equals -$6—i.e., the casino wins $6 from these 1,000 bets of $1 each. The sum of the Player Win – 20% Rebate column equals +$2.4, which translates to a total loss of -$2.40 for the casino. The point of this example is that a game with a positive house advantage can easily become a losing game for the casino with a rebate on actual loss program. Therefore, careful mathematical analysis must be performed before announcing a rebate on loss program.
The problem lies with not clearly understanding the EV of a player’s wager, which is a long-term concept. For example, a player will lose 1.235 cents of every $1 wager on the player hand in baccarat in a very long sequence of bets of $1 on the player hand. The situation is quite different if we are looking at a small number of hands played.
In order to clarify this issue, we designed the following simulation of baccarat games:
A total of n hands of $1 wagers on the player hand is simulated (as in Example 2), and the player win amount A1 in n hands is calculated. This process is repeated 1,000 times, resulting in a column of 1,000 player win values—A1, A2, …, A1,000. Dividing each of these A-values by 100 gives 1,000 estimates of EV of player win when no rebate is given and n hands are played. We also calculate a column of player win after applying a 20 percent rebate on player loss; this is accomplished by calculating .20 x Aj for every Aj that is < 0. The averages of 1,000 values of the Player Win – No Rebate and Player Win – 20% Rebate columns are calculated next, and these are estimates of EV(Player Win – No Rebate) and EV(Player Win – 20% Rebate).
From these two estimates, we can calculate the percent of win returned to the player as:
100 x (1 – EV Player Win 20% Rebate/EV Player Win No Rebate)
This experiment was run for n = 100, 200, … , 1,000; 2,000; 3,000; 5,000; 10,000; 150,000; and 20,000. The percent of win returned to the player is plotted versus the number of hands played (n) in Figure 1. It can be seen from Figure 1 that for n = 100 hands, the percent of win returned to the player (with the 20 percent rebate program) is close to 95 percent. This number comes down to 30 percent for n = 1,000, and unless the number of hands is increased above 5,000, this number is much higher than the intended 20 percent (see the blue line in Figure 1).
So, the real question is how do we apply statistical methods to determine the correct rebate amount? In this example, we will show how a 16.38 percent rebate is equivalent to a 25 percent rebate against theoretical loss (in a game of baccarat with 800 hands played).
To show this calculation, there are three dependencies:
1) the number of wagers
2) the house advantage per unit wager
3) the sd per unit wager
The formula for calculating a rebate percentage on actual loss that is equivalent to a specified percentage (r) of theoretical (long-term) loss4 is given as:
Suppose a casino wants to give a rebate on actual loss in baccarat that will be equivalent to a 25 percent rebate on a player’s theoretical loss. Let’s assume that the player has played 800 total hands, betting the same amount each time—half of the time on the banker hand and the other half on the player hand. The per unit house advantage then, counting ties, is the average of the banker and player house advantages (1.15 percent), and the sd is approximately 1.
Substituting N = 800, HA = .0115, and sd = 1, we first calculate:
z = 800 x .0115/(1 x √800) = 0.4243
UNLLI(z) = .2236 (from Table 2 above – Row 5, Column 3)
Finally, a rebate percent equivalent to a 25 percent rebate on player theoretical loss is calculated as:
Equivalent rebate = (800 x .0115 x .25)/[800 x .0115 + .2236 x √800 x 1) = 3/18.32 = .1638, or 16.38%.
Note that the above formula for calculating an equivalent rebate is based upon normal approximation to the probability distribution of total casino win in n identical wagers; this approximation should be reasonable for a game such as baccarat with almost equal win and loss probabilities, but it is not known if this formula can be used for an n value as small as 100.
Also note that these formulas should not be applied without specific knowledge of their limitations. For example, the use of these formulas for games in which the win probability is too small for a small number of hands could be disastrous, since the normal approximation to the probability distribution of total casino win will not be very good in such cases.
Expected Value vs. Game Play
The expected value of game play goes by other names, such as theoretical win. Expected win forms the basis on which many compensation are based, including rebate or points programs, and as we move forward in this series on game analysis, we will show more examples of the dangers of applying over-simplified models to characterize player returns. As the supply of gaming products continues to grow, it is critical that an understanding of the actual game play is part of the calculation of the player profitability. The operators who apply these deep techniques gain a mathematical edge over their competitors—an edge that has a direct impact on profitability.
In the past, the casino business was highly profitable, and tactical decisions were normally sufficient for the maximization of profit. In today’s world, it seems reasonable to question if these practices can continue and whether a casino should operate without mathematical optimization.
1 Lucas, Anthony F. and Kilby, Jim (2008). Principles of Casino Marketing. www.principlesofcasinomarketing.com.
2 Lucas, Anthony F., Kilby, Jim and Santos, Jocelina (2002). Assessing the Profitability of Premium Players.
3 Kilby, Jim, Fox, Jim and Lucas, Anthony F. (2005). Casino Operations Management, John Wiley & Sons
4 Robert C. Hannum and Anthony N. Cabot (2005). Practical Casino Math, 2nd Edition.