Authors’ Note: Most of the articles in this series on quantitative analysis are written from a casino’s perspective. In this installment, we take a look at a player’s experience at the casino and the effect that skewness has on that experience.
As we have shown in previous installments of this series, the house advantage, or hold, is a poor indicator of the players’ experience. This means that discussions about whether to “tighten” or “loosen” the floor should be taken with a proverbial grain of salt. Discussions about how the game play impacts the players’ experience are far more mathematically correct. In this article, we take aim at another commonly used metric describing gaming machines, the standard deviation. It is often quoted as a measure of the volatility of the game, but for some games it is quite misleading, as it is not designed to represent skewed distributions—and skewed distributions are very common in the gaming world.
In the April 2010 installment of this series, we used a Monte Carlo simulation and analyzed the results of playing a different number of hands with a single bet amount for the video poker game Deuces Wild. We found that, apart from being lucky, the most likely way for a player to have a positive experience was to play a limited amount of hands per trip. It is important to note that the house advantage (HA) of a game is not a guaranteed value. It can be interpreted as the average casino win over a very long number of hands or pulls of a slot game. The Central Limit Theorem (CLT) guarantees that the percentage of casino win for a large number of slot pulls will be very close to the HA. A wrinkle in this explanation is that no one knows how large the number of pulls has to be before actual percentage of casino win gets close to the HA; later in this article we will give an example from video poker simulations. The number of pulls needed to achieve the HA depends on the pay table of the slot game, a fact that people tend to forget. We will also consider the effect of standard deviation of payouts and the skewness of the probability distribution on the players’ trip experience. The standard deviation is used as a measure of the volatility of the game. Games with large jackpots have greater volatility and therefore a higher standard deviation.
The Effect on Players’ Trip Experience
Standard deviation tells you how much and how likely it is that some value (e.g., player win) will vary from the average value. If a casino wins an average of $10 million per day with a standard deviation of $1.5 million, it means that the chances of the casino making between $8.5 million and $11.5 million on any given day is about 68 percent, assuming a normal distribution. Figure 1 shows that within one standard deviation, there is a 34.1 percent chance of making between $8.5 and $10 million as well as a 34.1 percent chance of making between $10 million and $11.5 million. Normal distributions are used in many casino operations calculations, but sometimes the results are not very accurate because the distribution doesn’t fit the situation. Take the case of a fictitious player who plays a fixed number (n) of hands of the same game on each trip to the casino. According to the CLT, the actual percent of casino win from this player, in the long run (i.e., averaged over a large number of trips) will be close to the HA of the game. Our hypothesis is that higher the volatility (standard deviation) and skewness of the payout distribution, the greater n has to be for the actual percent of casino win from n pulls to be close to the game’s HA. Take a look at the probability distribution of payouts for 10/6 Double Double Bonus Poker. The “10/6” means that there is a 10X pay out for a full house and a 6X payout for a flush (see Figure 2).
Suppose we are interested in the probability distribution of the average casino win for a player who plays only five hands of this game and then cashes out. This can be obtained by simulating five hands of the game over a large number (M) of time, calculating the average casino win from five hands for each trip, and then determining the probability distribution from these average casino win values. We simulated 500 trips of a player who, on each trip, plays exactly five hands of Aces and Faces Video Poker and then quits. The results of the first and last 10 trips are shown in Table 1; the last two columns show the player’s per trip win and the actual HA calculated from the five hands played per trip. Figure 2 shows the probability distribution of these HA values.
(HA = 4.56, Standard Deviation = 27.8, Skewness = 81.4)
Table 1: Results of Five Hands of 10/6 Double Double Bonus Poker for 500 Trips
Trip Hand 1 Hand 2 Hand 3 Hand 4 Hand 5 Trip Win HA*
1 0 -5 -5 -5 -5 -20 80
2 0 0 -5 -5 0 -10 40
3 -5 -5 -5 -5 0 -20 80
4 0 0 -5 15 0 10 -40
5 -5 -5 -5 -5 -5 -25 100
6 -5 -5 0 -5 -5 -20 80
7 -5 0 -5 0 -5 -15 60
8 -5 0 0 -5 -5 -15 60
9 0 0 0 -5 -5 -10 40
10 -5 -5 -5 0 10 -5 20
. . . . . . . .
491 10 -5 -5 -5 -5 -10 40
492 0 -5 -5 10 -5 -5 20
493 0 -5 -5 0 10 0 0
* HA =100 x Trip Win/25, since the player wagers $25 on five hands per trip.
The probability distribution in Figure 3 does not resemble a bell-shaped curve by any stretch of the imagination, and is not even centered on the theoretical house edge of +4.56; the mean of the distribution, in fact, is -3.24. If the player were to play a larger number of hands, the mean would be closer to the theoretical 4.56. This is due to very high skewness of the payout probability distribution of Figure 2, as demonstrated by the following example.
To understand the effect of skewness on the distribution of actual HA from n hands, let’s look at a game that has a symmetric distribution (see Figure 4). If you were to flip a coin and bet on the outcome, the chances of winning that bet are 50/50. This is a fair game, as its expected value is equal to 0. Even with a small sample (the equivalent of a person playing a limited number of hands), we can still get a bell-shaped curve for trip win. Table 2 shows the trip HA calculated for five simulated hands for each of 500 trips. Figure 5 shows the probability distribution of the HA from five hands, estimated from 500 simulations.
(HA = 0, Standard Deviation = 5, Skewness = 0)
Table 2: Results of Five Hands of Coin Toss Game per Trip for 500 Trips
Trip Hand 1 Hand 2 Hand 3 Hand 4 Hand 5 Trip Win HA*
1 5 5 -5 5 5 15 -60
2 5 -5 -5 -5 -5 -15 60
3 5 -5 -5 -5 5 -5 20
4 -5 5 -5 5 5 5 -20
5 -5 5 -5 5 -5 -5 20
6 5 5 5 – 5 -5 5 -20
7 5 -5 5 5 -5 5 -20
. 0 0
491 5 5 -5 -5 -5 -5 20
492 -5 5 5 -5 5 5 -20
493 -5 -5 -5 5 5 -5 20
494 -5 -5 -5 -5 -5 -25 100
495 -5 -5 5 -5 -5 -15 60
496 5 5 5 -5 5 15 -60
497 -5 5 5 5 5 15 -60
498 -5 -5 5 -5 -5 -15 60
499 -5 5 5 -5 -5 -5 20
500 5 5 -5 5 -5 5 -20
* HA = -100 x Trip Win/25, since player wagers $25 on each trip of five hands.
The probability distribution in Figure 5 does resemble a bell-shaped curve, and it is centered on -0.66, which is quite close to the theoretical house edge of 0. The point of this example is that when the underlying probability distribution being sampled is symmetric, the probability distribution of average player win and HA per trip looks like the bell-shaped distribution centered around the theoretical value guaranteed by the CLT, even for a small number of hands played per trip and for a small number of trips. When the probability distribution is highly skewed, as is the case for 10/6 Double Double Bonus Poker, a very large number of trips would be needed for the average HA to be close to the theoretical HA; the probability distribution of the calculated trip HA would still not resemble the bell-shaped curve of the CLT.
Is it Standard Deviation or Skewness?
The obvious question that comes to mind is which parameter controls the player’s trip experience, the standard deviation or the skewness? In order to find an answer to this question, we simulated M trips of a player who plays exactly n hands of a game. The games selected for this experiment were: a coin toss game that pays 4-to-5, with an HA of 10 percent, standard deviation of 4.5 and skewness of 0; and 10/6 Double Bonus Poker, with an HA of -.1251 percent, standard deviation of 32.62 and skewness of 66.17.
For the number of hands (n) ranging from five to 50, we simulated M trips to play each of these two games. For each trip, n hands of the selected game were simulated and the trip HAs were calculated as shown in Tables 1 and 2. The percent relative error in the trip HA values were then calculated by the formula:
Relative Absolute % Error = 100 x (trip HA – game HA)/(game HA)
Figures 6 and 7 show the graphs of the relative absolute percent error versus the number of hands per trip (n) for number of trips M =1,000, 10,000, 100,000 and 1,000,000. Figure 8 shows this graph for 10/6 Double Bonus Poker, which has a highly skewed probability distribution of payouts.
It is clear from Figures 6 and 7 that increasing the standard deviation of the game from 4.5 to 32.62 does not have a significant effect on the relative absolute percent error. A comparison of Figures 7 and 8, on the other hand, shows that keeping the standard deviation at 32.62 and increasing the skewness from 0 to 66.17 has a huge impact on the relative absolute percent error. It is safe to conclude then that when the payout distribution is heavily skewed, the distribution of per trip HA takes a very large number of trips to stabilize around the theoretical value.
The Customer Perspective and Skewness
As we have demonstrated, the standard deviation is only related to game play when the skewness is low (such as with the coin toss game). Unfortunately, most games are not like the coin toss game and so we need to look much deeper to understand the player experience. As we have shown, if you are looking to understand the player experience, then you need to look past the simple measures—or the analysis may well be quite misleading.