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Mathematical models of Slot Games

Friday, 2 September 2011

I don’t know about you, but I do know how I order my food in an Indian restaurant. Rather than looking at the names of the starters and the main dishes, I select the type of food I want to eat (lamb or fish for example) and then I make my final selection based on the number of ´peppers´ behind the dish. The peppers typically indicate how spicy the food is. The titles of the dishes don´t mean that much to me (when I travel I usually am not capable of reading the names of the dishes in the first place). So I order based on selection of type of food and then I select how spicy I want to food to be.

Wouldn´t it be nice if we did the same thing with our slot menus? At the moment players can select the type of game they want to play; video games with stacked symbols or video games with free games, but they can´t select the peppers! We do not communicate with our players how spicy the games are.

Think about my Indian restaurant. If I order a dish I never had before, without the peppers I run the risk of burning my mouth or getting a dish which does not appeal to me because it is not spicy enough. Fair enough I can select a lamb dish (like a player can select a video game with stacked symbols), but there is no way of telling if the lamb dish is to my taste unless it tells me how spicy the dish is (like a player will not know how volatile a game is he never played before unless he spends quite a bit of time ´trying the game´)

It´s amazing really if you start to look for metaphors: bars do put the alcohol percentage on the various types of drinks, car makers do communicate how much gasoline a specific model car consumes, food products typically show the amount of calories per 100 grams; but slot manufacturers do not show the most important parameter for a player on the game. With all respect for the manufacturers, but for frequent slot machine players it is not the name of the game that attracts them, it is not the colors, nor is it the attract sound. All these elements will surely have a modest effect on the experience the player gets while playing, however it is the mathematical concept that is what keeps him on a machine (or makes him leave a machine).

I am increasingly cautious trying games I have never played before, because I burned my fingers too often. With the exception of WMS G-plus games, I never know what I get when I try a new machine. I find it amazing however to see WMS G-plus games mixed with WMS non G-plus games in the same bank of machines. I note that quite a few slot managers do not know that the mathematical concepts of G-plus and non G-plus are for different types of players. In my previous article I wrote about Time on Device Players and Gamblers. G-plus is for Gamblers, non G-plus more typically for Time on Device Players.

So, what do the peppers on the games have to communicate? What makes a game interesting for a Gambler and another type of game interesting for a Time on Device Player?

It is not the hit frequency; that I worked out a long time ago. Games with a very low hit frequency are not played by Time on Device Players, but also not by Gamblers. I hardly ever saw a Gambler play a ten line game with 10 credits on 1 line (which would be much more volatile than playing 10 lines with 1 credit per line). Players (Time on Device and Gamblers alike) play all the lines of a game. And they should; games are designed to be played on all lines. I do not understand why Slot Machines still have line buttons; that is so outdated! What we need is ‘forced bet’ (or ‘easy bet’ as some suppliers call it) on all machines. The minimum bet should no longer be the credit value, but all lines with one credit! If residual credits are the reason to keep the line buttons, better options are available.

The hit frequency has not really changed over the years. The majority of the games have a hit frequency somewhere between 33% and 20% (or between 1 out 3 to 1 out of 5 games).  Time on Device players would walk away from a machine with a lower hit frequency and Gamblers will only continue playing a game with a lower hit frequency if the mathematical model appeals to them.

Let me try to explain mathematical models for different types of players using an example with dice.

Using three dice with different colors, you can theoretically make 216 different combinations. On a slot machine we would call that the Game Cycle. Some of the combinations in the Game Cycle are Winning Combinations, some of them are Non-Winning Combinations. The percentage of Winning Combinations per Game Cycle is what we call the Hit Frequency.

Have a look at the pay-table below. I would call this a ´one pepper´ pay table

Hits per game cycle Probability Pay per game cycle Pay loading
Paytable (game cycle = 216)
SIX SIX SIX 9 1 0,00463 9 4,17%
SIX SIX 7 5 0,02315 35 16,20%
SIX 2 30 0,13889 60 27,78%
SIX FIVE FOUR 5 1 0,00463 5 2,31%
FIVE FIVE FIVE 5 1 0,00463 5 2,31%
FOUR FOUR FOUR 5 1 0,00463 5 2,31%
THREE THREE THREE 5 1 0,00463 5 2,31%
TWO TWO TWO 7 1 0,00463 7 3,24%
TWO TWO 5 5 0,02315 25 11,57%
TWO 2 30 0,13889 60 27,78%
76 0,35185 216 100,00%
35,19% 100,00%

The hit frequency is 35,19%. This means that 35,19% of the possible combinations in the game cycle are winning combinations. Theoretically the player throwing the dice will have a winning combination roughly once per three games. The theoretical Player Return Percentage is 100%. If the player plays long enough, he will not win and he will not lose.

By changing the pay table (or the symbols on the reels of a video game), the game designer can make the game more spicy.

The pay table below I would call a ´four pepper´ pay table

Hits per game cycle Probability Pay per game cycle Pay loading
Paytable (game cycle = 216)
SIX SIX SIX 150 1 0,00463 150 69,44%
SIX SIX 1 5 0,02315 5 2,31%
SIX 0,5 30 0,13889 15 6,94%
SIX FIVE FOUR 6 1 0,00463 6 2,78%
FIVE FIVE FIVE 5 1 0,00463 5 2,31%
FOUR FOUR FOUR 5 1 0,00463 5 2,31%
THREE THREE THREE 5 1 0,00463 5 2,31%
TWO TWO TWO 5 1 0,00463 5 2,31%
TWO TWO 1 5 0,02315 5 2,31%
TWO 0,5 30 0,13889 15 6,94%
76 0,35185 216 100,00%
35,19% 100,00%

Note that the hit frequency and the theoretical player return percentage are exactly the same as in the ´one pepper´ pay table. The difference is in the pay loading.

If a player plays any of the two games above long enough, he would not win and he would not lose. Yet there is a big difference in the mathematical concept.

If you had 100 RON in your pocket and each time you throw the dice you had to pay 5 RON, which of the two pay tables would you prefer? Gamblers would clearly prefer the ´four pepper pay table´, Time on Device players would prefer the ´one pepper pay table´.

Playing the one pepper pay-table you probably play for a very long time, you win a little, you loses a little; but chances are you can play very long before you run out of money (if at all; the pay table theoretically returns 100% to the player). This is what Time on Device Players like. The only downside of the ´one pepper pay table´ is that the player will never really win. He can play long, but it is very unlikely he will double his 100 RON. This explains the most heard complaint of Time on Device Players; “I never win”.

The Gambler, who plays the ´four pepper pay table´, has a chance to lose his 100 RON (and rather fast too). Although the game theoretically returns to the player 100%, if he is unlucky he will run out of money before he hits a combination that gets him ´out of trouble´. The flip-side is that if he gets lucky and throws three sixes, he can easily double his money (or more).

Let´s translate the logic from the dice game to a video slot game.

The pay table below I would call a ´one pepper´ pay table

Symbol Reel 1 Reel 2 Reel 3 PAY TABLE Hits Per Game Cycle Payment per game cycle Pay Loading
1 5 5 ¥ £ 1000 4 4.000 0,50%
£ 10 8 2 250 25 6.250 0,79%
¥ 10 2 10 £ £ £ 100 160 16.000 2,02%
BAR 19 18 16 ¥ ¥ ¥ 75 200 15.000 1,89%
SCATTER 1 2 4 SCATTER SCATTER SCATTER 50 216 10.800 1,36%
Blank 19 25 22 € / £ / ¥ € / £ / ¥ € / £ / ¥ 30 4.966 148.980 18,77%
BAR BAR BAR 25 5.472 136.800 17,23%
Total 60 60 59 BAR BAR 15 14.706 220.590 27,79%
Game Cycle 212.400 BAR 5 47.082 235.410 29,65%
72.831 793.830 100%
Hit Frequency Player return %
34,29% 93,44%

The yellow part on the picture above shows the reel stops. Reel 1 has 60 stops, of which 1 is the euro symbol, 10 are the pound symbol, 10 are the yen symbol, 19 bars, 1scattered symbol and 19 blanks. Total number of stops on reel 1 is 60, stops on reel two are also 60 and stops on reel 3 are 59. The game cycle (total number of combinations that can made with these three reels equals 60 x 60 x 59 = 212.400)

The pay table shows the winning combinations. In total there are 72.831 winning combinations. This represents 34,29% of the total combinations per game cycle. Theoretically this game will give the player a winning combination roughly once per 3 games. The Theoretical Player Return Percentage is 93,44%.

What makes this game a ´one pepper game´ is the pay loading. Playing this game, players will have frequent small wins. 93,44% of the total player return is in the smallest wins on the pay table (the yellow boxes under pay loading). Even if the player does not win any of the ´bigger wins´ of the pay table, he can play for a long time with his money. This game is typical for Time on Device Players. They can play very long with a limited amount of money; however they will almost never win a very large amount.

The next pay table I would call a ´four pepper pay table´

Symbol Reel 1 Reel 2 Reel 3 Pay table base game Hits Per Game Cycle Payment per game cycle Pay Loading
1 3 4 ¥ £ 500 1 500 0,25%
£ 8 7 1 100 12 1.200 0,60%
¥ 10 1 10 £ £ £ 75 56 4.200 2,10%
BAR 18 18 18 ¥ ¥ ¥ 50 100 5.000 2,50%
Blank 20 29 24 € / £ / ¥ € / £ / ¥ € / £ / ¥ 10 2.966 29.660 14,85%
BAR BAR BAR 4 5.832 23.328 11,68%
Total 60 60 60 BAR BAR 2 13.608 27.216 13,63%
Game Cycle 216.000 BAR 1 46.440 46.440 23,26%
69.744 199.691 100%
Hit Frequency Player return %
32,29% 92,45%

When the player plays the pay table above and he wins the free games scatter, the pay table will change.  The Theoretical Payback Percentage of the game (including the free games) is 92,45%.

During the free games the symbols on the reels and the pay table will show following:

Symbol Reel 1 Reel 2 Reel 3 Pay table free games Hits Per Cycle Payment per cycle Pay Loading
1 4 2 ¥ £ 100 48 4.800 7,04%
£ 5 4 4 75 8 600 0,88%
¥ 14 12 14 £ £ £ 50 80 4.000 5,87%
¥ ¥ ¥ 25 2.352 58.800 86,22%
Total 20 20 20 2.488 68.200
Game Cycle 8.000 31,10% 852,50%

What makes the game above a ´four pepper game´ is that 31,12% of the total Theoretical Return to Player is paid through the free games. As long as the player does not win the free games, the game does not return to player 92,45%, but only 63,68%! Theoretically the player will win the free games once per 296 games (729 times per game cycle).  During the free games the player will have a large win, guaranteed!

On the four pepper pay table, the player gets into trouble (financially) as long as the free games don´t hit. He knows however, that he will be ´out of trouble´ if the free games hit; and that´s exactly why he hangs in there.

Games with the mathematical concept above are –amongst others- Book of Ra (Austrian Gaming) and Zeus (WMS). They both have theoretical hit cycles for free games of around 300 games and they both have around 30% of the total player return in the free games. Now who likes to play these games with a significant bet? I do (and I am a gambler)!

The next course of The Slot Academy is October 30 to November 5 in the Hilton Hotel in Warsaw, Poland. Information through . Discounts apply for readers of Casino Inside Magazine.

Author: Editor

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