Gaming is the embodiment of math as entertainment. While gaming can entertain in ways apart from math, at its core gaming is about winning and losing money and the manner in which winning and losing money is accomplished. This is particularly true for slot machines because there is little social interaction or strategy involved in playing a slot. It is just a random outcome. Both operators and manufacturers would like to provide slots with entertaining math models, but today there is no consensus about what those entertaining math models look like, nor is there even a language through which such mathematical ideas are communicated. Indeed, most of what is communicated by operators to manufacturers about the mathematics of a slot game has little value in determining how players will receive the game.
Slot math is communicated by means of the PAR sheet, which provides an answer to the question, “If a slot is played many times, how much can the house expect to win, and how far can the house expect the actual result to vary from the programmed result?” This is valuable information for the house and for the tax man, but they say very little about what an individual player will experience. This is because PAR sheets still follow the same format as 20 years ago when most slot machines played only a single line and most players played only max bet. Back then one could get the feel of a game from examining the PAR sheet. Today slots have many pay lines, players wager in differing amounts, and there are dozens if not hundreds of different bonus features. The PAR sheet while still accurate in determining the theoretical return of the game does not give the reader much clue as to how the game will play out for the average customer.
One way to analyze math so as to get an idea of how a game will play out is to run a “Road to Ruin” analysis. The Road to Ruin analysis consists of a large number of computer simulations, each of which represents a single player with a predetermined bankroll playing a given game. The simulation plays until each virtual player runs out of money and records events that the programmer deems to be important (hence the name “Road to Ruin,” although in practice the analysis will include a “walk away” parameter for simulated players who do very well). For example, a Road to Ruin analysis may simulate 20,000 players with a bankroll of say, $100, wagering $1 per play. The simulation would then report back data on things like what percentage of players that doubled their bankroll during the simulation, what percentage got to play the bonus, what percentage had a win in excess of 100X their bet, how long did the luckiest 25 percent get to play or how long the unluckiest 25 percent get to play. An example of the output of the above analysis shown in graph 1:
Looking at the graph 1, the reader can quickly and easily see that a player with average luck can expect to play the game in question for about 500 games, will likely be ahead at sometime during his play and has a small chance of doubling their money and winning a $50 but a good chance of hitting a bonus and winning $20. The table also shows what a comparatively lucky and unlucky player can expect. The information presented in the table is very useful to a person that is seeking to evaluate the game and can’t be obtained by looking at the PAR sheet.
The graph 1 displays just a small slice of what can be analyzed using the Road to Ruin. The analysis can be easily be scaled up to include more parameters, percentages and players in a manner that is simple to communicate. Additionally, the analysis is ideally suited to make comparisons. For example a game manufacturer or operator might want to know how loosening or tightening a slot will affect the players experience. The Road to Ruin could be re-run to include different payback and the data shown on graph 2.
Once again, the graph 2 only displays a small slice of what the analysis can reveal but even this limited comparison gives the analyst a good idea of how the player’s experience changes when a game is tightened or loosened.The comparisons that can be made go beyond payback percentages. The Road to Ruin can also be used to segment between customer demographics. For example, by varying the parameters for the starting stake and the wager amount, the manufacturer can see how a game will played for a high roller, low roller or anybody in between. Because of this ability to simulate the game play as well as the players, the Road to Ruin analysis can be tailored to almost any casino setting.
In addition to revealing useful information the Road to Ruin has the added benefit of being easy to set up. The programming involved was never complex to begin with but as computers have become faster and programming environments have become more user-friendly (I personally do mine within using Visual Basic for Applications, which is contained in the version of Excel that I use to calculate the math in the first place), which means that analyses such as the Road to Ruin can be executed by people with very basic computer knowledge. However, even if analyses such as the Road to Ruin were difficult to execute, they should still be done because the days of simple, easy-to-analyze, single-line slot games have ended.
It should be self-evident that both game designers and operators demand greater sophistication in the manner that the game math is created and evaluated. Slot machines have better graphics, can offer more features and can interface directly with a player’s slot club and generally exceed in almost every way their ancestors of a few decades ago. Unfortunately this has not translated into a corresponding increase in the amount of money that they bring into the house, nor has it translated into an increased popularity of the slot genre as a whole. Today’s slots have a lot more going, on so it follows that the analysis of today’s slots should have more going on as well, and the Road to Ruin analysis is a good place to start.