Modeling Reality in Role-Playing Games:
Insights from Mathematical Psychology

by Simon Dennis

In which the author shows how a century of scientific research can finally prove useful to roleplayers

Inside every role-player is a budding game designer. Role-players tend to be an opinionated lot at the best of times, but nothing generates quite as much controversy as the rules of the game. Most players have dabbled in creating their own systems and the ones that haven't can certainly propound on the virtues and limitations of the different systems they have played. On more than one occasion my own group has given up on actually playing the game as we descended into mortal combat over the system.

The contention and acrimony associated with game systems is almost certainly generated by the fact that different people are looking for different things. Some players want a rich array of options and outcomes. They want the system to model the cinematic images they form in vivid detail and they want that model to be in some sense realistic. Other players feel the weight of the dice and tables acutely and prefer the system to provide just the bones upon which they hang their evolving sense of the scene. It takes a particularly clever system to satisfy both styles of player and most games fall somewhere on a continuum between simplicity and precision.

One of the problems facing game designers and players is that typically they have little understanding of the reality they are trying to model. They are forced to rely on their own vague intuitions about how likely people are to succeed, how fast they can do things, and how that will change as they gain experience. Fortunately, these topics have been the subject of over a century of scientific study and we can draw on the accumulated understanding of mathematical psychology to provide some insight into the reality that designers are attempting to capture.

In this article, I will summarize what is known about human attributes, success rates, reaction times and the effects of learning - all critical components of most role-playing systems - with the objective of informing both the construction of new systems and the discussion of already existing ones. Sometimes existing game mechanics are a good reflection of how humans perform and sometimes they aren't. Consequently, I will also look at how different phenomena can be captured with simple combinations of die rolls, so as to move towards the ultimate goal of a simple, but realistic role-playing system.

Before diving into the psychological theory, however, let's start by introducing some simple ideas in probability theory by reviewing some of the ways in which RPGs typically use dice to model reality.

Many RPGs use a simple die role in order to decide an action (e.g. the original D&D combat system). Rolling above a certain number typically indicates success. When only a single die is used all values on that die are equally likely. For instance, if you roll a D6 then you have the same chance of rolling a 1 as a 6 and the probability of rolling any given value is 1/6. We can graph this by indicating all of the possible outcomes on the x axis and plotting each of their probabilities.

This is called a uniform distribution and is probably the easiest distribution to understand. If the system requires that you roll a 5 or a 6 to succeed then you just add the probabilities of those two results and you know that you have a 1/3 chance of success.

Another mechanic that RPGs often use is to roll multiple dice and to add their values. The next figure shows the probability distribution that results when you roll three D6 and add the values (as done in the standard D&D attribute generation). Unlike the uniform distribution above, the probabilities of the possible outcomes are not equal. The probability of rolling either a 3 or an 18 is quite low (1/216), while the probability of rolling one of the middle numbers, either a 10 or an 11 is 27 times as great. Rolling many dice and adding the values approximates a normal distribution.

When modifying an added roll many games will simply add the modifier onto the total of the dice (e.g. GURPS). This sort of mechanism moves the whole probability distribution up (or down if the modifier is negative), but does not change the shape of the curve in anyway.

The final mechanism that deserves a mention is the dice pool, as used in all White Wolf games, and which is becoming more and more common in recent years. In games using this method, generally a difficulty level is set (either situationally or for the entire system) and players roll a set of dice noting how many exceed the difficulty level. As a character's skill increases they are allowed to roll additional dice, hence increasing their chance of more success. Some systems take the highest number rolled, or the sum of all the dice, rather than number of successes, but the process is much the same.

There are several different takes on this mechanism, each having different probability distributions. However, in general the number of successes (or magnitude of the roll) will approximate the normal curve with both small and high numbers of successes being improbable. Note, however, that adding a dice to the dice pool is not the same as adding a modifier to an added set of dice (see above). It moves the probability distribution up and spreads it out. In particular, it is always possible for a highly skilled character to fail with all of their dice regardless of how many they have, so the distribution is in some sense anchored at 0.

Generating Character Attributes

When it comes to the generation of character attributes such intelligence, strength and size most systems do a good job, helped perhaps by the fact that the mother of all role-playing games, D&D, got it pretty right from the start. D&D uses a 3D6 mechanism, which approximates a normal distribution. Physical attributes such as height and weight have been shown to be approximately normally distributed and standard tests of intelligence are also based on the assumption that a normal distribution underlies intelligence scores.

Modeling other (and more subjective) attributes such as dexterity, endurance, charisma, attractiveness, wisdom, technical aptitude, strength of will etc, is a little more difficult as there aren't any good ways for measuring these qualities. However, using a normal mechanism like 3D6 is probably a good guess. The fact that the normal distribution fits human attributes so well in such a wide variety of situations probably occurs because characteristics such as these are the result of large numbers of processes all of which add to the final result. For instance, an intelligence test contains a variety of items each designed to test a different competency. The final score that is calculated is the addition of the scores on these items. When large numbers of random variables are added they produce an approximate normal distribution, even if each of them is not normal itself. As a consequence, when there is no good way for deciding what to use, the normal is a good default.

Skill Success Rates

When modeling the skill with which a character performs an action such as swinging a sword or picking a lock the normal distribution is also a good choice for deciding upon success. In the psychological literature, the normal distribution is often found to be an accurate model of human performance whether talking about detecting stimuli such as very dim lights or very soft sounds, or for higher level cognitive tasks such as recognition memory.

In fact, the Theory of Signal Detection (TSD) is probably one of the most successful theories in the history of psychology, and thus makes a great model. Developed during the second world war to help understand the performance of radar operators, the theory states that when you are trying to detect a stimulus, like a very dim light, a sample is taken from a normal distribution (i.e. the dice are rolled and added) and the result is compared against a criterion. If the score is above the criterion you successfully detect the light, otherwise you say there is no light. If this dice mechanism sounds familiar it should. It is essentially the same procedure that is used in games like GURPS to resolve skills.

So, in the case of simple skill resolution at least some games are doing a good job at modeling human performance. However, in detection style tasks, like perception, there is another distinction that is made in Signal Detection Theory that is usually missing in RPGs. This is the distinction between accuracy and bias. If you are trying to detect a dim light in an experimental situation then a good strategy is to always indicate that it is present. If the experimenter always showed a light then the canny subject would always be right without actually detecting the light. For this reason, wily experimenters introduced the catch trial - trials on which there is no light. The subject who is always indicating that the light is present will then be incorrect on the trials on which the light did not appear - that is they will false alarm. These subjects are said to have low accuracy and high bias. In the TSD, this is modeled by assuming that subjects also sample from a normal distribution when there is no light, but that this normal distribution is shifted down, making it less likely that it will now exceed the criterion. The accuracy (or lack there of) of each person is indicated by the distance between the "light" and "no light" distributions, while their bias is indicated by where they set their own perceived criterion.

Translating that technical explanation into game terms, the high bias character is the one who is always super alert. They are unlikely to ever be ambushed, but they will also spend a lot of time and effort beating empty bushes. To model this you would throw in some catch trials - occasions when there isn't anything out there, but asking the players to roll anyway. If their roll (with whatever skill adjustment) exceeds the set criterion then they believe there is something out there any way. Even if you don't want to incorporate this level of detail into the formal mechanics of a game, it can be a useful thing for a GM to keep in mind if they have a recalcitrant player claiming they are always alert and therefore can never be ambushed. Once they have woken the party five or six times on wild goose chases they will start to understand the disadvantages of hyper-vigilance.

Reaction Time

While RPGs typically do a reasonable job of modeling skill success they generally do rather poorly at modeling the time those actions take to complete. Initiative systems typically involve the rolling of a single dice (uniform distribution) with perhaps some modifier to indicate character's speed, and taking into account the type of weapon, the relative positions of the fighters etc. However, when someone gets faster at performing a skill, either through practice or because they are in a heightened state of arousal, there typically aren't large differences in their fastest responses, as your typical initiative system would suggest. Rather, skilled protagonists have their average speed decreased, because they make smaller numbers of very slow responses. Furthermore, the distribution of reaction times is typically not symmetrical but is pushed to the left, that is, towards the fast responses (see the figure).

We can get a rough approximation of this distribution by rolling two die and subtracting the smaller from the larger. The distribution that this creates when the die is a D6 is shown in the next diagram.

Notice how the distribution is pushed to the left, with low numbers being predominant. Using this method, the lowest roll would win iniative. Slower players should roll dice with more sides (say d20s rather than d12s) and the preternaturally fast should roll dice with few sides. This gives every character the possibility of a very fast action, but ensures that those highly skilled have few very slow actions.

In the discussion thus far speed and accuracy have been treated separately. Typically, however, people show a speed/accuracy tradeoff. They can either focus on accuracy at the expense of speed or they can focus on speed at the expense of accuracy. This shift of focus can have a major effect on the success or speed of the action, but again, this is a factor that is often missing from RPGs. One could imagine how a mechanic based on this phenomena could help to build the intensity of combat as players are forced to choose between speed or accuracy, based on the current circumstances. Consider the following vignette:

Tokanaka, assassin of the Hamoi Tong, waited patiently within the eves of the Acoma ancestral home. He knew that he was the ideal choice for this sort of job. Unlike those sworn to house colours he could take a Lord or Lady without so much as a quickening of the pulse. He had been selected for his cold resolve and it served him well this moment. Footsteps in the room next door alerted him to the arrival of the Lady of the Acoma. He readied his dart and waited. The Lady entered and Tokanaka took aim. The dart flew true and Lady Mara had only enough time to look startled before the dart bisected her eyes and ended the Acoma line. Tokanaka dropped to the floor and dashed towards the open window as the Acoma guard rushed into the room. He turned briefly and released his remaining three darts in quick succession. He knew that without the time to take aim his shots would be wild as best, but he hoped to gain the time necessary to secure his escape.

The difference between the Tokanaka's first shot and the subsequent shots is substantial but many game systems do not reflect this difference. However, it should be noted that West End Games' D6 system is one that goes someway towards this, allowing players multiple actions in a round, at the cost of one dice from their pool for every action above one.

The Effects of Learning

Finally, one other area in which psychology can provide insight is the nature and effects of learning and improving skills. Almost all RPGs provide a method by which characters can advance in proficiency and aquire new skills, and interestingly, they are often quite realistic.

Many modern games have some mechanism by which learning becomes more difficult as the skill level of the character increases. Such a mechanism is not only useful because it tempers the speed of advancement and prevents high level characters from becoming overwhelmingly skilled, but is also a reasonable expression of how learning works.

The most successful model of simple associative learning, the Rescorla Wagner model, states that the amount that is learned on a given trial is proportional to the amount still to be learned. When learning begins, there is much to be learned and so progress is fast. As learning progresses there is less to be learned and the rate of learning decreases. This model gives rise to a exponential curve (see the figure below).

This sort of learning function applies well to simple tasks, such as a dog learning to salivate when a bell is rung, but is not as good a description of what happens in more complex cognitive tasks such as categorization or remembering. The common tasks of human performance are better modeled with more complex, high level functions. In practice, however, an exponential progression has been shown to be a sufficient approximation. It is also easy to implement, by simply making increases in the level of a skill contingent on the current level of that skill, as mentioned above.

One good example of this is in Chaosium's Call of Cthulhu RPG. There skills are percentage based, with a set range from 1 to 100. The exponential method is thus implemented by requiring that a player roll above their current skill level on d% in order to increase that skill. In this way the probability of increasing is proportional to the amount left to learn.

Conclusions

In this article, I have given a very brief outline of some of the major psychological phenomena that are relevant to the construction of an RPG system along with some simple mechanisms for implementing these principles within a game. What has been presented, however, is only the tip of the iceberg and I would encourage anyone interested in these issues to grab a cognitive psychology text and take a look.

Simon has a PhD in Computer Science, a Bachelor in Psychology, and is currently lecturing and doing research with the Key Centre for Human Factors and Applied Cognitive Psychology, at the University of Queensland. He has a brain the size of a planet and more cold strategic cunning than Sun Tzu and Machiavelli combined. Thus he tends to make the rest of us very, very nervous. He also has a kitten called Min.