Experience CurvesSergio Mascarenhas May 30, 2000
In my last column I wrote about ability curves, graphical means of depicting how different members of a population distribute among the different levels of the abilities they have in common. This month I'm going to address a related concept: Experience Curves. These are graphical means to show the evolution of PCs, NPCs or any other type of entities in their abilities. While the ability curve allows us to compare different members of a given population in a given moment in time, the experience curve allows us to compare the different ability levels of a single creature in different moments in time.
Why look at experience curves? Because, as we will see, they allows us to gauge whether the game system models how the characters advance in their abilities in a consistent way.
I'll deal with the Experience Curve in three parts, according to the next plan:
Since RFF#04 was a really long file, I decided to divide it: This week I present the Experience Curve concept. Next week I'll analyze the experience curve of several games. I hope you enjoy this column.
EXPERIENCE CURVE: THE ESSENTIALS
Experience became a key concept in any RPG since D&D got into the market. Gaining experience is one of the major objectives of any PC/player. Rules about experience usually take up a large part of any game system. So, let's look at how different game systems model experience. To do so, we will use a graphical representation which I've called the Experience Curve. But before we look at this curve, we need to understand the concepts that underly it.
For the purposes of the present column, Experience means the level of any individual in a particular ability. As I mentioned in previous columns, I make a distinction between two types of abilities, attributes and skills. The former are more or less set. The latter change upon time. This means that usually experience is more relevant when we're looking at skills than at attributes. (1) The way abilities change is through experience. A given level of experience tells us how well a character is able to perform in a given field.
Experience is acquired. This means that characters start with low levels of experience, and somehow are able to advance them. The absolute departing level in most abilities is close to 0 at the moment of birth. A newborn is almost unable to do anything. (2) Through his life he'll develop a wide ranging set of abilities, each one at different levels. So, the character's level in each of his abilities is contingent on the moment in his life that we chose to look at. Usually, the more mature and the older he is, the better off he will be. Eventually, he may reach a decaying age, and his abilities may start to regress.
A further point relates to the "how" of experience. How does one gain experience. This may result from several alternative possibilities: pure talent or fitness; tutored training and learning; learning by doing (or "experience", in the strict sense); etc. While discussing the Experience Curve we will not care to distinguish among these, and will qualify them collectively as "Experience".
We can represent experience by defining a set of experience levels, each corresponding to a given platform of performance in the ability. For now, let's assume that the set is close: it starts at 0 (the lowest level, corresponding to no experience at all, in other words, complete inability to use the ability); and ends at a given maximum level (which value is contingent on each game system, and corresponds to full domain of the ability). (3)
To gain experience and advance an ability one must apply a certain set of resources. These include time, attention, equipment, tutoring, etc. Once more, we will not discuss in detail the nature of those resources. All we need to assume is that they're needed, and are somehow mensurable, in order to allow us to design rules to model it. (4)
Based on the discussion so far, we reach the next proposition: To model experience we need a scale of experience levels. Advancing from an experience level to the next is done by expending a certain amount of adequate resources. There's a certain relationship between the marginal investment of resources in the ability, and the marginal increase in the ability. (5) The graphical representation of the relationship between the investment in resources, and the level attained in the ability is no other than the ... Experience Curve!
The Experience Curve
To draw our curve we need to define the scales required to draw it. Let's assume for the present purposes that both these scales go from 0 to 1:
Is that all we need to know? No! There's a further consideration we must address before we're able to draw our curve: What's the relationship between a marginal investment in resources, and a marginal increase in ability? In other words, we know that a certain spending of resources leads to a corresponding increase in ability, but is the rate of increase in the ability directly proportional to the rate of spending of resources?
If there's a direct relationship between the marginal spending of resources and the marginal increase of ability, we don't get a curve but a straight line:
So, once more the question is: Does the relationship between the investment of resources in increasing an ability, and the marginal gain in experience vary in a direct proportion?
Other people discussed this question before me, and reached the conclusion that this does not hold true. They consider that there's a "diminishing return" on the investment in resources to increase an ability:
"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", states Simon Dennis in Places to Go, People to Be (6). For instance, this reasoning was the foundation of the free RPG Alternate Realities (it can be found in the free RPGs section of RPGnet), which refers to this as the Law of Diminishing Returns, a terminology that we will use from now on.
Coupling this with the guidelines above on how to draw the Experience curve, we get the next curve:
I don't know how well informed you are on the Rescorla Wagner model (RWm, from now on), but if you're like me, you would not have a clue about it if it wasn't for Simon. So, I did a little research on the web to get as much info about it as I could find. (Since I did this months ago, the links I could find may no longer be accessible. And since you folks can do as much, I'll not bore you with the details of what I could understand about it.) The main finding was that... the RWm is only able to explain a very narrow set of cognitive processes, a set that does not include human learning. In fact, that's something that Simon stated in his article: "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."
Result, I was left without a clear scientific model on which to represent experience. (The RWm was hard enough for me to try to move into the "more complex, high level functions".) So, I decided to try to understand experience based on my empirical observations (7) about my own learning processes, what happens with my students, and my two-years old daughter, Maria, among others. When I started to think about it, I noticed a pattern that does not conform to the curve above. Let me provide a couple of examples:
By now you must be able to see where I'm going: The way I see it, learning - another way of saying "gaining experience" - takes place in two different phases, mediated by a "turning point":
This results in the next S-shaped curve:
The first half of the curve corresponds to the learning of the basics in the ability. In this phase we can see that learning is subject to increasing returns. The second part of the curve represents specialized learning, subject to diminishing returns. Of course, it's very hard to define exactly where the turning point - the comfortable ability level where one moves from the learning of the basics to the expert learning - resides: This is an abstract concept.
There's a further aspect that we must have in mind: In the representation above the investment in resources is constant, while the marginal advances in ability (how much one gains in terms of experience from that investment) are not. This means that for a constant investment in resources (each more "0.1" in the resources scale) we get variable increases in experience. It's intuitive that we may do things differently:
So far, I presented the way I look at experience. Next week I'll attempt to draw the experience curves of several games, so that we may gauge how well they handle experience according to my conceptions about it . Of course, I look forward to your comments on what has been presented so far.
(1) Of course, this is not to say that attributes can't change. For instance, if we define muscular mass as an attribute, it may change through exercise. So, gaining experience may have impact both in the skill (what we are able to do), and the attribute (what we use to do it).
(2) Yet, there are some abilities that he acquired and developed while in his mother's womb, like the ability to suck.
(3) The fact that the set is closed just means that there's a maximum reachable level of performance in the ability. Moving past that maximum level means that the definition of the ability changed, so it's no longer the same ability. This is something I'll discuss in RFF#05.
(4) To simplify the analysis we will assume that we're dealing with an "average person". We will not care to know how individual differences impact on gaining experience. These individual differences can be seen as differences in the set of "resources" any given character has. We will have time to look at this next month.
(5) What's more, keeping a certain level in the ability may require a marginal investment in resources lower to the one needed to advance from level to level. And conversely, a marginal deinvestment of resources in the ability may lead to a drop in the experience level. Do I need to say these are issues that I'll leave to a future occasion?
(6) I'm referring here to at excellent article by Simon Dennis published in Places to Go, People to Be that you can find at http://ptgptb.humbug.org.au/0009/simon.html, where he discusses the effects of learning.
(7) In fact, buried within those web files about the RWm are some hints that signal that the diminishing returns curve, derived from the RWm, does not even represent correctly the learning function of simple tasks. Instead, the learning of these tasks requires an S-shaped curve... somewhat like the one I propose.
(8) I'm applying here a standard rate of learning based on two, two-hour classes weekly.