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Experience Curves

Sergio 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 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.

Understanding experience

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:

I have a daughter, and she's 21 months old by now. It's a wonderful thing to see the development of a child. For instance, the way she learned to walk: first she had to learn how to stand up without external support; next how to walk with support; finally, how to walk without support. Once one knows how to walk, there are all other things one wants to learn: how to run; how to move up and down steps; how to jump; etc. What I noticed was that each new step in this progress was more rapid, and less painful then the previous ones. In other words, for the same investment in resources (time and effort), my child was able to reach more than in the previous development steps.

In my life I was able to learn Portuguese as my mother tongue, and latter French, and English. I also studied a little Italian, and can communicate with Spanish speakers, even if I was never taught that language. Finally, I had a couple of months on Mandarin Chinese, and on Kimbundu ( a Bantu language) before I dropped off. When analyzing my progress on those languages, it's important to rate them in terms of how similar they are to Portuguese (my mother tongue, and the one I learned first).

This leads us to the next ordering (from closer to more different): Spanish, Italian, French, English, Kimbundu and Mandarin Chinese. It took me one year to be able to entertain simple conversations in French. It required me about two years learning English to get to the same level. I only needed six months to do this in Italian (specially since I could leverage this learning in my knowledge of both Portuguese and French). As I said, I never learned Spanish, but this language is so similar to Portuguese that I don't even need to learn  it to be able to communicate with Spanish speakers.

On the other hand, several months studying Kimbundu or Mandarin Chinese only allowed me to understand the basics of the structure of the language. I'm sure that I would need some years of studying until the day I would feel confident enough to communicate in those languages. (8) Notice that I only learned to speak, read and write. I didn't move into advanced and specialized abilities like the artistic usage of the language, linguistics, literature, etc. Well, I did this to a certain extent in what concerns Portuguese.

One thing I noticed was that to be able to know more and more than what I already know in Portuguese, I would need more and more... and more time and effort. In other words, to move past my present ability in the languages I feel confident with (Portuguese, but also French and English), I would be subject to the law of diminishing returns.

I never forgot the way I learned how to drive a bicycle when I was 6 years old. It was a lengthy process, requiring me to go through several steps. I had to learn how to maintain the equilibrium of the thing; how to use the pedals to move forward, and keep the equilibrium; how to turn while moving; etc, etc. One of the things that I recall is that each new step was easier to achieve them the previous step. In other words, each new step required less resources (in time and effort) then the previous ones. Eventually I reached the level of comfort, where I was able to ride a bicycle without risk of falling in normal situations. I did not decide to try to advance my skills past that level, though.

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.

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What do you think?

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