12 October 2009

Games and Reality are Probably Different, Part 2

In Part 1 I tried to describe what I consider to be a basic difference between games and reality in the scale used to represent the difficulty of tasks. When you are playing a game and need to roll a certain target number or higher on the dice to succeed, this defines a scale of difficulty that is additive, and adding or subtracting 1 from the target number changes the probability of success in a certain way. I would argue that real world difficulty and probabilities for success are better represented on a proportional scale.
I have prepared some graphs to illustrate what I'm getting at. It's a bit of a difficult concept, and I find it hard to describe in simple terms. Hopefully this will help get my idea across.

Below is a graph of the probability density functions (PDF) for some common probability distributions, including several dice distributions (1), with two important changes: On the X-axis, the units here are not in the numbers you might roll on the dice, but instead in standard deviations as are used to describe the spread of the standard normal distribution (represented by Z). All the these have been "centered" so the most likely roll (or event) is at 0 (zero), and "scaled" so that the are spread out in an equal way. The other change is I have "inflated" the distributions by re-scaling the probability inversely proportional to the standard deviation of that distribution (2).  By standardizing these distributions onto the same X & Y scale, I hope it will make them easier to compare. You might want to consider popping the image out to another window for reference.


Working my way down the legend:

Uniform distribution - This represents random numbers between 0 and 1 (any range, actually) where every number is equally likely to occur. Thus, this line is perfectly flat across the graph. Compared to the other distributions, the most extreme events (easiest and hardest) are much more probable, but the scale of difficulty doesn't go out as far, limiting the range of smallest and largest probabilities.

1d10 - The dots on top of the uniform distribution represent the distribution of a 10-sided die. This a discrete uniform distribution with a range from 1 to 10. In fact, the distribution of probabilities for any single die roll will fall on this line, though the dots would of course be spread differently (any regular sided die, that is).

2d10 - this represents the distribution of the sum of two dice, and you might recognize the distinctive triangular shape from Kit's recent post about the Math of 2DX Systems. This random distribution is much more "central" than the uniform, but it also extends out to smaller probabilities.  Note this distribution is closer to the normal distribution than any other presented here.

2d6 - Due to the way I have standardized the distributions this has just the same shape as the 2d10 distribution. The sum of any two regular dice will look much the same.

(Standard) Normal distribution - This is here partly as a reference for comparison, because I have tweaked the other distributions to the same scale. It's also a useful reference because it shows up in many real world applications. This distribution can describe very extreme events, but the probabilities becomes very close to zero rapidly as you move away from the middle of the distribution.

Laplace distribution - This is the the mathematical relation I originally worked out for my shooting a target example in Part 1 to demonstrate proportional probabilities (the problem that started me thinking about all this in the first place). This distribution is very "central"; if you could have a die that rolled numbers with a Laplace distribution, most the the rolls would be fairly close to the average.Most, but not all, because the the remainder of the rolls would tend to be very high or very low. This distribution has "heavy tails", meaning that the probability of the most extreme events gets smaller very slowly as you move out from the middle (the tails are "heavier" than the normal distribution).

The next chart are the cumulative density functions (CDF) for the same distributions (3). By statistical convention I have created this so the probabilities accumulate from left to right, so if you think about this as trying to roll your dice higher than a target number, the higher number start on the left and go down to the right.
This manner of presenting distributions tends to squish everything together in the middle, but you can see that the heavy tails of the Laplace distribution really stand out from those of the normal.


So far I've shown these probabilities on a the usual scale from zero to one. However, to demonstrate the proportional relationship, it helps to present it on a logarithmic scale; just the thing for presenting proportional relationships. This requires converting from the usual 0-1 probability scale to an "odds scale" than ranges from 0 to infinity, and then taking the natural log. This really requires a separate description to fully explain this, which is the reason for my previous post on Probability versus Odds. Reading that first may be helpful.

Here is the previous chart again, except that now instead of probability, the Y-axis is the log-odds:


On the logarithmic scale proportional relationships appear as straight lines, and look what has happened here with the Laplace distribution; it is very nearly a straight line. Everything is crunched together in the middle, so I made a "zoom in" of the middle portion:


It looks as if this post is headed for Part 3, because it's getting late and I have to get up very early. I still need to discuss what I think this really means about the differences between games and reality, and this is a good place to break for comments. Stay tuned for Part 3.

Footnotes:
  1. For dice, which can only generate discrete numbers in a limited range, these are technically probability mass functions
  2. This is a weird thing to do, but I have re-scaled probability by dividing by the standard deviation. I could come up with distributions that looked like this in the first place, but they would be harder to compare directly. I really ought to redo that plot to labeled the Y-axis correctly.
  3. With one additional tinker: I shifted the discrete distributions so that a 0.50 probability lines up at Z=0 for all distributions.
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