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Conclusions And Future Ideas

I feel we have learned all we can from Trash Idle's use of the Cauchy distribution. In summary, this is what I feel I have learned:

  • The Cauchy distribution isn't quite able to recreate some of the most common RNG scenarios like random hit chances or loot drops. It may instead work best as a source for some kind of currency because the Cauchy samples can take any positive or negative value on the number line.
  • The distribution can be used to create unique random-walks throughout a parameter space. The walkers will explore a particular location for some time, before making an extreme jump somewhere else.
  • The Cauchy distribution has fat tails. These massively amplify the impact of rare rolls.
  • While often not explicitly included, the Normal distribution is present in practically all games as a consequence of the Central Limit Theorem.
  • Because the Cauchy distribution is a stable distribution with infinite variance, it does not obey the CLT. Player's of a game using the Cauchy distribution are free from the CLT, and can be distributed in a fat-tailed manner - thus the rarest of lucky players might benefit from extreme boons.

An early draft for Trash Idle

What Next?

There is a final thing I discovered while making the game. The Cauchy distribution is not the only fat-tailed stable distribution...

In fact, there is a whole family of curves known as the Lévy alpha-stable distributions. Most of these distributions cannot be written down in neat forms, but along with the Cauchy distribution there is one other such distribution: the Lévy distribution:

Levy Distribution

Stable Distributions Normal distribution is blue, Cauchy is red

The Levy distribution is also fat-tailed, and is stable. Thus it is also free from the Central Limit Theorem, and can throw out some extreme values for the luckiest of rolls. Comparing it to both the Cauchy distribution and Normal distribution however, we've now got such a distribution which is strictly positive and asymmetric. Unlike the Cauchy, it does have a well-defined means value. Well, it actually has infinite mean!

Because it is strictly positive, this might be the more interesting distribution to make use of. A friend of mine suggested a shotgun where the number of pellets is Levy distributed, which is an idea that I love. I think this will be the next thing I explore.

Thanks for reading! Get in touch if you have questions.