Is it possible to generate a Pareto distribution with dice?

Question

So I know that there's a really easy way to generate a normal distribution with dice (simply add them). Is there a way to generate a Pareto distribution?

Answer 1

If you are looking to use physical dice in a tabletop game, here are a couple approximations that may be reasonably aesthetically pleasing.

Tail behavior: "stackable critical hits"

• Roll a die with $s$ sides labeled with a "critical hit" on one face, and the rest positive integers.
• If you roll a critical hit, roll again and multiply the result by $c$.
• Keep multiplying as long as you keep rolling critical hits.

What is the asymptotic chance of reaching at least a threshold $x$?

• You need $\log_c x$ critical hits (plus some constant) to reach $x$.
• Each critical hit has a chance of occurring $\frac{1}{s}$.

All in all, the chance is proportional to

$$ P\left(X \geq x \right) \propto \left( \frac{1}{s} \right)^{\log_c x} = x^{- \log s / \log c} $$

Compare the survival function of a Pareto distribution

$$ P\left(X \geq x\right) = \left( \frac{x_m}{x} \right)^\alpha $$

and we have a match with $\alpha = \frac{\log s}{\log c}$. For example:

• The St. Petersburg game corresponds to the case of $s = 2, c = 2$ which makes $\alpha = 1$.
• A Dungeons & Dragons-like system with $s = 20, c = 2$ makes $\alpha \approx 4.32$.

You can make the distribution look more or less like the Pareto distribution in details and/or try to hit a particular $x_m$ by choosing the numbers on the faces of the die, or by using some sort of compound dice roll rather than a single standard die. Though you'll eventually run into tradeoffs between making a better approximation, the complexity of the rolling process, and other considerations.

Threshold-only: "step dice"

If you don't need the exact value of a sample, but only whether it exceeds a target threshold (e.g. binary pass/fail resolution in a tabletop game), then you can use the survival function

$$ P\left(X \geq x\right) = \left( \frac{x_m}{x} \right)^\alpha $$

directly by rolling $\alpha$ dice, each with $x$ sides labeled $1 \ldots x$, taking the highest, and seeing if the result is less than or equal to $x_m$.