# Is it possible to generate a Pareto distribution with dice?

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?

• Maybe multiplication/division? Commented Feb 14, 2023 at 8:13
• Applying the inverse cdf + cdf transform turns a Normal into a Pareto. Commented Feb 14, 2023 at 8:26
• More seriously, using (only) a dice cannot produce a generation from a continuous distribution. See this stats.stackexchange.com/a/247250/7224 entry. Commented Feb 14, 2023 at 8:31
• You tagged with discrete-distributions, but the Pareto is continuous? Commented Feb 14, 2023 at 23:02
• My guess is that the "dice" are implied to be physical with discrete faces. Though IMX this sort of problem tends to end up in a grey area where any concrete solution must be discrete, but we are not really committing to any specific target discretization. Commented Feb 15, 2023 at 0:24

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 $$x_m$$.