Non-monotonic dice

Given a pair of dice, $A$ and $B$, let $A_{k}$ and $B_{k}$ be defined as the random variables which represent the sum of $k$ rolls of $A$ and $B$ correspondingly. Consider the arithmetic function $f(k)$ for $k=1,2,3,...$ that indicates which dice has a higher probability of rolling a higher sum for $k$ rolls. For example, $f$ can be defined as$f\left(k\right)=1$ if $P\left(A_{k}>B_{k}\right)>P\left(A_{k}<B_{k}\right)$, $f\left(k\right)=-1$ if $P\left(A_{k}>B_{k}\right)<P\left(A_{k}<B_{k}\right)$ and$f\left(k\right)=0$ otherwise.

If $f$ is a non-monotonic function, we say that $A$ and $B$ are non-monotonic dice.

Example

The David vs Goliath Dice

The David vs Goliath Dice

David die has sides 1, 1, 4, 4, 5, 6.

Goliath die has sides 0, 1, 2, 6, 6, 6.

Discovered by Ivo Fagundes David de Oliveira and Yogev Shpilman in 2023. In this pair of non-monotonic dice, one die, named Goliath, is more likely to have a higher score than the second die, named David, for any number of rolls $k$, except for $k=4$. In other words, $f\left(k\right)=1$ for any $k\neq 4$, and $f\left(k\right)=-1$ for $k=4$.

For $k=1$ Goliath has an advantage over David as depicted by the following comparison matrix:

This pattern repeats itself for any value of $k$, except for $k=4$. At this value of $k=4$ David has 789,540 winning states and Goliath has 789,407 winning states and therefore David wins in 133 more ways than Goliath.

Other properties of the David vs Goliath dice

This pair of dice is balanced, meaning they are 6-sided dice with a sum of faces of $21$, just like a standard die. Goliath is demonic - meaning it contains a 6, 6, 6 sub-sequence of its faces.

It is conjectured that no other balanced dice with integer face falues between 0 and 6 can produce a single inversion of which die is stronger at $k=5$ or more. If this conjuecture is true than David vs Goliath are maximal in this sense.

The paradoxical nature of non-monotonic dice

Non-monotonic dice produce a seemingly paradoxical relations. This is summarized with the folloiwing explanation of David vs Goliath dice: for every value of $k\neq 4$ we seem to confirm that Goliath is a stronger die than David, it is therefore unreasonable to expect that David would be stronger than David at $k=4$.

Another argument that enhances the paradox is captured when realizing that $k=4$ is nothing more than $k=3$, where Goliath has the advantage, plus one roll, i.e. $k=1$ where Goliath also has the advantage. This seems to intuitively violate principles of mathematical induction as well as principles of inductive reasoning.

Other related dice

Grime dice are a set of 5 intransitive dice known to invert the intrantisive relation when you roll one or two pairs of dice.

See also

• Intransitive dice
• Blotto games
• Go First Dice
• Nontransitive game

External links

• David and Goliath Dice - Non-transitive, Go First, and Sicherman Dice @ mathartfun.com