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arXiv · 2503.16306

The Paradox of Anti-Inductive Dice

Summer Eldridge·Ivo David de Oliveira·Yogev Shpilman
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Abstract

We identify a new type of paradoxical behavior in dice, where the sum of independent rolls produces a deceptive sequence of dominance relations. We call these ``anti-inductive dice". Consider a game with two players and two non-identical dice. Each rolls their die $k$ times, adding the results, and the player with the highest sum wins. For each $k$, this induces a dominance relation between dice, with $A[k]\succ B[k]$ if $A$ is more likely than $B$ to win after $k$ rolls, and vice versa. For certain classes of dice, the limiting behavior of these relations is well-established in the literature, but the transient behavior, the subject of this paper, is less well-understood. This transient behavior, even for dice with only 4 faces, contains an immensely rich parameter space with fractal-like behavior.

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Summer Eldridge, Ivo David de Oliveira, Yogev Shpilman. 2025-03-20. The Paradox of Anti-Inductive Dice. https://arxiv.org/abs/2503.16306

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