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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.16306 |
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| _version_ | 1866909545819799552 |
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| author | Eldridge, Summer de Oliveira, Ivo David Shpilman, Yogev |
| author_facet | Eldridge, Summer de Oliveira, Ivo David Shpilman, Yogev |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16306 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Paradox of Anti-Inductive Dice Eldridge, Summer de Oliveira, Ivo David Shpilman, Yogev Probability 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. |
| title | The Paradox of Anti-Inductive Dice |
| topic | Probability |
| url | https://arxiv.org/abs/2503.16306 |