The Paradox of Anti-Inductive Dice

• Published in 2025
• Added on 2025-05-23

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.

Links

• http://arxiv.org/abs/2503.16306v1
• http://arxiv.org/pdf/2503.16306v1

Other information

BibTeX entry

@article{TheParadoxofAntiInductiveDice,
	key = {TheParadoxofAntiInductiveDice},
	type = {article},
	title = {The Paradox of Anti-Inductive Dice},
	author = {Summer Eldridge and Ivo David de Oliveira and Yogev Shpilman},
	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.},
	comment = {},
	date_added = {2025-05-23},
	date_published = {2025-05-23},
	urls = {http://arxiv.org/abs/2503.16306v1,http://arxiv.org/pdf/2503.16306v1},
	collections = {easily-explained,fun-maths-facts,things-to-make-and-do},
	url = {http://arxiv.org/abs/2503.16306v1 http://arxiv.org/pdf/2503.16306v1},
	year = 2025,
	urldate = {2025-05-23},
	archivePrefix = {arXiv},
	eprint = {2503.16306},
	primaryClass = {math.PR}
}