The Paradox of Anti-Inductive Dice

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] 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.