The Paradox of Anti-Inductive Dice

TL;DR

This paper identifies anti-inductive dice, where the dominance relation between two dice over k independent rolls can invert non-monotonically, and formalizes the concept with the David vs Goliath example. Using Edgeworth-type expansions and a careful analysis of the difference die Δ, the authors prove that D[k] ≻ G[k] only at k = 4, despite Goliath leading for all other k, and they provide a computational framework to map and visualize transient dominance for low-dimensional dice spaces. They show that even small dice (3- and 4-sided) exhibit intricate, fractal-like regions of dominance when considering sums, and they identify families of 5-sided dice with arbitrarily late inversions, suggesting deeply rich transient behavior beyond limiting results. The work combines exact probabilistic tools with brute-force exploration to reveal a surprisingly complex landscape of dominance that challenges conventional transitivity assumptions in dice and probability.

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.

Figures (4)

• Figure 1: The truncated sequence of dominance relations for repeated rolls of a 3-sided die with maximum value 1, against the 0 die with one side. The centers of the "peaks" where behavior remains periodic take a particular form, with dice that win on rolls with residue 1 and 2 mod 3 clustering around dice of the form $x=1/(2+3n)$ and dice that win on rolls with residue 1 mod 3 clustering around dice of the form $x=2/(1+6n)$.
• Figure 2: Dominance relations between $\Delta[19]$ and the 0 die are plotted in the $\{1,x,y,-1-x-y\}$ parametrization. Even after less than 20 rolls, the regions where a die wins or loses against the zero dice are very elaborate. We suspect, but have not yet proved, that the limiting shape is infinitely rough.
• Figure 3: Given a die $\Delta$, there is a sequence $\{B_k\}_{k=1}$, where $B_k$ encodes its dominance relation with the $\{0\}$ die, using $0,1,2$ for $\Delta[k]\prec \{0\},\; \Delta[k]\approx \{0\},\; \Delta[k]\succ \{0\}$. The sequence is converted to a number in trinary by concatenating the values, which here are represented with shade. This expresses which areas in the parameter space have similar outcomes, with more weight placed on earlier rolls
• Figure 4: Approximately quadratic inversion delay, which, if it continues, would indicate a die can win for arbitrarily many rolls until a certain point, losing forever after.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2: David vs Goliath Dice