Intransitive Dice

Abstract

We consider $n$-sided dice whose face values lie between $1$ and $n$ and whose faces sum to $n(n+1)/2$. For two dice $A$ and $B$, define $A \succ B$ if it is more likely for $A$ to show a higher face than $B$. Suppose $k$ such dice $A_1,\dots,A_k$ are randomly selected. We conjecture that the probability of ties goes to 0 as $n$ grows. We conjecture and provide some supporting evidence that---contrary to intuition---each of the $2^{k \choose 2}$ assignments of $\succ$ or $\prec$ to each pair is equally likely asymptotically. For a specific example, suppose we randomly select $k$ dice $A_1,\dots,A_k$ and observe that $A_1 \succ A_2 \succ \ldots \succ A_k$. Then our conjecture asserts that the outcomes $A_k \succ A_1$ and $A_1 \prec A_k$ both have probability approaching $1/2$ as $n \rightarrow \infty$

Generalized intransitive dice II: Partition constructions

“…The example (3) models the Rock-Paper-Scissors-Lizard-Spock tournament as a 5-partition of [30] and we will see in the next section that we can do even better. 4. Examples.…”

Possible Probability and Irreducibility of Balanced Nontransitive Dice

“…It is called nontransitive because if we define the relation X≻Y as P(X > Y) > (1/2) for X, Y ∈ A, B, C { }, then the relation ≻ is not transitive. For example, the following set of 6-sided dice is nontransitive since P(A > B) � P(B > C) � P(C > A) � (19/36): Note that nontransitive sets of dice are first introduced by Gardner [1], further studied in [2,3], and have been generalized in several directions (see [4][5][6][7][8][9][10][11][12][13][14]). In [14], Schaefer and Schweig constructed balanced nontransitive sets of n-sided dice for any positive integer n ≥ 3 (see below for the definition of "balanced" and other terms).…”

Generalized intransitive dice: Mimicking an arbitrary tournament

“…Such N -sided dice have been considered by Gowers in his blog and by Corey et al [4]. Considering large numbers of such dice leads us to the theory of tournaments.…”

“…In Corey et al [4] the authors' numerical work led them to the much stronger conjecture that, with n fixed, and letting N tend to infinity, every tournament on [n] becomes equally likely.…”