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Intransitive dice tournament is not quasirandom

Elisabetta Cornacchia, Jan Hązła

Abstract

We settle a version of the conjecture about intransitive dice posed by Conrey, Gabbard, Grant, Liu and Morrison in 2016 and Polymath in 2017. We consider generalized dice with $n$ faces and we say that a die $A$ beats $B$ if a random face of $A$ is more likely to show a higher number than a random face of $B$. We study random dice with faces drawn iid from the uniform distribution on $[0,1]$ and conditioned on the sum of the faces equal to $n/2$. Considering the "beats" relation for three such random dice, Polymath showed that each of eight possible tournaments between them is asymptotically equally likely. In particular, three dice form an intransitive cycle with probability converging to $1/4$. In this paper we prove that for four random dice not all tournaments are equally likely and the probability of a transitive tournament is strictly higher than $3/8$.

Intransitive dice tournament is not quasirandom

Abstract

We settle a version of the conjecture about intransitive dice posed by Conrey, Gabbard, Grant, Liu and Morrison in 2016 and Polymath in 2017. We consider generalized dice with faces and we say that a die beats if a random face of is more likely to show a higher number than a random face of . We study random dice with faces drawn iid from the uniform distribution on and conditioned on the sum of the faces equal to . Considering the "beats" relation for three such random dice, Polymath showed that each of eight possible tournaments between them is asymptotically equally likely. In particular, three dice form an intransitive cycle with probability converging to . In this paper we prove that for four random dice not all tournaments are equally likely and the probability of a transitive tournament is strictly higher than .

Paper Structure

This paper contains 37 sections, 37 theorems, 194 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof strategy
  3. Organization
  4. Notation
  5. Acknowledgments
  6. Proof outline
  7. Reduction from four to three dice
  8. Function g_ A
  9. Central limit theorem for g_ A
  10. Proof strategy for Theorem \ref{['thm:conditional-clt']}
  11. Bounding the covariance
  12. Proof strategy for Lemmas \ref{['lem:var-upper-bound']} and \ref{['lem:cv-lower-bound']}
  13. Proof of Theorem \ref{['thm:two-fixed-equivalent']}
  14. CLT for two dice
  15. Characteristic functions and joint Gaussians
  16. ...and 22 more sections

Key Result

Theorem 1

If $A,B,C$ are three random dice with $n$ faces iid uniform in $[n]$ and conditioned on all face-sums equal to $n(n+1)/2$, then the probability that $A,B,C$ are intransitive is $1/4+o(1)$.

Figures (3)

  • Figure 1: Three types of tournaments on four elements. From the left: A transitive ordering (there are 24 in total), an overall winner on top of a 3-cycle (16 in total together with a symmetric case of overall loser) and a 4-cycle (24 in total).
  • Figure 2: Illustration of the proof of Claim \ref{['cl:one-event']}. The circle represents the set $\faktor{\mathbb{R}}{\mathbb{Z}/|\alpha|}$ and its circumference is at least $6\sqrt{m}$. The black dot represents zero.
  • Figure 3: A graphical illustration of claims in the proof of Theorem \ref{['thm:conditional-clt']}. We are using the fact that $g(ta_0,tb_0,0)$ is decreasing in $|t|$ for every direction $(a_0,b_0)$.

Theorems & Definitions (76)

  • Theorem 1: Pol17
  • Theorem 2: Pol17
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Claim 6
  • Lemma 7
  • proof
  • Remark 8
  • Claim 9
  • ...and 66 more