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Metric Dimensions of March Madness Brackets

Sam Spiro

TL;DR

This work reframes reconstructing March Madness-style tournament outcomes from scored brackets as a metric-dimension problem on single-elimination tournaments. It proves universal bounds on the metric dimension $ ext{dim}(S,\sigma)$ that are independent of the scoring system and gives exact results for standard balanced tournaments, notably $ ext{dim}(S,\sigma)=n/2$ when $n$ is a power of two. A central technical tool is a lifting technique that derives upper bounds by transferring resolving sets from sub-tournaments to the whole tournament, yielding the general bound $ ext{dim}(S,\sigma)\le n-1$ and the constructive $n/2$-bracket resolution for standard cases; the paper also analyzes the resolving number $ ext{res}(S,\sigma)$, showing it concentrates near the total number of brackets via $q_{\max}$. Together, these results quantify how much information brackets must carry to uniquely identify the tournament's actual progression, with implications for bracket-based inference and metric-dimension problems on tournament-like graphs.

Abstract

Say you and some friends decide to make brackets for March Madness and are told how each of your brackets scored. The question we ask is: when can you determine how the actual tournament went given your scores? We determine the exact minimum number of brackets needed to do this for any March Madness-style tournament regardless of the scoring system used, and more generally we prove effective bounds for the problem for arbitrary single-elimination tournaments.

Metric Dimensions of March Madness Brackets

TL;DR

This work reframes reconstructing March Madness-style tournament outcomes from scored brackets as a metric-dimension problem on single-elimination tournaments. It proves universal bounds on the metric dimension that are independent of the scoring system and gives exact results for standard balanced tournaments, notably when is a power of two. A central technical tool is a lifting technique that derives upper bounds by transferring resolving sets from sub-tournaments to the whole tournament, yielding the general bound and the constructive -bracket resolution for standard cases; the paper also analyzes the resolving number , showing it concentrates near the total number of brackets via . Together, these results quantify how much information brackets must carry to uniquely identify the tournament's actual progression, with implications for bracket-based inference and metric-dimension problems on tournament-like graphs.

Abstract

Say you and some friends decide to make brackets for March Madness and are told how each of your brackets scored. The question we ask is: when can you determine how the actual tournament went given your scores? We determine the exact minimum number of brackets needed to do this for any March Madness-style tournament regardless of the scoring system used, and more generally we prove effective bounds for the problem for arbitrary single-elimination tournaments.
Paper Structure (13 sections, 31 theorems, 50 equations, 1 figure)

This paper contains 13 sections, 31 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Main Results
  4. Proof Sketch
  5. Preliminaries
  6. Lower Bounds
  7. Upper Bounds
  8. Additional Preliminaries
  9. Tightness of Lower Bounds
  10. Metric Dimension Upper Bounds
  11. Resolving Number Upper Bound
  12. Concluding Remarks
  13. More on Resolving Numbers

Key Result

Theorem 1.1

If $\vec{S}$ is a standard single-elimination tournament on $n$ players with $n\ge 2$ a power of 2, then for every scoring system $\sigma$ we have Moreover, for all $n$ there exists a set of $n/2$ brackets $\mathcal{B}$ which is $\sigma$-resolving for every $\sigma$.

Figures (1)

  • Figure 1: A standard single-elimination tournament $\vec{S}$ with 4 players $a,b,c,d$, along with two brackets $B_1,B_2$. Here $\mathrm{score}_\sigma(B_1,B_2)=\sigma(y)$ for any scoring system $\sigma$ since the only match which $B_1,B_2$ agree on is $y$. Moreover, one can check that $\{B_1,B_2\}$ is a $\sigma$-resolving set for every $\sigma$.

Theorems & Definitions (101)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1.1
  • Theorem 1.2
  • Definition 5
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 3.1
  • ...and 91 more