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Fair Schedules for Single Round Robin Tournaments with Ranked Participants

Sten Wessel, Cor Hurkens, Frits Spieksma

TL;DR

This paper addresses fairness in single round robin tournaments with a prespecified ranking by formalizing a ranking fairness measure that distributes home/away advantages evenly across opponents ranked by strength. It provides an explicit, circle-method–like construction proving the existence of ranking-fair, minimum-break schedules when the number of teams is a multiple of four, and a mathematical programming formulation to find ranking-fair schedules when they exist. It further shows nonexistence results for the canonical pattern set beyond small instances and demonstrates practical relevance through real-world examples from chess and football leagues, plus a public repository for replication. Overall, the work delivers both constructive schedules and computational tools to ensure fairness and efficacy in SRR tournaments with ranked participants.

Abstract

We introduce a new measure to capture fairness of a schedule in a single round robin (SRR) tournament when participants are ranked by strength. To prevent distortion of the outcome of an SRR tournament as well as to guarantee equal treatment, we argue that each participant should face its opponents when ranked by strength in an alternating fashion with respect to the home/away advantage. Here, the home/away advantage captures a variety of situations. We provide an explicit construction proving that so-called ranking-fair schedules exist when the number of participants is a multiple of 4. Further, we give a formulation that outputs ranking-fair schedules when they exist. Finally, we show that the most popular method to come to a schedule for an SRR tournament, does not allow ranking-fair schedules when the number of teams exceeds 8. These findings impact the type of schedules to be used for SRR tournaments.

Fair Schedules for Single Round Robin Tournaments with Ranked Participants

TL;DR

This paper addresses fairness in single round robin tournaments with a prespecified ranking by formalizing a ranking fairness measure that distributes home/away advantages evenly across opponents ranked by strength. It provides an explicit, circle-method–like construction proving the existence of ranking-fair, minimum-break schedules when the number of teams is a multiple of four, and a mathematical programming formulation to find ranking-fair schedules when they exist. It further shows nonexistence results for the canonical pattern set beyond small instances and demonstrates practical relevance through real-world examples from chess and football leagues, plus a public repository for replication. Overall, the work delivers both constructive schedules and computational tools to ensure fairness and efficacy in SRR tournaments with ranked participants.

Abstract

We introduce a new measure to capture fairness of a schedule in a single round robin (SRR) tournament when participants are ranked by strength. To prevent distortion of the outcome of an SRR tournament as well as to guarantee equal treatment, we argue that each participant should face its opponents when ranked by strength in an alternating fashion with respect to the home/away advantage. Here, the home/away advantage captures a variety of situations. We provide an explicit construction proving that so-called ranking-fair schedules exist when the number of participants is a multiple of 4. Further, we give a formulation that outputs ranking-fair schedules when they exist. Finally, we show that the most popular method to come to a schedule for an SRR tournament, does not allow ranking-fair schedules when the number of teams exceeds 8. These findings impact the type of schedules to be used for SRR tournaments.

Paper Structure

This paper contains 9 sections, 9 theorems, 7 equations, 3 figures, 7 tables.

Table of Contents

  1. Prologue
  2. Introduction
  3. Problem Description
  4. A Measure for Ranking Fairness
  5. Results
  6. The Case $n=4k$
  7. The Case $n=4k+2$
  8. The Canonical Pattern Set
  9. Examples from Practice

Key Result

Lemma 2.3

In a ranking-fair schedule, consider the match between the two teams $i, j \in T$. If $i$ and $j$ have the same parity (i.e., both odd or both even), the strongest team plays away. If $i$ and $j$ have a different parity, the strongest team plays at home.

Figures (3)

  • Figure 1: The ranking-fair schedule generated by a modified circle method.
  • Figure 2: The fixed matches in a single-break, ranking-fair schedule for $n=6$, depending on which pattern is assigned to team $1$.
  • Figure 3: The imbalance value $F$ for the schedule of several editions of the Tata Steel Chess tournament in which the tournament was organized with 14 players. The horizontal line indicates the optimal $F$-value for 14 players.

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 3.1
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 9 more