How to Design a Stable Serial Knockout Competition

TL;DR

This paper explores the design of a new tournament format called Serial Knockout Competition (SKC), ensuring fairness by equalizing match opportunities among players, and demonstrates its feasibility for certain player counts using combinatorial structures.

Contribution

It formalizes the concept of stable SKCs, proves their existence for 8 players using Fano plane connections, and provides stable schedules for 16 and 32 players.

Findings

Stable SKCs exist for 8 players using Fano plane structures.

Stable schedules can be constructed for 16 and 32 players.

The proposed format promotes fairness by equalizing matchups across rounds.

Abstract

We investigate a new tournament format that consists of a series of individual knockout tournaments; we call this new format a Serial Knockout Competition (SKC). This format has recently been adopted by the Professional Darts Corporation. Depending on the seedings of the players used for each of the knockout tournaments, players can meet in the various rounds (eg first round, second round, ..., semi-final, final) of the knockout tournaments. Following a fairness principle of treating all players equal, we identify an attractive property of an SKC: each pair of players should potentially meet equally often in each of the rounds of the SKC. If the seedings are such that this property is indeed present, we call the resulting SKC stable. In this note we formalize this notion, and we address the question: do there exist seedings for each of the knockout tournaments such that the resulting SKC is stable? We show, using a connection to the Fano plane, that the answer is yes for 8 players. We show how to generalize this to any number of players that is a power of 2, and we provide stable schedules for competitions on 16 and 32 players