A Real-World Markov Chain arising in Recreational Volleyball

TL;DR

This paper introduces a realistic Markov chain model for mixing players in recreational volleyball, analyzing its effectiveness and proving its irreducibility through combinatorial methods.

Contribution

It presents a novel Markov chain model based on real-world volleyball team mixing, with numerical analysis and a combinatorial proof of irreducibility.

Findings

The chain effectively mixes players within 7-8 steps.

The model accurately reflects real-world volleyball team changes.

The chain is proven to be irreducible.

Abstract

Card shuffling models have provided simple motivating examples for the mathematical theory of mixing times for Markov chains. As a complement, we introduce a more intricate realistic model of a certain observable real-world scheme for mixing human players onto teams. We quantify numerically the effectiveness of this mixing scheme over the 7 or 8 steps performed in practice. We give a combinatorial proof of the non-trivial fact that the chain is indeed irreducible.