The Speed and Threshold of the Biased Perfect Matching Game

TL;DR

This paper analyzes the speed and bias thresholds in the biased perfect matching game, establishing conditions under which Maker can win efficiently as the game parameters grow.

Contribution

It provides new bounds on the bias and turn count for Maker to secure a perfect matching in the biased game, extending previous results.

Findings

Maker wins in approximately n/2 turns under specified bias conditions

The bias threshold for Maker's victory is at least n/log n minus a lower order term

The results apply for large n with a bias growing faster than n/log n

Abstract

We show that Maker wins the Maker-Breaker perfect matching game in $\frac{n}{2}+o(n)$ turns when the bias is at least $\frac{n}{\log{n}}-\frac{f(n)n}{(\log{n})^{5/4}}$, for any $f$ going to infinity with $n$ and $n$ sufficiently large (in terms of $f$).