A remark on the Tournament game

TL;DR

This paper investigates the Maker-Breaker tournament game on graphs, establishing thresholds for Maker's win in both complete and random graphs, and compares these results with the clique game to analyze underlying intuitions.

Contribution

It determines the threshold bias for the tournament game on complete graphs and the threshold probability on random graphs, providing new insights into positional game strategies.

Findings

Threshold bias for Maker's win on $K_n$ is established.

Threshold probability for Maker's win on ${ m f G}_{n,p}$ is identified.

Comparison with the clique game reveals insights into random graph intuition.

Abstract

We study the Maker-Breaker tournament game played on the edge set of a given graph $G$. Two players, Maker and Breaker claim unclaimed edges of $G$ in turns, and Maker wins if by the end of the game she claims all the edges of a pre-defined goal tournament. Given a tournament $T_k$ on $k$ vertices, we determine the threshold bias for the $(1:b)$ $T_k$-tournament game on $K_n$. We also look at the $(1:1)$ $T_k$-tournament game played on the edge set of a random graph ${\mathcal{G}_{n,p}}$ and determine the threshold probability for Maker's win. We compare these games with the clique game and discuss whether a random graph intuition is satisfied.