A Remark on the Tournament Game

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, while Maker additionally assigns orientations to the edges that she claims. If by the end of the game Maker claims all the edges of a pre-defined goal tournament, she wins the game. 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.