The random graph intuition for the tournament game

TL;DR

This paper analyzes the threshold for Maker to win the tournament game on a complete graph, showing Maker's advantage for k up to approximately 2 log_2 n vertices, and explores the applicability of the random graph intuition.

Contribution

It improves the known bounds for Maker's winning threshold in the tournament game and examines the validity of the random graph intuition in this context.

Findings

Maker wins for k = (2 - o(1)) log_2 n, nearly tight with known upper bounds.

The random graph intuition is only partially valid; a more nuanced version applies.

Breaker wins the orientation game for k = (4 + o(1)) log_2 n.

Abstract

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph on n vertices and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament T_k on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of T_k; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2-o(1))log_2 n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2-o(1))log_2 n Breaker can prevent that the underlying graph of Maker's graph contains a k-clique. Moreover the precise value of our lower bound differs from the upper bound only by an additive constant of 12. We also discuss the question whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two "clever" players and the game played by two "random" players, is supported by the tournament game: It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid. Finally, we consider the orientation-game version of the tournament game, where Maker wins the game if the final digraph -- containing also the edges directed by Breaker -- possesses a copy of T_k. We prove that in that game Breaker has a winning strategy for k = (4+o(1))log_2 n.