Multistage Positional Games

TL;DR

This paper introduces and analyzes a new multistage variant of Maker-Breaker positional games, estimating the maximum game duration for various graph properties on complete graphs, revealing different behaviors in probabilistic and non-probabilistic cases.

Contribution

The paper defines the multistage Maker-Breaker game, provides bounds on its duration for key graph properties, and distinguishes between probabilistic and non-probabilistic behaviors in these games.

Findings

Maximum game duration estimates for several graph properties.

Probabilistic intuition applies to some games but not others.

Different behaviors observed in probabilistic versus non-probabilistic games.

Abstract

We initiate the study of a new variant of the Maker-Breaker positional game, which we call multistage game. Given a hypergraph $\mathcal{H}=(\mathcal{X},\mathcal{F})$ and a bias $b \ge 1$, the $(1:b)$ multistage Maker-Breaker game on $\mathcal{H}$ is played in several stages as follows. Each stage is played as a usual $(1:b)$ Maker-Breaker game, until all the elements of the board get claimed by one of the players, with the first stage being played on $\mathcal{H}$. In every subsequent stage, the game is played on the board reduced to the elements that Maker claimed in the previous stage, and with the winning sets reduced to those fully contained in the new board. The game proceeds until no winning sets remain, and the goal of Maker is to prolong the duration of the game for as many stages as possible. In this paper we estimate the maximum duration of the $(1:b)$ multistage Maker-Breaker game, for biases $b$ subpolynomial in $n$, for some standard graph games played on the edge set of $K_n$: the connectivity game, the Hamilton cycle game, the non-$k$-colorability game, the pancyclicity game and the $H$-game. While the first three games exhibit a probabilistic intuition, it turns out that the last two games fail to do so.