Transitive Avoidance Games

TL;DR

This paper investigates transitive avoidance games, revealing that contrary to expectations, some are first-player wins for non-prime, non-power-of-2 board sizes, with various properties like quick wins and small winning sets.

Contribution

It demonstrates the existence of transitive avoidance games that are first-player wins for certain board sizes, challenging previous assumptions.

Findings

Existence of transitive avoidance games with first-player wins for non-prime, non-power-of-2 sizes

Such games can have stronger transitivity, fast winning times, and small winning sets

Contradicts the expectation that transitive avoidance games cannot be first-player wins

Abstract

Positional games are a well-studied class of combinatorial game. In their usual form, two players take turns to play moves in a set (`the board'), and certain subsets are designated as `winning': the first person to occupy such a set wins the game. For these games, it is well known that (with correct play) the game cannot be a second-player win. In the avoidance (or mis\`{e}re) form, the first person to occupy such a set \emph{loses} the game. Here it would be natural to expect that the game cannot be a first-player win, at least if the game is transitive, meaning that all points of the board look the same. Our main result is that, contrary to this expectation, there are transitive games that are first-player wins, for all board sizes which are not prime or a power of 2. Further, we show that such games can have additional properties such as stronger transitivity conditions, fast winning times, and `small' winning sets.