Maker playing against an invisible Breaker

TL;DR

This paper introduces the phantom version of Maker-Breaker positional games where one player's moves are hidden, analyzing strategies and biases for Maker to win in various classical graph games.

Contribution

It is the first to study the phantom variant of Maker-Breaker games, providing randomized strategies and bias characterizations for Maker's victory.

Findings

Characterized biases for Maker's asymptotic win

Developed randomized strategies for four classical games

Analyzed the impact of hidden moves on game outcomes

Abstract

We initiate the study of the phantom version of Maker-Breaker positional games. In a phantom game, the moves of one of the players are hidden from the other player, who still has the complete information. We look at the biased $(a:b)$ Maker-PhantomBreaker games where the board is the edge set of the complete graph on $n$ vertices, $K_n$, and Maker has no information about PhantomBreaker's choices of edges. We give randomized strategies for both players in four classical games: connectivity game, perfect matching game, mindegree-$k$ game and Hamiltonicity game. In particular, we focus on characterizing all biases $(a:b)$ for which Maker wins asymptotically almost surely.