Maker playing against an invisible Breaker
Dennis Clemens, Fabian Hamann, Mirjana Mikala\v{c}ki, Yannick Mogge, Milo\v{s} Stojakovi\'c

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 Maker-PhantomBreaker games where the board is the edge set of the complete graph on vertices, , 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- game and Hamiltonicity game. In particular, we focus on characterizing all biases for which Maker wins asymptotically almost surely.
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