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arXiv · 2507.22519

Maker playing against an invisible Breaker

Dennis Clemens·Fabian Hamann·Mirjana Mikalački·Yannick Mogge·Miloš Stojaković
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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.

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Dennis Clemens, Fabian Hamann, Mirjana Mikalački, Yannick Mogge, Miloš Stojaković. 2025-07-30. Maker playing against an invisible Breaker. https://arxiv.org/abs/2507.22519

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