Erd\H{o}s-Szekeres Maker-Breaker Games
Aleksa D\v{z}uklevski, D\"om\"ot\"or P\'alv\"olgyi, Alexey Pokrovskiy, Csaba D. T\'oth, Tom\'a\v{s} Valla, Lander Verlinde

TL;DR
This paper investigates Maker-Breaker games inspired by the Erd ext{o}s-Szekeres problem, establishing winning strategies for Maker in various biased and monochromatic/bichromatic settings, and resolving open questions about the game's dynamics.
Contribution
It provides new results on Maker-Breaker games related to convex polygons, proving Maker's winning strategies under different biases and settings, and settles an open problem from prior research.
Findings
Maker always wins in the monochromatic game.
Maker wins in the bichromatic game if Breaker places fewer than 2 points per round.
Breaker can win when she plays 12 or more points per round with k ≥ 8.
Abstract
We present new results on Maker-Breaker games arising from the Erd\H{o}s-Szekeres problem in planar geometry. This classical problem asks how large a set in general position has to be to ensure the existence of points that are the vertices of a convex -gon. Moreover, Erd\H{o}s further extended this problem by asking what happens if we also require that this -gon has an empty interior. In a 2-player Maker-Breaker setting, this problem inspires two main games. In both games, Maker tries to obtain an empty convex -gon, while Breaker tries to prevent her from doing so. The games differ only in which points can comprise the winning -gons: in the monochromatic version the points of both players can make up a -gon, while in the bichromatic version only Maker's points contribute to such a polygon. Both settings are studied in this paper. We show that in the monochromatic…
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Taxonomy
TopicsArtificial Intelligence in Games · Digital Games and Media
