On the optimality of the uniform random strategy

TL;DR

This paper establishes general criteria for biased Maker-Breaker games on hypergraphs, determining thresholds and optimal strategies, and applies these to hypergraph generalizations of classical combinatorial games.

Contribution

It provides the first unified, hypergraph-level criteria for Maker-Breaker games, extending classical results to more complex structures and biased settings.

Findings

Derived hypergraph generalizations of H-building and van der Waerden games.

Established threshold biases for Rado games within constant factors.

Demonstrated the effectiveness of a deterministic approach in complex hypergraph scenarios.

Abstract

The concept of biased Maker-Breaker games, introduced by Chv\'atal and Erd{\H o}s, is a central topic in the field of positional games, with deep connections to the theory of random structures. For any given hypergraph ${\cal H}$ the main questions is to determine the smallest bias $q({\cal H})$ that allows Breaker to force that Maker ends up with an independent set of ${\cal H}$. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies a couple of natural `container-type' regularity conditions about the degree of subsets of its vertices. This will enable us to derive a hypergraph generalization of the $H$-building games, studied for graphs by Bednarska and {\L}uczak. Furthermore, we investigate the biased version of generalizations of the van der Waerden games introduced by Beck. We refer to these generalizations as Rado games and determine their threshold bias up to constant factors by applying our general criteria. We find it quite remarkable that a purely game theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, when the generalizations to hypergraphs in the analogous setup of sparse random discrete structures are usually quite challenging.