Optimal strategies in fractional games: vertex cover and domination

TL;DR

This paper studies a two-player game on hypergraphs where players construct fractional transversals aiming to minimize or maximize their size, proving the existence of rational strategies for both.

Contribution

It introduces a strategic framework for fractional transversals in hypergraphs and proves both players can achieve their optimal goals with rational strategies.

Findings

Both players have strategies to reach their optimal fractional transversal.

Optimal strategies can be implemented with rational weights.

The game equilibrium aligns with the fractional transversal number.

Abstract

In a hypergraph with vertex set $V$ and edge set $E$, a real-valued function $f: V \to [0, 1]$ is a fractional transversal if $\sum_{v\in e} f(v) \ge 1$ for every edge $e \in E$. Its size is $|f| := \sum_{v \in V} f(v)$, and the fractional transversal number is the smallest possible $|f|$. We consider a game scenario where two players with opposite goals construct a fractional transversal incrementally, trying to minimize and maximize $|f|$, respectively. We prove that both players have strategies to achieve their common optimum, and they can reach their goals using rational weights.