Impartial Avoidance Games on Convex Geometries

TL;DR

This paper studies a two-player avoidance game on convex geometries, developing a theoretical framework and calculating nim numbers for various cases, including tree geometries and extreme points.

Contribution

It introduces a new theoretical approach to analyze avoidance games on convex geometries and computes nim numbers for specific cases.

Findings

Nim numbers are determined for avoidance games on tree vertex and edge geometries.

The framework applies to cases with predefined sets of extreme points.

The analysis extends understanding of combinatorial game theory on convex structures.

Abstract

We analyze a two-player game in which players take turns avoiding the selection of certain points within a convex geometry. The objective is to prevent the convex closure of all chosen points from encompassing a predefined set. The first player forced into a move that results in the inclusion of this set loses the game. We redevelop a theoretical framework for these avoidance games and determine their nim numbers, including cases involving vertex geometries of trees, edge geometries of trees, and scenarios where the predefined set consists of extreme points.