Impartial Achievement Games on Convex Geometries

TL;DR

This paper analyzes a strategic game played on convex geometries, developing a structure theory to determine winning strategies and nim numbers for various classes of these geometries.

Contribution

It introduces a new structure theory for impartial achievement games on convex geometries and computes nim numbers for specific classes.

Findings

Determined nim numbers for one-dimensional affine geometries

Analyzed vertex geometries of trees

Studied games with extreme points as winning sets

Abstract

We study a game where two players take turns selecting points of a convex geometry until the convex closure of the jointly selected points contains all the points of a given winning set. The winner of the game is the last player able to move. We develop a structure theory for these games and use it to determine the nim number for several classes of convex geometries, including one-dimensional affine geometries, vertex geometries of trees, and games with a winning set consisting of extreme points.