Impartial Achievement Games on Convex Geometries
Stephanie McCoy, N\'andor Sieben

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.
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