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.

中文翻译：

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凸几何上的公正成就博弈

我们研究了一个游戏，其中两个玩家轮流选择凸几何图形的点，直到共同选择的点的凸闭合包含给定获胜集的所有点为止。游戏的获胜者是最后一个能够移动的玩家。我们为这些游戏开发了一种结构理论，并用它来确定几类凸几何的nim值，包括一维仿射几何，树木的顶点几何以及具有由极值组成的获胜集合的游戏。