Maker-Breaker games on random geometric graphs BeveridgeAndrew DudekAndrzej FriezeAlan MullerTobias StojakovicMilos 2014 <p>In a Maker-Breaker game on a graph <em>G</em>, Breaker and Maker alternately claim edges of <em>G</em>. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four Maker-Breaker games played on random geometric graphs. For each of our four games we show that if we add edges between <em>n</em> points chosen uniformly at random in the unit square by order of increasing edge-length then, with probability tending to one as <em>n</em> <em>∞</em>, the graph becomes Maker-win the very moment it satisfies a simple necessary condition. In particular, with high probability, Maker wins the connectivity game as soon as the minimum degree is at least two; Maker wins the Hamilton cycle game as soon as the minimum degree is at least four; Maker wins the perfect matching game as soon as the minimum degree is at least two and every edge has at least three neighbouring vertices; and Maker wins the <em>H</em>-game as soon as there is a subgraph from a finite list of “minimal graphs.” These results also allow us to give precise expressions for the limiting probability that <em>G</em>(<em>n, r</em>) is Maker-win in each case, where <em>G</em>(<em>n, r</em>) is the graph on <em>n</em> points chosen uniformly at random on the unit square with an edge between two points if and only if their distance is at most <em>r</em>.</p>