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As a supplement to Joel's answer, you may want to look at this nice paper of Bollobas, Leader, and Walters concerning continuous games. As a starting point they discuss the classical Lion and Man game introduced by Rado. In this game there is a lion chasing a man inside the unit disk. Both have identical maximum speeds. The lion wins if he catches the man, and the man wins if he is never caught by the lion. If the lion chooses to always run directly toward the man, then he will get arbitrarily close to the man, but never catch him. On the other hand, if the lion instead moves at top speed so that he is always on the radial vector from the centre to the man, it was 'clear' that this was a winning strategy. Proof: without loss of generality, the man stays on the boundary of the disk. However, in 1952, Besicovitch exhibited an ingenious winning strategy for man! Thus, staying on the boundary is with loss of generality for man. Nonetheless, one can ask the perplexing question if man also has a winning strategy? In this particular game, it turns out that the answer is no. But by changing the metric space, Bollobas, Leader, and Walters prove that there are games in a similar vein where both lion and man have winning strategies!