We investigate simulation hemi-metrics between certain forms of turnbased 2 1 2 -player games played on infinite topological spaces. They have the desirable property of bounding the difference in payoffs obtained by starting from one state or another. All constructions are described as the special case of a unique one, which we call the Hutchinson hemi-metric on various spaces of continuous previsions. We show a directed form of the Kantorovich-Rubinstein theorem, stating that the Hutchinson

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... i-metric on spaces of continuous probability valuations coincides with a notion of trans-shipment hemi-metric. We also identify the class of so-called sym-compact spaces as the right class of topological spaces, where the theory works out as nicely as possible.