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The paper considers a voting model where each voter's type is her preference. The type graph for a voter is a graph whose vertices are the possible types of the voter. Two vertices are connected by an edge in the graph if the associated types are “neighbors.” A social choice function is locally strategy-proof if no type of a voter can gain by misrepresentation to a type that is a neighbor of her true type. A social choice function is strategy-proof if no type of a voter can gain by misrepresentation to an arbitrary type. Local-global equivalence (LGE) is satisfied if local strategy-proofness implies strategy-proofness. The paper identifies a condition on the graph that characterizes LGE. Our notion of “localness” is perfectly general. We use this feature of our model to identify notions of localness according to which various models of multidimensional voting satisfy LGE. Finally, we show that LGE for deterministic social choice functions does not imply LGE for random social choice functions.