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Abstract We prove a representation formula that gives a new characterization of the coalitionally strategy-proof binary social choice functions. We give this characterization in the case of social choice functions selecting one of two alternatives (i.e., binary social choice). The domain of the functions we consider consists of profiles of preferences over a society of arbitrary cardinality and indifference is admitted. Strategy proofness is meant to be coalitional: No group of agents has incentives to form a coalition that can manipulate the social choice for their own advantage with false reporting.