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We extend the reduced games introduced by Moulin (J Econ Theory 36:120–148, 1985) to multi-choice games, and define related properties of consistency and its converse. Since the unit-level-core proposed by Hwang and Liao (J Glob Optim 2009) violates (weak) converse consistency, we propose to minimally enlarge the unit-level-core so as to recover (weak) converse consistency.
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