This paper studies strategic manipulations of multi-valued solutions in the problem of fairly allocating homogeneous indivisible objects with monetary transfers. We provide various extensions of strategy-proofness to multi-valued solutions and examine their impact on standard solutions. We show that some efficient and fair solutions, such as the envy-free solution, satisfy certain extensions of strategy-proofness. We also establish an impossibility result on extended strategy-proofness that is defined in terms of expected utility.
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