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.
Bochet, O. L. A., & Sakai, T. (2007). Strategic Manipulations of Multi-Valued Solutions in Economies with Indivisibilities. Mathematical Social Sciences, 53(1), 53-68. https://doi.org/10.1016/j.mathsocsci.2006.07.001