Strategic manipulations of multi-valued solutions in economies with indivisibilities

We study manipulability of multi-valued solutions in fair allocation of homogeneous indivisible objects with monetary transfers. We show that the no-envy solution satis…es some extension of strategy-proofness to multi-valued solutions. Since no-envy implies Pareto e¢ ciency in this literature, this result implies that we can escape many impossibility results on strategy-proofness by allowing a solution to be multi-valued and extending strategy-proofness in an appropriate way. Nevertheless, we show that no-envy is not immune to all forms of manipulations. In that sense, we are able to identify what kind of manipulations occur. Next, we derive impossibility results on the egalitarian-equivalence and e¢ cient solution. The intuition that manipulability comes from singlevaluedness is not con…rmed by this solution. Finally, we show that there is no subsolution of the Pareto and identical preferences lower bound satisfying the notion of extended strategy-proofness used by TadenumaThomson (1995). Keywords: Indivisible goods, Extended strategy-proofness, Noenvy, Egalitarian-equivalence, Identical preferences lower bound, Fair allocation. The …rst version was written while T. Sakai was visiting the University of Namur. He gratefully acknowledges the hospitality of the school. yUniversity of Namur and CORE, 8 rempart de la vierge, 5000 Namur, Belgium; [email protected]. zDepartment of Economics, University of Rochester, Rochester, NY 14627, USA; [email protected]; http://troi.cc.rochester.edu/ toyo.