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We study a fair division problem with indivisible items, namely thecomputation of maximin share allocations. Given a set of $n$ players, themaximin share of a single player is the best she can guarantee to herself, ifshe would partition the items in any way she prefers, into $n$ bundles, andthen receive her least desirable bundle. The objective then is to find anallocation, so that each player is guaranteed her maximin share. Previous workshave studied this problem mostly algorithmically, providing constant factorapproximation algorithms. In this work we embark on a mechanism design approachand investigate the existence of truthful mechanisms. We propose three modelsregarding the information that the mechanism attempts to elicit from theplayers, based on the cardinal and ordinal representation of preferences. Weestablish positive and negative (impossibility) results for each model andhighlight the limitations imposed by truthfulness on the approximability of theproblem. Finally, we pay particular attention to the case of two players, whichalready leads to challenging questions.

On Truthful Mechanisms for Maximin Share Allocations