  We study a fair division problem with indivisible items, namely the
computation of maximin share allocations. Given a set of $n$ players, the
maximin share of a single player is the best she can guarantee to herself, if
she would partition the items in any way she prefers, into $n$ bundles, and
then receive her least desirable bundle. The objective then is to find an
allocation, so that each player is guaranteed her maximin share. Previous works
have studied this problem mostly algorithmically, providing constant factor
approximation algorithms. In this work we embark on a mechanism design approach
and investigate the existence of truthful mechanisms. We propose three models
regarding the information that the mechanism attempts to elicit from the
players, based on the cardinal and ordinal representation of preferences. We
establish positive and negative (impossibility) results for each model and
highlight the limitations imposed by truthfulness on the approximability of the
problem. Finally, we pay particular attention to the case of two players, which
already leads to challenging questions.
