Steven J. Brams, D. Marc Kilgour, and Christian Klamler have published Two-Person Fair Division of Indivisible Items: An Eﬃcient, Envy-Free Algorithm, Notices of the AMS, 61, 130-141 (2014).
Here is the abstract:
Many procedures have been suggested for the venerable problem of dividing a set of indivisible items between two players. We propose a new algorithm (AL), related to one proposed by Brams and Taylor (BT), which requires only that the players strictly rank items from best to worst. Unlike BT, in which any item named by both players in the same round goes into a “contested pile”, AL may reduce, or even eliminate the contested pile, allocating additional or more preferred items to the players. The allocation(s) that AL yields are Pareto-optimal, envy-free, and maximal; as the number of items (assumed even) increases, the probability that AL allocates all the items appears to approach infinity if all possible rankings are equi-probable. Although AL is potentially manipulable, strategizing under it would be difficult in practice.
Here is the conclusion:
[…] The fact that only AL requires that the players indicate their preference rankings is clearly an advantage, but in some applications it may be desirable to elicit and use information about the intensity of the players’ preferences. But when obtaining such information is diﬃcult, AL oﬀers a compelling alternative—for example, in allocating the marital property in a divorce or the items in an estate, especially when the players have diﬀerent tastes (e.g., for memorabilia or artworks).