I think your first problem is similar to the fair-division/cake-cutting problem as mentioned in one of the other answer and comments(I don't think it is out in the left field): find an allocation of parts of a continuous good among n players, with various optimality criterions criteria (max sum, max min, leximin). Two papers very related to your work are one by Dubins and Spanier(http://www.jstor.org/stable/10.2307/2311357) and another by Marco Dall'Aglio(http://adsabs.harvard.edu/abs/2001JCoAM.130...17D), , based on the Dubins-Spanier paper.
In the work by Dall'Aglio, they state relationships between the max min and the max sum problem. In one of the more recent papersby him (http://arxiv.org/pdf/1110.4241.pdf) , they provide an algorithm for finding max min allocation: the solution is a max sum allocation weighted by the dual parameters of the max min problem. They search for the dual parameters using a subgradient method. I think this wor is related to your post, you can solve the max min problem by getting an equitable solution to the max sum problem. I guess that by taking the logarithm of replacing the utility with a concave utility, you are driving the optimal solutions towards the equitable solutions.
