User juan camilo gamboa - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:18:35Z http://mathoverflow.net/feeds/user/24882 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100638/equitable-division-of-a-contiguous-resource/101275#101275 Answer by Juan Camilo Gamboa for Equitable division of a contiguous resource Juan Camilo Gamboa 2012-07-04T00:59:01Z 2012-07-04T18:33:41Z <p>I think your first problem is similar to the fair-division/cake-cutting problem as mentioned in the other answer and comments: find an allocation of parts of a continuous good among n players, with various optimality criteria (max sum, max min, leximin). Two papers very related to your work are one by <a href="http://www.jstor.org/stable/10.2307/2311357" rel="nofollow">Dubins and Spanier</a> and another by <a href="http://adsabs.harvard.edu/abs/2001JCoAM.130...17D" rel="nofollow">Marco Dall'Aglio</a>, based on the Dubins-Spanier paper. </p> <p>In the work by Dall'Aglio, they state relationships between the max min and the max sum problem. In one of the more <a href="http://arxiv.org/pdf/1110.4241.pdf" rel="nofollow">recent papers</a>, 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 replacing the utility with a concave utility, you are driving the optimal solutions towards the equitable solutions.</p>