To Give or Not to Give: Fair Division for Single Minded Valuations

Single minded agents have strict preferences, in which a bundle is acceptable only if it meets a certain demand. Such preferences arise naturally in scenarios such as allocating computational resources among users, where the goal is to fairly serve as many requests as possible. In this paper we study the fair division problem for such agents, which is harder to handle due to discontinuity and complementarities of the preferences. Our solution concept—the competitive allocation from equal incomes (CAEI)—is inspired from market equilibria and implements fair outcomes through a pricing mechanism. We study the existence and computation of CAEI for multiple divisible goods, cake cutting, and multiple discrete goods. For the first two scenarios we show that existence of CAEI solutions is guaranteed, while for the third we give a succinct characterization of instances that admit this solution; then we give an efficient algorithm to find one in all three cases. Maximizing social welfare turns out to be NP-hard in general, however we obtain efficient algorithms for (i) divisible and discrete goods when the number of different types of players is a constant, (ii) cake cutting with contiguous demands, for which we establish an interesting connection with interval scheduling, and (iii) cake cutting with a constant number of players with arbitrary demands. Our solution is useful more generally, when the players have a target set of desired goods, and very small positive values for any bundle not containing their target set. ∗The authors aknowledge support from the Danish National Research Foundation and The National Science Foundation of China (under the grant 61361136003) for the Sino-Danish Center for the Theory of Interactive Computation and from the Center for Research in Foundations of Electronic Markets (CFEM), supported by the Danish Strategic Research Council. A part of this work was done while Simina Brânzei and Ruta Mehta were visiting the Simons Institute for the Theory of Computing. Simina Brânzei was also supported by ISF grant 1435/14 administered by the Israeli Academy of Sciences and Israel-USA Bi-national Science Foundation (BSF) grant 2014389, as well as the I-CORE Program of the Planning and Budgeting Committee and The Israel Science Foundation. †Hebrew University of Jerusalem, Israel. E-mail: [email protected] ‡Tsinghua University, China. E-mail: [email protected] §University of Illinois Urbana-Champaign, USA. E-mail: [email protected] 1 ar X iv :1 60 2. 09 08 8v 2 [ cs .G T ] 2 8 A pr 2 01 6