Abstract
I examine the classic problem of fair division of a piecewise homogeneous good. Previous work developed algorithms satisfying various combinations of fairness notions (such as proportionality, envy-freeness, equitability, and Pareto-optimality). However, this previous work assumed that all utility functions are additive. Recognizing that additive functions accurately model utility only in certain situations, I investigate superadditive and subadditive utility functions. Next, I propose a new division protocol that utilizes nonlinear programming and test it on a sample instance. Finally, I prove theoretical results that address (a) relationships between fairness notions, and (b) the orthogonal issue of division efficiency (i.e., the price of satisfying particular fairness notions).
Highlights
The problem of fair division of goods dates back to ancient times (Note 1) and occurs frequently in the social sciences, law, economics, game theory, and other fields
One player may prefer a third of the strawberry portion to half of the chocolate portion, and a second player may prefer a fourth of the chocolate portion to the entire strawberry portion, while a third player might derive positive utility from only the rhubarb portion. (Such a set of player-dependent utility functions for each homogenous piece defines an instance of the cake-cutting problem.) The objective is to obtain a fair division of the cake
Aumann and Dombb (2011) extend this result to the case where distributions of the cake must be contiguous. Both these papers consider only additive utility functions–one of the results in the present paper demonstrates that the price is unbounded if utility functions are nonadditive
Summary
The problem of fair division of goods dates back to ancient times (Note 1) and occurs frequently in the social sciences, law, economics, game theory, and other fields. (Such a set of player-dependent utility functions for each homogenous piece defines an instance of the cake-cutting problem.) The objective is to obtain a fair division of the cake. Steinhaus (1948), Brams and Taylor (1995), and Peterson and Su (2002) are a few among many who have found division protocols, which yield allocations satisfying various combinations of the fairness notions for instances with all additive utility functions. A second difficulty in the cake-cutting problem arises: while an allocation may be, e.g., envy-free, it may be inefficient; in other words, the total social welfare derived from the allocation may be lower than the maximum possible social welfare for the given instance.
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