Fair allocation based on diminishing differences

Ranking alternatives is a natural way for humans to explain their preferences. It is being used in many settings, such as school choice (NY, Boston), course allocations, and the Israeli medical lottery. In some cases (such as the latter two), several "items" are given to each participant. Without having any information on the underlying cardinal utilities, arguing about fairness of allocation requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is at least as large as the difference between two items down the list. This assumption implies a preference extension which we call diminishing differences (DD), where a X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possiblyproportional according to this assumption. Based on this characterization, we present a polynomialtime algorithm for finding a necessarily-DDproportional allocation if it exists. Simulations based on a simple random model show that with high probability, a necessarily-proportional allocation does not exist but a necessarily-DDproportional allocation exists. Moreover, that allocation is proportional according to the underlying cardinal utilities.