Fair and efficient allocation with few agent types, few item types, or small value levels

Abstract

In fair division of indivisible goods, allocations that satisfy fairness and efficiency simultaneously are highly desired but may not exist or, even if they do exist, are computationally hard to find. Conditions under which such allocations, or allocations satisfying specific levels of fairness and efficiency simultaneously, can be efficiently found have thus been explored. Following this line of research, this study is concerned with the problem in a high-multiplicity setting where instances come with certain parameters, including agent types, item types, and value levels. Particularly, we address two computational problems. First, we wish to compute fair and Pareto-optimal allocations, w.r.t. any of the common fairness criteria: proportionality, maximin share, and max-min fairness. Second, we seek to find a max-min fair allocation that is efficient in the sense of maximizing utilitarian social welfare. We show that the first problem is tractable for most of the fairness criteria when the number of item types is fixed, or when at least two of the three parameters are fixed. For the second problem, we model it as a bi-criteria optimization problem that is solved approximately by determining an approximate Pareto set of bounded size. Our results are obtained based on dynamic programming and linear programming approaches. Our techniques strengthen known methods and can be potentially applied to other notions of fairness and efficiency.