Fairness and Rank-Weighted Utilitarianism in Resource Allocation

In multiagent resource allocation with indivisible goods, boolean fairness criteria and optimization of inequality-reducing collective utility functions (CUFs) are orthogonal approaches to fairness. We investigate the question of whether the proposed scale of criteria by Bouveret and Lemaitre [5] applies to nonadditive utility functions and find that only the more demanding part of the scale remains intact for k-additive utility functions. In addition, we show that the min-max fair-share allocation existence problem is NP-hard and that under strict preferences competitive equilibrium from equal incomes does not coincide with envy-freeness and Pareto-optimality. Then we study the approximability of rank-weighted utilitarianism problems. In the special case of rank dictator functions the approximation problem is closely related to the MaxMin-Fairness problem: Approximation and/or hardness results would immediately transfer to the MaxMin-Fairness problem. For general inequality-reducing rank-weighted utilitarianism we show (strong) NP-completeness. Experimentally, we answer the question of how often maximizers of rank-weighted utilitarianism satisfy the max-min fair-share criterion, the weakest fairness criterion according to Bouveret and Lemaitre's scale. For inequality-reducing weight vectors there is high compatibility. But even for weight vectors that do not imply inequality-reducing CUFs, the Hurwicz weight vectors, we find a high compatibility that decreases as the Hurwicz parameter decreases.