Tight Subsidy Bounds for Weighted Proportional Allocation of Mixed Manna

RESUMO

We study the problem of fairly allocating m indivisible items among n agents with possibly unequal entitlements in the mixed manna setting, where each item may be perceived as a good or a chore by different agents. We focus on the fundamental fairness notion of proportionality. Since proportional allocations need not exist in this setting, we allow monetary subsidies to restore proportionality while minimizing the total subsidy. When each item's (dis)utility is bounded by 1, a total subsidy of at least \tau(n) \approx n/4 may be necessary. For goods-only or chores-only instances, the best previously known upper bound was n/3-1/6 due to Wu and Zhou~(2024). We close this gap by proving that a total subsidy of at most \tau(n) always suffices, thereby establishing the tight subsidy bound. Our results hold even in the more general setting of weighted mixed manna, resolving an open question posed by~Wu et al. (2023) and Garg et al. (2026). The allocation also satisfies weighted proportionality up to one item (WPROP1). Our proof develops a novel application of the Knaster-Kuratowski-Mazurkiewicz (KKM) fixed-point theorem, extending the KKM framework to share-based fairness notions. Finally, we design a polynomial-time algorithm to compute such allocations for any fixed number of agents.