Subsidies with heterogeneous divisible goods

Determine, under random valuations of indivisible goods supplemented by heterogeneous divisible goods, how preference diversity affects the minimum divisible endowment value B required for exact envy-freeness with high probability and when this quantity is asymptotically smaller than the required monetary budget.

Background

The paper studies exact envy-freeness when indivisible goods are allocated together with nonnegative monetary subsidies, assuming that all agents value money identically. Its main results characterize the asymptotic minimum monetary subsidy under i.i.d. additive valuations, including the effects of the ratio and divisibility of the number of goods relative to the number of agents.

The conclusion proposes replacing homogeneous money with heterogeneous divisible goods. In that setting, each agent values the entire divisible endowment at most B, but may value different parts of that endowment differently. Prior worst-case work is cited as providing EF guarantees related to agents’ indivisibility ratios, but the paper leaves unresolved how random valuations of the indivisible goods and diversity in preferences over divisible goods jointly determine the required endowment and whether divisible compensation can asymptotically outperform monetary compensation.

References

A natural future direction is to replace money by heterogeneous divisible goods. For comparison with a monetary budget $B$, assume that each agent values the entire divisible endowment at most $B$, while valuations of its parts may differ. This setting has been studied in the worst case by Li et al., who give EF guarantees up to a fraction of one good depending on each agent's indivisibility ratio, namely the fraction of her total value attributable to indivisible goods. Under random valuations of the indivisible goods, it remains to determine how preference diversity affects the minimum $B$ needed for exact envy-freeness with high probability and when this quantity is asymptotically smaller than the required monetary budget.

— Asymptotic Subsidies for Envy-Free Allocation  (2610.10272 - Wu, 7 Oct 2026) in Section Conclusion