Finding Fair Allocations under Budget Constraints

We study the fair allocation of indivisible goods among agents with identical, additive valuations but individual budget constraints. Here, goods--each a specific size and value--need to be allocated such that bundle assigned each agent is total at most agent's budget. Since envy-free allocations do not necessarily exist in context, compelling relaxations--in particular, notion envy-freeness up k (EFk)--have received significant attention recent years. In an EFk allocation, prefers its own over any other agent, removal goods, have similarly bounded envy against charity (which corresponds set all unallocated goods). It has been shown prior work satisfies constraints maximizes Nash social welfare 1/4-approximately EF1. However, computation (or even existence) exact remained intriguing open problem. make notable progress towards this by proposing simple, greedy, polynomial-time algorithm computes EF2 under Our algorithmic result implies universal existence division context. The analysis exploits intricate structural properties envy-freeness. Interestingly, same also provides EF1 guarantees for important special cases. Specifically, we settle instances which: (i) value good proportional size, (ii) or (iii) value. extends setting wherein goods' sizes are specific.