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Almost Envy-Free Allocations of Indivisible Goods or Chores with Entitlements
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Shows WEFX and XWEF allocations of indivisible goods/chores with weighted agents may not exist for 2-3 agents, gives positive WEFX procedures for two agents (integer weights exact, normalized weights approximate) and a polynomial-time 1WEF chore algorithm.
Problem
Fair allocation of indivisible goods or chores to agents with unequal weights (entitlements), where standard EFX/EF1 notions are generalized to weighted envy-freeness.
Approach
Generalizes EFX and EF1 to weighted (entitlement-scaled) envy; proves impossibility counterexamples for 2 and 3 agents; adapts I-cut-you-choose for integer weights, a moving-knife procedure for normalized weights, and a weight-dependent round-robin picking sequence for chores yielding 1WEF.
Key results
- WEFX allocations may not exist for n=2 or 3 with additive valuations; counterexamples are at best 0.786-WEFX for n=2 and 0.795-WEFX for n=3.
- For two agents with integer weights, a modified I-cut-you-choose algorithm computes an exact WEFX allocation.
- A polynomial-time (O(l^2)) weighted round-robin for chores guarantees a complete 1WEF allocation for any number of agents and arbitrary positive weights.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use weighted envy-cycle elimination with identical additive valuations | Goods allocation · Goods, any number of agents | WEFX existence | Helped |
| Assume identical ordinal preferences | Weighted envy-cycle elimination · Goods, any number of agents | WEFX existence | Helped |
| Allow arbitrary additive valuations and unequal weights | Goods allocation · Goods, 2 agents | WEFX factor: 0.786 | Hurt |
| Allow arbitrary additive valuations and unequal weights | Goods allocation · Goods, 3 agents | WEFX factor: 0.795 | Hurt |
| Use integer-valued weights and the modified I-cut-you-choose procedure | Goods allocation · Goods, 2 agents | WEFX existence | Helped |
| Use normalized weights and the moving-knife-based procedure | Goods allocation · Goods, 2 agents | approximate WEFX factor: \frac{w}{2\sqrt[3]{m}} | Helped |
| Allow arbitrary additive cost functions and unequal weights | Chores allocation · Chores, 2 agents | XWEF factor: 1.272 | Hurt |
| Allow arbitrary additive cost functions and unequal weights | Chores allocation · Chores, 3 agents | XWEF factor: 1.214 | Hurt |
| Use a weight-dependent picking sequence and final allocation loop | Chore-Division Round-Robin · Chores, any number of agents | 1WEF existence | Helped |
Surprising findings
- Envy-cycle elimination, which yields EFX for identical valuations, may fail to give WEF1 even for two agents with additive valuations.
- Chore XWEF counterexamples exist for n=2 and n=3 (at best 1.272-XWEF and 1.214-XWEF), mirroring the goods case.
Limitations
- Existence of WEFX for general additive valuations and n>3 remains open.
- Approximate WEFX for normalized weights is only guaranteed with factor w/(2 m^{1/3}), which degrades with the number of items.
- Results are theoretical; no empirical evaluation.
Experimental setup
Baselines: WEF1 algorithm (Chakraborty et al.) weighted MMS envy-cycle elimination
Builds on: envy-cycle elimination I-cut-you-choose moving-knife procedure weighted EF1 algorithm
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