Revisiting Fair and Efficient Allocations for Bivalued Goods

Abstract: This paper re-examines the problem of fairly and efficiently allocating indivisible goods among agents with additive bivalued valuations. Garg and Murhekar (2021) proposed a polynomial-time algorithm that purported to find an EFX and fPO allocation. However, we provide a counterexample demonstrating that their algorithm may fail to terminate. To address this issue, we propose a new polynomial-time algorithm that computes a WEFX (Weighted Envy-Free up to any good) and fPO allocation, thereby correcting the prior approach and offering a more general solution. Furthermore, we show that our algorithm can be adapted to compute a WEQX (Weighted Equitable up to any good) and fPO allocation.

• The paper corrects previous polynomial-time EFX/fPO claims by presenting a counterexample and introducing a new, robust algorithm.
• The paper introduces an innovative agent grouping and sequential price-lifting technique that maintains Fisher market invariants while targeting WEFX/WEQX outcomes.
• The paper achieves a polynomial runtime of O(min{k,m} n² m²), extending fair division methods to weighted, bivalued allocation settings.

Revisiting Fair and Efficient Allocations for Bivalued Goods: Technical Analysis

Problem Setting and Motivation

This paper investigates fair division of indivisible goods among agents whose valuations are bivalued—each good for each agent takes one of two possible positive values ($1$ or $k>1$). It focuses on allocations that satisfy high standards of both fairness and efficiency, under the framework of additive utilities. Weighted fairness—where agents have (possibly non-uniform) entitlements—adds to the generality and technical challenge.

The work is situated at the intersection of recent developments in relaxations of envy-freeness (notably EFX—envy-free up to any good—and its weighted variant WEFX), fractional Pareto optimality (fPO), and the algorithmic construction of allocations via Fisher market mechanisms. Recent literature claimed polynomial-time EFX and fPO allocations for bivalued agents, but this paper identifies a critical error in that claim, provides a counterexample, and proposes new, provably polynomial-time algorithms for WEFX/fPO and WEQX/fPO allocations.

Critical Review and Correction of Prior Work

Garg and Murhekar (2021) developed a Fisher market-based combinatorial algorithm for EFX and fPO allocation in the bivalued context. However, by explicitly constructing a parameterized family of instances, this paper demonstrates that their main routine may cycle forever, never terminating or guaranteeing output. This is a significant technical correction—since the literature had treated EFX/fPO for bivalued additive instances as a solved case in polynomial time.

The specific counterexample exploits circumstances where the greedy or iterative price-raising steps of the original algorithm alternate indefinitely, without ever meeting their termination criteria. The oscillation between least-spender and big-spender without convergence proves the prior polynomial-time claim invalid.

Weighted EFX/fPO: New Algorithm and Theoretical Guarantees

To circumvent the flaw, the authors construct a new, provably polynomial-time algorithm. Core novel features:

• Explicit Agent Grouping: Agents are partitioned into ordered groups using an initial welfare-maximizing assignment and analysis of the maximum bang-per-buck (MBB) Fisher market graph. This structure is essential for efficient subsequent reallocation and price adjustment.
• Relaxed Termination and Sequential Price Lifting: The new approach relaxes the termination condition, allowing the process to halt as soon as a specific weighted EFX property is met, rather than requiring universal pairwise envy mitigation at every step. Groups are processed in sequence with price multiplications, ensuring eventual progress and avoiding infinite cycling.
• Transfer Graphs with Invariant Maintenance: Substantial effort is devoted to the maintenance of technical invariants that guarantee Fisher market equilibrium, MBB property, group-wise and global WEFX, monotonicity of key quantities, and the absence of regressions in fairness or efficiency with each operation. These invariants support the strong guarantee that at each step, the computed allocation is both a Fisher market equilibrium (thus fPO) and progressively closer to the WEFX target.
• Runtime Analysis: The reallocation and price-raising algorithm achieves $\mathcal{O}(\min\{k,m\} n^2 m^2)$ worst-case runtime, polynomial in all relevant parameters, with $n$ agents, $m$ items, and maximum item value $k$.
• Completeness of Fairness/Efficiency Guarantees: The output allocation is weighted envy-free up to any good (WEFX) and fractionally Pareto optimal (fPO). Notably, the algorithm works for non-uniform weights, generalizing both BuLLLT24 and NeohT25 to bivalued and weighted domains.

Weighted Equitability up to Any Good

Beyond envy-avoidance, the paper also addresses weighted equitability up to any good (WEQX), combining it with fPO. It extends the group-formation and reallocation framework to equitability-based fairness—where an agent’s bundle value (normalized by weight) is not less than the normalized value of any other's bundle, up to the removal of any good from that bundle.

The WEQX/fPO algorithm follows much of the EFX/fPO machinery, with necessary modifications for utility (as opposed to price) comparisons. The same level of algorithmic detail, invariant maintenance, and complexity analysis is presented, establishing polynomial-time computability.

Numerical and Theoretical Highlights

• Algorithm Termination and Correctness: The new approach always terminates in polynomial time, in contrast to the previously claimed EFX/fPO routine for bivalued instances, which can cycle indefinitely.
• Generality: All results extend to the weighted agent case, not just unweighted fairness, and hold for all bivalued additive valuation instances.
• Existence: The constructive approach in both fairness frameworks (envy and equitability) implicitly proves the existence of WEFX/fPO and WEQX/fPO allocations in all weighted bivalued settings.

Implications and Future Directions

This work sharpens the boundary of algorithmic tractability in fair division: for additive, bivalued (hence, low-complexity) utilities, both WEFX/fPO and WEQX/fPO allocations can be efficiently computed, even under agent weights.

Theoretical implications: The result strengthens the connection between Fisher markets as equilibrium computation devices and combinatorial fairness mechanisms. It suggests that subtle choices in graph-based transfer routines and group handling are critical for progress and correctness.

Practical relevance: The ability to guarantee polynomial runtime for weighted, bivalued EFX/fPO allocations strengthens the case for deploying these mechanisms in practical settings where agent priorities (weights) and simple utility models are present—inheritance, resource scheduling, course allocation, etc.

Follow-up challenges: The paper identifies two directions:

• Efficient algorithms for personalized bivalued utilities (where the two allowed values differ by item and agent, not just globally).
• WEFX/fPO in the domain of chores (with negative item utilities), where fairness and efficiency goals are more subtle.

Conclusion

This work provides a rigorous correction to the literature on fair division for bivalued utilities, clarifying existence and algorithmic boundaries for WEFX and WEQX in conjunction with efficiency (fPO). By constructing robust, polynomial-time Fisher market-based routines, the paper advances the understanding of nuanced fairness in multi-agent allocation and sets the stage for future exploration of more expressive domains and fairness notions.

References:

• Garg and Murhekar (2021): “EFX and fPO allocation for bivalued goods” — see counterexample and discussion.
• Bu, Li, Li, Liu, and Tang (2024): Existence of EFX and fPO for bivalued unweighted settings.
• Neoh and Teo (2025), Lin et al. (2025): Related results for binary/chore cases.