Envy-Freeness for Mixed Resources (EFM)

• EFM is a fairness notion that unifies the allocation of divisible and indivisible resources by ensuring that no agent strongly envies another's bundle.
• It relaxes classic envy-freeness through conditions similar to EF1 and EFX, providing structured guarantees even when resource types differ in nature and valuation.
• Recent research establishes existence results, efficient algorithms, and tight price-of-fairness bounds, linking EFM to Nash welfare optimization and mechanism design.

Envy-freeness for mixed resources (EFM) is a central fairness concept in the allocation of collections involving both divisible and indivisible resources, such as combinations of goods, chores, and cake. EFM unifies and generalizes classic envy-freeness and its relaxations—such as EF1 and EFX—to the mixed-resource setting, guaranteeing that no agent has strong reason to prefer another's allocation even when the resources have fundamentally different structures and valuation properties. Recent research has established existence theorems, algorithmic constructions, price-of-fairness bounds, and structural connections to Nash welfare and mechanism design.

1. Formal Framework: The EFM Model

Consider a resource allocation setting with a set of agents $N = \{1,\dots,n\}$, a finite set of indivisible items $M$ (which may be goods, chores, or mixed), and one or more divisible resources ("cake") $C$, often represented as $[0,1]$. Each agent $i$ possesses an additive utility function over both $M$ and $C$. For indivisible items, $u_i:2^M \to \mathbb R$ is additive: $u_i(S) = \sum_{t \in S} u_i(t)$. For the divisible cake, $u_i(X) = \int_{x \in X} f_i(x)dx$, where $f_i$ is a measurable density.

A mixed-resource allocation is a pair $(A_i, C_i)$ for each agent, where $A_i$ is the bundle of indivisible items and $C_i \subseteq [0,1]$ is the piece of cake assigned to $i$. The total utility is $u_i(A_i, C_i) = u_i(A_i) + u_i(C_i)$.

EFM is defined via the following desideratum: for every ordered pair $(i,j)$,

• if agent $j$ receives some divisible resource valued positively by $i$, then $i$ must not envy $j$:

$u_i(A_i, C_i) \ge u_i(A_j, C_j)$

• otherwise (i.e., $j$'s bundle is purely indivisible and $i$ does not value $j$'s cake), $i$ can envy $j$ by at most one item:

$$\exists t \in A_j : u_i(A_i, C_i) \ge u_i(A_j \setminus \{t\}, C_j)$$

This generalizes EF (for pure cake), EF1 (for pure indivisibles), and captures both chores and goods as well as their combinations.

2. Theoretical Guarantees: Existence Results

EFM existence is guaranteed under various settings:

• For indivisible chores plus a divisible 'bad cake' (i.e., all resources are undesirable), an EFM allocation always exists if agents' preferences are additive (or more generally, doubly-monotone) and valuations over the cake are given by continuous, non-positive densities. The proof uses a two-phase algorithm: first, compute an EF1 division of chores by top-trading envy-cycle elimination; then, allocate the cake using iterative perfect partitions among carefully selected agent subsets, ensuring that assigning small bad-cake prefixes does not introduce new envy edges (Bhaskar et al., 2020).
• For the general mixed goods model (goods, chores, and cake), EFM allocations always exist for any number of agents with additive utilities (Aziz et al., 7 Nov 2025). The existence proof uses a reduction to a "core" discrete problem: finding an allocation that is both EF1 and envy-freeable (i.e., can be made envy-free with some payments). Stromquist–Woodall’s cake-cutting theorem enables the translation from payments to cake pieces.
• In the model of indivisible goods plus divisible goods (good cake), EFM allocations exist for any $n$ and additive utilities (Bei et al., 2019).

The EFM property thus provides a robust relaxation of envy-freeness that remains achievable in rich heterogeneous environments.

3. Algorithmic Construction and Complexity

Algorithmic constructions for EFM vary by setting:

Resource Setting	Algorithmic Technique	Complexity
Chores + bad cake (Bhaskar et al., 2020)	EF1 via top-trading envy-cycle elimination + incremental bad-cake partitioning	$O(n^3)$ steps + oracles
Goods + good cake (Bei et al., 2019)	EF1 allocation + envy-graph maintenance + perfect-EF cake splits	$O(n^3)$ with oracles
Mixed items + cake (Aziz et al., 7 Nov 2025)	Core EF1 + envy-freeable allocation via meta-item bundling and matchings; Stromquist–Woodall for cake	$O(n^3 m^2)$
Two agents (Bei et al., 2019)	Cut-and-choose for cake; EF1 allocation	$O(m \log m)$

Key subroutines include:

• Envy-graph-based approaches: Repeatedly allocate indivisible items via sources in the envy graph, with cycle eliminations to maintain EF1.
• Sink-addable/source-addable set identification: Critical for incremental cake allocation without introducing envy.
• Perfect partition oracles: Used to split the cake (bad or good) among a subset of agents so every agent in that subset gets an equal share of (dis)utility.
• Meta-item bundling and matching: Bundles subjective goods and objective chores to ensure matching-based allocation satisfies discrete envy-freeness constraints.

In the absence of perfect-partition oracles, approximate EFM allocations ($\epsilon$-EFM) can be computed in fully polynomial time for piecewise-linear valuations (Bei et al., 2019).

4. Relationship to Other Fairness Notions

EFM sits within a hierarchy of envy-based and equitable allocations:

• Exact envy-freeness (EF): $u_i(A_i, C_i) \ge u_i(A_j, C_j)$ for all $i, j$; only possible with divisible goods.
• EF1 (envy-free up to one item): On indivisibles only.
• EFX (envy-free up to any item): Stronger than EF1 but often unachievable.
• EFM: Mixed setting, generalizes EF and EF1; EFX always implies EFM, but not conversely.
• EFXM: Envy-freeness up to any indivisible good in mixed settings; strictly stronger than EFM (Nishimura et al., 2023, Li et al., 3 Jan 2024).
• Pareto optimality (PO): Some EFM allocations can be Pareto-optimal, but the simultaneous existence of both is not guaranteed in all mixed-manna settings (Aleksandrov, 2020).

For chore-division and mixed manna (goods and chores in the same instance), EFM shares many foundational algorithmic features with EF1 and its relatives. However, EFM's enforcement across both indivisible and divisible resources distinguishes it from classical notions.

5. Price of Fairness for EFM

Imposing EFM can entail significant welfare loss compared to unconstrained social-welfare maximization. Research provides tight upper and lower bounds on this price of fairness:

Setting	$\mathbf{POF_n(\mathrm{EFM})}$ (Scaled)	$\mathbf{POF_n(\mathrm{EFM})}$ (Unscaled)
$n=2$ agents	$3/2$	$2$
$n$ agents	$\Theta(\sqrt{n})$	$\Theta(n)$

Here, "scaled" means all total utilities normalized to one per agent; "unscaled" allows arbitrary positive utilities. For EFM, the worst-case price of fairness matches that for EFX among purely indivisibles, suggesting that introducing divisible resources does not dramatically improve worst-case welfare loss (Li et al., 3 Jan 2024).

Concrete worst-case instances are constructed by bundling a highly-valued indivisible with a divisible; enforcing EFM prevents “winner-takes-all” allocations and thus limits total welfare.

6. Variations, Extensions, and Connections

Variants and extensions include:

• EFXM: A stricter relaxation where any envy toward a purely-indivisible bundle must be removable by deleting each indivisible good for which the envier has positive marginal utility (Nishimura et al., 2023). Maximum Nash Welfare (MNW) allocations for binary-linear instances are both EFXM and Pareto-optimal.
• Truthful Mechanisms: It is impossible to have a deterministic allocation mechanism satisfying both EFM and strategyproofness in general, even with just two agents and one indivisible plus one divisible good (Li et al., 2023). However, restricted settings (binary valuations or identical value on divisibles) admit truthful, EFM-satisfying algorithms.
• Best-of-both-worlds Guarantees: For two agents (and in certain bi-valued multi-agent settings), one can simultaneously guarantee ex-ante envy-freeness and ex-post EFM via randomized algorithms that combine EFX (or EF1) solutions with adjustments to the division of divisible resources (Bu et al., 9 Oct 2024).
• Approximate and Algorithmic Frontiers: For general non-additive (e.g., submodular) valuations or settings lacking access to perfect partitions, the existence and efficient computability of EFM allocations remain major open questions.

7. Open Problems and Future Directions

Key unresolved questions and research directions include:

• EFM with Nonlinear Valuations: The existence of EFM under general submodular or arbitrary monotone preferences remains open (Bhaskar et al., 2020, Aziz et al., 7 Nov 2025).
• EFM-plus-Pareto-optimality: The compatibility of efficient (high-welfare) allocations with EFM in mixed-manna instances is unsettled in general (Aleksandrov, 2020, Li et al., 3 Jan 2024).
• Discrete Cake Algorithms: Developing algorithms for the allocation of bad cake (or heterogeneous divisible chores) without appealing to perfect-partition oracles is an active challenge (Bhaskar et al., 2020).
• Stronger Fairness Notions: Achieving EFXM (or stronger) guarantees in the mixed domain, especially together with PO and incentive compatibility.
• Truthful EFM Mechanisms: Complete characterization of settings where strategyproofness and EFM are simultaneously achievable, and exploration of relaxed notions such as maximin-strategyproofness (Li et al., 2023).
• Social Welfare Guarantees under EFM: Tightening worst-case POF, resource monotonicity, and the trade-off with efficiency, especially in large-scale or heterogeneous domains (Li et al., 3 Jan 2024).
• Empirical and Applied Directions: Comparison of greedy, matching-based, and Nash-welfare methods for EFM in practical settings (e.g., food bank allocations, chore divisions) (Aleksandrov, 2020).

EFM thus marks a boundary in the theory of fair division—balancing between the impossibility of achieving categorical envy-freeness for indivisibles and the desideratum of strong fairness guarantees in complex, heterogeneous resource settings.