Fair Allocation of Indivisible Goods

• Fair allocation of indivisible goods is a method of dividing limited, discrete items among agents while ensuring envy-freeness up to one good for every possible bundle permutation.
• The symEF1 concept extends traditional fairness notions by requiring that each agent's value for any bundle is nearly as high as for any other, achieved through n-colorable item graphs and combinatorial techniques.
• Algorithmic and empirical studies validate symEF1's practical applications in humanitarian logistics, game balancing, and other settings where assignment uncertainty and robust fairness are critical.

Fair allocation of indivisible goods addresses the problem of distributing a finite set of distinct items among multiple agents in a manner that satisfies robust fairness guarantees, even when the goods cannot be subdivided and each agent's preferences may be heterogeneous. The "symmetrically envy free up to one good" (symEF1) notion strengthens standard guarantees by requiring that any agent be EF1‐satisfied regardless of which bundle in the partition they receive. This concept draws on techniques from combinatorics and combinatorial optimization to ensure worst-case fairness, and is motivated by settings where bundle assignments may subsequently occur by lottery or adversarially.

1. Definition and Formal Model of symEF1 Allocations

The symEF1 property generalizes classical envy-freeness up to one good (EF1) to a "bundle-symmetric" setting. Given $m$ indivisible goods and $n$ agents, let a partition of the goods into $n$ bundles $(A_1, \ldots, A_n)$ be sought. Each agent $i$ possesses an additive utility function $v_i : 2^{[m]} \to \mathbb{R}$, assigning values to subsets of goods.

A partition $(A_1, ..., A_n)$ is said to be symEF1 if, for every agent $i$ and every pair of bundles $A_k$, $A_\ell$, the following condition holds: $$v_i(A_k) \geq v_i(A_\ell) - \max_{g \in A_\ell} v_i(\{g\}),$$ for all $i,k,\ell \in [n]$. That is, the agent's value for any bundle is at least the value of any other bundle after removing the item from that bundle which is most valuable to $i$.

Unlike the standard EF1 notion, which applies to a specific assignment of bundles to agents, symEF1 requires that the EF1 guarantee hold for every possible reassignment of bundles—that is, for every possible permutation of bundles to agents. This makes symEF1 a robust and highly symmetric fairness property.

2. Existence Results and Combinatorial Characterization

The existence of symEF1 allocations is linked to a combinatorial coloring problem on a graph constructed from the instance. Each agent's items are organized into $n$-tuples ("indexed $n$-tuples") representing their preferences as if they were assigned bundles in a hypothetical round-robin order. An "item graph" is constructed with:

• Vertices corresponding to goods,
• Edges joining two items if they share an $n$-tuple for any agent.

A partition is said to separate the $n$-tuples if no bundle contains two items from the same $n$-tuple according to any agent. The central result states: if the item graph is $n$-colorable, then a partition can be constructed such that every agent's $n$-tuples are separated, which by construction yields a symEF1 allocation.

This sufficient condition always holds for two agents (since any such item graph is bipartite), as well as for agents with identical, disjoint, or binary valuations. In these settings, symEF1 allocations are guaranteed to exist. For $n > 2$, the coloring condition is not always satisfied—there exist instances (even with binary valuations) where symEF1 allocations do not exist, demonstrating tightness of the characterization.

3. Algorithmic Construction and Enumeration of symEF1 Allocations

To construct a symEF1 allocation when the $n$-colorability condition is satisfied, assign each color class to a unique bundle. Since the coloring ensures no two items from the same $n$-tuple are assigned together, the separation property and symEF1 immediately follow.

Furthermore, the number of distinct symEF1 allocations can be exponential in the number of connected components $C$ of the item graph. Specifically, there are at least $(n!)^{C-1}$ symEF1 allocations in such instances, as each component can be assigned a separate permutation of the $n$ bundles independently.

For general instances, finding a symEF1 allocation is NP-hard in the absence of the coloring property. Heuristic algorithms and integer programming (IP) models are used in practice. The former incrementally builds a candidate partition using greedy item assignments, item moves, and swaps, while the latter formalizes symEF1 as IP constraints.

4. Empirical Findings and Structural Properties

Extensive computational experiments support the prevalence of symEF1 allocations in random instances. For two agents, the greedy heuristic alone finds a symEF1 allocation in virtually all tested cases, with direct assignment accounting for over 90% of all item assignments. For $n = 3,4,5$, the proportion of instances admitting a symEF1 allocation rapidly increases as the number of goods exceeds the number of agents, often reaching 100% with moderate margin ($m \gg n$). Larger agent counts require more backtracking via IP, but the density of positive instances remains high.

The granularity of valuations (i.e., the maximum individual item utility $M$) modulates the incidence: with larger $M$, symEF1 allocations may be less common for fixed $n$ and $m$. Theoretical lower bounds confirm that for two agents and four items at least two distinct symEF1 allocations always exist, suggesting significant non-uniqueness even in small instances.

5. Applications and Context of symEF1 in Practice

The symEF1 property has compelling application scenarios where bundles are prepared in advance and the eventual mapping of bundles to agents is uncertain or random. For example:

• Humanitarian logistics (e.g., disaster relief): Pre-packaged boxes must be assigned without fine control over which recipient obtains which bundle; symmetry ensures all recipients feel fairly treated.
• Game balancing: Character presets or loot bundles are created to be broadly acceptable regardless of assignment.
• Assignment-robust mechanisms: When practical assignment protocols (e.g., random draws, decentralized selection) mean final allocation cannot be finely tailored to preferences, symEF1 provides strong worst-case guarantees.

The robust symmetry of the symEF1 requirement avoids disputes and strategic manipulation based on knowledge of assignment order and allows more transparent and trustworthy division protocols.

6. Future Directions, Open Problems, and Limitations

Several compelling research directions remain:

• Characterization: A complete necessary and sufficient condition for the existence of symEF1 remains open. Current results are only sufficient (via $n$-colorable item graphs), and negative instances are known.
• Symmetric Generalizations: Extending symmetric fairness to stronger relaxations (e.g., symEF$k$ for $k > 1$, where up to $k$ goods may be removed) is conjectured to facilitate existence in all instances (e.g., any instance admits a symEF$(n-1)$ allocation).
• Algorithmic Improvements: More efficient heuristic and exact approaches for finding symEF1 allocations in large, high-dimensional settings are needed, as well as comprehensive enumeration of all such partitions for small $n$.
• Connections to Welfare and Pareto Optimality: Understanding trade-offs between symEF1 and classical objectives such as maximizing Nash social welfare or achieving Pareto optimality is an open technical domain.
• Empirical Analysis: Whether the high probability of symEF1 allocations in randomly generated instances generalizes to structured, real-world instances with richer valuation domains (e.g., submodular or heterogeneous preferences) remains to be systematically analyzed.

In summary, symEF1 is a strong symmetry-based relaxation of envy-freeness up to one good for indivisible settings, supported by both combinatorial theory and empirical evidence. It provides a compelling fairness guarantee for practical applications where uncertainty in assignment is inevitable, and forms a foundation for further exploration into robust, permutation-independent notions of fair division (Johnston et al., 19 Jun 2024).