Symmetrically Fair Allocations of Indivisible Goods

TL;DR

SymEF1 allocations address symmetry in envy-free up to one good for indivisible goods with additive valuations. The authors introduce a graph-theoretic sufficient condition based on $n$-colorability of an item graph $G(\mathcal{T})$, proving existence of symEF1 allocations for two agents and under certain valuation classes, while showing nonexistence for some instances with three or more agents. They also develop the concept of separating $\mathcal{T}^{i}$, establish a lower bound on the number of symEF1 allocations when the condition holds, and analyze two-agent four-item cases. Complementing theory with simulations for up to five agents, they study incidence, algorithmic strategies (integer programming and a greedy heuristic), and the effect of item counts and valuation granularity. The work advances symmetric fairness in fair division and suggests avenues for broader symmetric relaxations such as symEF$k$.

Abstract

We consider allocating indivisible goods with provable fairness guarantees that are satisfied regardless of which bundle of items each agent receives. Symmetrical allocations of this type are known to exist for divisible resources, such as consensus splitting of a cake into parts, each having equal value for all agents, ensuring that in any allocation of the cake slices, no agent would envy another. For indivisible goods, one analogous concept relaxes envy freeness to guarantee the existence of an allocation in which any bundle is worth as much as any other, up to the value of a bounded number of items from the other bundle. Previous work has studied the number of items that need to be removed. In this paper, we improve upon these bounds for the specific setting in which the number of bundles equals the number of agents. Concretely, we develop the theory of symmetrically envy free up to one good, or symEF1, allocations. We prove that a symEF1 allocation exists if the vertices of a related graph can be partitioned (colored) into as many independent sets as there are agents. This sufficient condition always holds for two agents, and for agents that have identical, disjoint, or binary valuations. We further prove conditions under which exponentially-many distinct symEF1 allocations exist. Finally, we perform computational experiments to study the incidence of symEF1 allocations as a function of the number of agents and items when valuations are drawn uniformly at random.

Figures (5)

• Figure 1: An instance with three agents and six items in which the item graph is not $3$-colorable, though a symEF1 allocation exists, showing that the sufficient condition in \ref{['thm:symEF1']} is not necessary. The boxed items represent indexed $n$-tuple $\mathcal{T}^{i}_{2}$ for each agent $i \in [3]$ and lead to the dashed edges in the item graph. A 5-coloring of the graph is given.
• Figure 2: MNW and symEF1 allocations for \ref{['ex:MNW']}. The MNW allocation is EF1, and in fact entirely envy free, with agent $1$ receiving bundle $1$, but not symEF1.
• Figure 3: Proportion of instances with symEF1 allocations quickly tends to 1 with more items.
• Figure 4: The proportion of instances with symEF1 allocations decreases with higher granularity of valuations.
• Figure :

Theorems & Definitions (34)

• Definition 1: Symmetrically Envy-Free up to One Good
• Definition 2: Agent's Round-Robin Allocation
• Lemma 3
• proof
• Proposition 4
• proof
• Proposition 5
• Example 6
• Definition 7
• Definition 8