Maximin Share Guarantees for Few Agents with Subadditive Valuations

TL;DR

The paper addresses fair division of indivisible goods under Maximin Share fairness for subadditive valuations with few agents, proving a tight $1/2$-MMS guarantee for up to four agents and extending similar guarantees to settings with many agents under two valuation types. It introduces the $\boldsymbol{\alpha}$-MMS$(\mathbf{d})$ framework to unify and facilitate inductive proofs across partitions, and employs a cut-based maximum desirable half approach to achieve the results. The work provides complete characterizations for three-agent cases and tight impossibility results for three submodular agents, bridging gaps in the MMS literature beyond additive valuations. It also develops a new inductive model and analyzes multiple valuation classes (subadditive, submodular, XOS) to push the boundary of what constant-factor MMS guarantees are possible in the few-agents regime. The findings offer both theoretical insights and methodological tools that may pave the way for extending constant-factor MMS guarantees to larger numbers of agents.

Abstract

We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an improved lower bound of $1/2$ (which is tight) for the case of subadditive valuations when the number of agents is at most four. We also provide a tight lower bound for the case of multiple agents, when they are equipped with one of two possible types of valuations. Moreover, we propose a new model that extends previously studied models in the area of fair division, which will hopefully give rise to further research. We demonstrate the usefulness of this model by employing it as a technical tool to derive our main result, and we provide a thorough analysis for this model for the case of three agents. Finally, we provide an improved impossibility result for the case of three submodular agents.

Figures (13)

• Figure 1: Let $P=(S_1,S_2,S_3)$ be a partition for agent $S$. The maximum desired half over set $C$ is depicted by thick circle. For each bundle $S_i$, plus bundles (+) attain value at least $v_S(S_i)/2$, while minus (-) attain value at most $v_S(S_i)/2$. Hence $\mathcal{X}_{S}(C,P)=\{S_3\cap C\}$, $\mathcal{X}_{S}({M\setminus C},P)=\{S_1\setminus C,S_2 \setminus C \}$ and $\mathcal{X}^*_{S}(C,P)=\mathcal{X}_{S}({M\setminus C},P)$.
• Figure 2: We illustrate the partition for both $\boldsymbol{d}=(1,2)$ and $\boldsymbol{d}=(2,2)$. The set of items $S_1$ can be divided into $T_1$ and $T_2$. If agent $S$ values bundle $T_1 \cap S_1$ (represented with a thick line in (a)) more than $\frac{v_S(S_1)}{2}$ then allocation (b) has the desired properties (the blue bundle for agent $S$ and the red bundle for agent $T$). If this is not the case, then due to the subadditivity, the same holds for agent $S$ and bundle $T_2 \cap S_1$; then (b) has the desired properties.
• Figure 3: We use blue, red, and green to denote the bundles from which we will allocate to agents $S,T$ and $Q$, respectively. The thickened lines illustrate the first and second cuts, while (c) demonstrates the application of Corollary 2.
• Figure 4: The first candidate allocation $A=(S_2^*,T_1^*)$ for four agents and $\boldsymbol{d}=(3,3,4,4)$. We use blue to denote $S_2^* \subseteq S_2$ and red to denote $T_1^* \subseteq T_1$. We use a thick line to illustrate the cut $C=\{S^*_2 \cup R_4\}$. The allocation is valid and none of the bundles intersects with $R_4$. The cuts are symmetric, i.e. if we had $\mathcal{X}_S^*(C) =\mathcal{X}_S^*(M\setminus C)$ for cut $C=\{R_1 \cup R_2\}$ or/and $\mathcal{X}_T^*(C)=\mathcal{X}_T(C)$ for the corresponding cut $C$, we could construct the same allocation by renaming the bundles.
• Figure 5: The second candidate allocation $A'=(T_2^*,Q_1^*)$. We use red for bundle $T_2^* \subseteq T_2$ and green for $Q_1^*\subseteq Q_1$. The cut $C=\{T_1^* \cup S_2^*\}$ is shown with a thick line. We try to apply Corollary \ref{['cor:2agents']} for agents $S$ and $R$ and the set of items $M \setminus (T_2^*\cup Q_1^*)$. We could construct the same allocation for any $3$ bundles $Q_i^*$ by renaming the bundles.

Theorems & Definitions (51)

• Definition 1: $\boldsymbol{\alpha}$-MMS$(\mathbf{P})$, $\boldsymbol{\alpha}$-MMS$(\mathbf{d})$
• Definition 2: Maximum Desired Half
• proof
• Theorem 1
• proof
• Lemma 1: Two agents
• proof
• Corollary 1
• Corollary 2
• Lemma 2: Three agents