Approximately Envy-free and Equitable Allocations of Indivisible Items for Non-monotone Valuations

TL;DR

This work addresses fair allocation of indivisible items among agents with non-monotone valuations, focusing on existence and computation of allocations that are approximately envy-free or equitable. The authors introduce $\text{EF1}_{g}^{c}$ and $\text{EQ1}_{g}^{c}$ and prove existential results via Sperner’s Lemma, including when bundles must form connected subpaths, and they provide a polynomial-time dynamic programming algorithm to compute $\text{EQ1}_{g}^{c}$ under non-negative valuations. They extend the framework to non-positive valuations using a novel multi-coloring Sperner variant and establish existential and computational results for $\text{EQ1}_{g}^{c}$ and $\text{EF1}_{g}^{c}$ under path constraints, with additional results for objective valuations yielding $\text{EQX}_{g}^{c}$ in pseudo-polynomial time. The paper also shows that under monotone valuations, stronger guarantees (EF1/EQ1) can be obtained along paths, and it outlines several open problems, including the computational complexity (PPAD) of $\text{EF1P}_{g}^{c}$ allocations and the existence of stronger up-to-item variants for broader valuation classes. Overall, the work advances fair division by establishing existence and computability of relaxed fairness notions in non-monotone, indivisible-item settings and by extending topological methods to new valuation regimes.

Abstract

We revisit the setting of fair allocation of indivisible items among agents with heterogeneous, non-monotone valuations. We explore the existence and efficient computation of allocations that approximately satisfy either envy-freeness or equity constraints. Approximate envy-freeness ensures that each agent values her bundle at least as much as those given to the others, after some (or any) item removal, while approximate equity guarantees roughly equal valuations among agents, under similar adjustments. As a key technical contribution of this work, by leveraging fixed-point theorems (such as Sperner's Lemma and its variants), we establish the existence of {\em envy-free-up-to-one-good-and-one-chore} ($\text{EF1}^c_g$) and {\em equitable-up-to-one-good-and-one-chore} ($\text{EQ1}^c_g$) allocations, for non-monotone valuations that are always either non-negative or non-positive. These notions represent slight relaxations of the well-studied {\em envy-free-up-to-one-item} (EF1) and {\em equitable-up-to-one-item} (EQ1) guarantees, respectively. Our existential results hold even when items are arranged in a path and bundles must form connected sub-paths. The case of non-positive valuations, in particular, has been solved by proving a novel multi-coloring variant of Sperner's Lemma that constitutes a combinatorial result of independent interest. In addition, we also design a polynomial-time dynamic programming algorithm that computes an $\text{EQ1}^c_g$ allocation. For monotone non-increasing valuations and path-connected bundles, all the above results can be extended to EF1 and EQ1 guarantees as well. Finally, we provide existential and computational results for certain stronger {\em up-to-any-item} equity notions under objective valuations, where items are partitioned into goods and chores.

Figures (5)

• Figure 1: The figure on the left (a) illustrates an application of Sperner's Lemma to a triangulation $T$ of a 2-simplex $\Delta = \text{conv}(\boldsymbol{v}_1, \boldsymbol{v}_2, \boldsymbol{v}_3)$. The coloring function assigns a color from set {1 (Red), 2 (Green), 3 (Blue)} to each vertex $\boldsymbol{x} \in V(T)$, and it is special. Specifically, for each $i\in [3]$, the color $i$ does not appear on the 1-face of $\Delta$ that does not contain $\boldsymbol{v}_i$ (i.e., on the edge opposite to vertex $\boldsymbol{v}_i$). Sperner's Lemma guarantees the existence of an odd number of fully-colored simplices. In this case, there are three such simplices (that is, there exists at least one such simplex), and they are highlighted in gray. On the right (b), we show Kuhn's triangulation of the 2-simplex $\{(x_1, x_2) : 0 \leq x_1 \leq x_2 \leq 3\}$, which is based on an allocation instance with $n = 3$ agents and $m = 3$ items. The colors are assigned to vertices based on the special coloring $L$ derived from an arbitrary non-negative virtual valuation functions. One of the fully-colored elementary $(n-1)$-simplices in $T$, whose existence is guaranteed by Sperner's Lemma, is highlighted in gray.
• Figure 2: The figure represents the fractional allocations $\mathcal{\tilde{A}}(\boldsymbol{x}_1^*), \mathcal{\tilde{A}}(\boldsymbol{x}_2^*), \mathcal{\tilde{A}}(\boldsymbol{x}_3^*)$ associated with the vertices $\boldsymbol{x}_1^*, \boldsymbol{x}_2^*, \boldsymbol{x}_3^*$ of the fully-colored simplex $\Delta^*$ in Figure \ref{['fig:1']}(b) (highlighted in gray), which are colored blue (label 3), red (label 1), and green (label 2) according to the coloring function $L$; the permutation $\sigma:[3]\rightarrow [3]$ such that $L(\boldsymbol{x}_{\sigma(i)})=i$ is defined by $(1,2,3)\xrightarrow{\sigma}(2,3,1)$. The integral items are represented by ellipses, the fractionality levels of each item (integral, 1-fractional, 2-fractional) are determined by the gray vertical lines, that cut each item in three parts, and the knives that determine the fractional allocations are represented by the black triangles. In such example, as indicated by the vertex colors, the virtual valuation determining $L$ is maximized by the third (resp. first, resp. second) fractional bundle from the left, of the allocation associated with the blue (resp. red, resp. green) vertex of $\Delta^*$, i.e., $\boldsymbol{x}_1^*$ (resp. $\boldsymbol{x}_2^*$, resp. $\boldsymbol{x}_3^*$). Finally, as an illustrative example of left-first or right-first bundles, we observe that the central bundle is right-first in $\Delta^*$.
• Figure 3: Given $j \in [n]$, we describe how the fractional bundle $\tilde{A}_j = [a_j, b_j]$ of the main allocation of $\Delta^*$ can be rounded to obtain the integral bundle $A_j$ in each of the following nine cases, assuming that $A_{j+1}, \ldots, A_n$ have already been determined: 1: $a_j\equiv 0,b_j\equiv 0$: $A_j\gets \llbracket a^-_j,b^-_j\rrbracket$; 2: $a_j\equiv 0,b_j\equiv 1$: (i) $A_j\gets \llbracket a^-_j,b^-_j\rrbracket$ if $b^+_j\in A_{j+1}$, and (ii) $A_j\gets \llbracket a^-_j,b^+_j\rrbracket$ if $b^+_j\not\in A_{j+1}$; 3: $a_j\equiv 0,b_j\equiv 2$: $A_j\gets \llbracket a^-_j,b^+_j\rrbracket$; 4: $a_j\equiv 1,b_j\equiv 0$: $A_j\gets \llbracket a^-_j,b^-_j\rrbracket$; 5: $a_j\equiv 1,b_j\equiv 1$: (i) $A_j\gets \llbracket a^-_j,b^-_j\rrbracket$ if $b^+_j\in A_{j+1}$, and (ii) $A_j\gets \llbracket a^+_j,b^+_j\rrbracket$ if $b^+_j\not\in A_{j+1}$; 6: $a_j\equiv 1,b_j\equiv 2$: (i) $A_j\gets \llbracket a^+_j,b^+_j\rrbracket$ if $A_j$ is left-first, and (ii) $A_j\gets \llbracket a^-_j,b^+_j\rrbracket$ if $A_j$ is right-first; 7: $a_j\equiv 2,b_j\equiv 0$: $A_j\gets \llbracket a^+_j,b^-_j\rrbracket$; 8: $a_j\equiv 2,b_j\equiv 1$: (i) $A_j\gets \llbracket a^+_j,b^-_j\rrbracket$ if $b^+_j\in A_{j+1}$, (ii) $A_j\gets \llbracket a^+_j,b^+_j\rrbracket$ if $b^+_j\not\in A_{j+1}$; 9: $a_j\equiv 2,b_j\equiv 2$: $A_j\gets \llbracket a^+_j,b^+_j\rrbracket$. The figure illustrates each of the nine cases as follows: each item and its three associated fractionality levels are represented by an ellipse divided into three parts (similarly to Figure \ref{['fig:2']} in Appendix \ref{['app:pictures']}); the two black triangles in each case represent the positions ($a_j$ and $b_j$) of the two knives that determine the $j$-th fractional bundle $\tilde{A}_j=[a_j,b_j]$ (by possibly cutting the two board items of the considered bundle); the red rectangle encloses all items that are fully included in the $j$-th bundle $A_j$ of the rounded (integral) allocation $\mathcal{A}$; the right-hand item, when marked with crossed lines, indicates that it was included in bundle $A_{j+1}$ during the previous step of the rounding procedure; the blue circle on the left (resp. right) knife indicates that bundle $[a_j,b_j]$ is left-first (resp. right-first) in $\Delta^*$.
• Figure 4: The figure on the left (a) illustrates an application of the Multi-coloring Sperner's lemma to a triangulation $T$ of a 2-simplex $\Delta = \text{conv}(\boldsymbol{v}_1, \boldsymbol{v}_2, \boldsymbol{v}_3)$. The multi-coloring function ${\cal L}$ assigns a non-empty subset from {1 (Red), 2 (Green), 3 (Blue)} to each vertex $\boldsymbol{x} \in V(T)$, and it is special; furthermore, in this specific case, ${\cal L}$ satisfies the assumption done w.l.o.g. in the proof of Theorem \ref{['multiSperner']}, that is: ${\cal L}(\boldsymbol{x}) = \{i \in [3] : \boldsymbol{x} \in F_i\}$ holds for any vertex $\boldsymbol{x}$ located on the boundary of $\Delta$, and $| {\cal L}(\boldsymbol{x}) | = 1$ holds for any internal vertex $\boldsymbol{x}$ not located on the boundary. The four gray triangles represent the fully-colored simplices in $T$ (w.r.t. the multi-coloring function ${\cal L}$); we observe that, differently from the standard Sperner's lemma, their number is not necessarily odd. The minimal restriction $L: V(T) \rightarrow [3]$ of ${\cal L}$, as defined in the proof of Theorem \ref{['multiSperner']}, assigns the same color as ${\cal L}$ when that color is unique, and assigns an internal color when it is not (i.e., Green for $\boldsymbol{v}_1$, and Red for $\boldsymbol{v}_2$ and $\boldsymbol{v}_3$). The figure on the right (b) illustrates the $1$-simplex $\Delta'$ obtained from the instance of figure (a), following the steps outlined in the proof of Theorem \ref{['multiSperner']}. $\Delta'$ is homeomorphic to the topological space $F'=[\boldsymbol{v}_2,\boldsymbol{v}_3]\cup [\boldsymbol{v}_2,\boldsymbol{v}_3]$ obtained by the union of the $1$-dimensional faces $[\boldsymbol{v}_2,\boldsymbol{v}_3]$ and $[\boldsymbol{v}_3,\boldsymbol{v}_1]$ of $\Delta$. The figure also illustrates the triangulation (equivalent to) $T'$ and the vertices of $V(T')$ are colored following the minimal restriction $L$ of ${\cal L}$. Finally, the two yellow paths connecting some elementary 2-simplices in the left figure (a) form the graph $G = (V, E)$ used in the proof of Theorem \ref{['multiSperner']} to establish the existence of an odd number of fully-colored $2$-simplices under the minimal restriction $L$, namely, the three gray triangles filled with black lines. These triangles are also fully colored with respect to the multi-coloring function ${\cal L}$; we observe that, in this case, there are four such triangles, which are shown in gray.
• Figure 5: The figure illustrates the projection $f$, where $\Delta$ is a 3-simplex. The left image shows the original simplex $\Delta$, along with the topological space of $F' = F_1 \cup F_2 \cup F_3$. Specifically, the 2-face $F_4 = \text{conv}(\boldsymbol{v}_1, \boldsymbol{v}_2, \boldsymbol{v}_3)$ opposite to $\boldsymbol{v}_4$ is represented with a yellow filling, while the other 2-faces $F_1, F_2, F_3$ (i.e., those constituting $F'$) are filled with light gray. The right image depicts the 2-simplex $\Delta'$, which is obtained by applying the projection $f$ of $F'$ onto $F_4$. The red arrow on the left indicates the axis and orientation associated with the projection.

Theorems & Definitions (50)

• Theorem 3.1: Sperner's Lemma Sperner1928
• Theorem 3.2: Generalized Sperner's Lemma Bapat1989
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• proof
• Theorem 3.4
• Theorem 3.5
• Theorem 4.1: Multi-coloring Sperner's Lemma
• Theorem 4.2