Reachability of Fair Allocations via Sequential Exchanges

TL;DR

It is shown that two EF1 allocations may not be reachable from each other even in the case of two agents, and deciding their reachability is PSPACE-complete in general.

Abstract

In the allocation of indivisible goods, a prominent fairness notion is envy-freeness up to one good (EF1). We initiate the study of reachability problems in fair division by investigating the problem of whether one EF1 allocation can be reached from another EF1 allocation via a sequence of exchanges such that every intermediate allocation is also EF1. We show that two EF1 allocations may not be reachable from each other even in the case of two agents, and deciding their reachability is PSPACE-complete in general. On the other hand, we prove that reachability is guaranteed for two agents with identical or binary utilities as well as for any number of agents with identical binary utilities. We also examine the complexity of deciding whether there is an EF1 exchange sequence that is optimal in the number of exchanges required.

Figures (5)

• Figure 1: The exchange of goods $g_x$ and $g_y$. The edges $e_x$ and $e_y$ correspond to the respective goods in $G_{\mathcal{A}, \mathcal{B}}$, while the edges $e_x'$ and $e_y'$ correspond to those in $G_{\mathcal{A}', \mathcal{B}}$.
• Figure 2: The graph $H_4$ and an example of a $T$-triangle and an $F$-triangle.
• Figure 3: A $T$-patch and an $F$-patch. The center $T$-triangle and $F$-triangle are denoted by bold lines, and the exteriors are denoted by non-bold solid lines. The dotted lines are not part of the patches.
• Figure 4: An $F$-$F$-$F$ join on $H_p^1$ (solid lines), $H_p^2$ (dashed lines) and $H_p^3$ (dotted lines). All three graphs share the exterior of the patch $P_F$ (bold lines).
• Figure 5: (a) An $F$-$F$ join on $H_p^1$ (solid lines) and $H_p^2$ (dashed lines). Both graphs share the patch $P_F$ (bold lines). (b) The association of vertices between $P_F^1$ and $P_T^2$ in an $F$-$T$ join on $H_p^1$ and $H_p^2$---note the labelling of vertices.

Theorems & Definitions (43)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• Theorem 3.4
• proof : Proof of \ref{['thm:iden_2_optimal']}
• proof : Proof of \ref{['thm:binary_2_optimal']}
• Proposition 4.1
• proof