Local envy-freeness in house allocation problems

Abstract

We study the fair division problem consisting in allocating one item per agent so as to avoid (or minimize) envy, in a setting where only agents connected in a given network may experience envy. In a variant of the problem, agents themselves can be located on the network by the central authority. These problems turn out to be difficult even on very simple graph structures, but we identify several tractable cases. We further provide practical algorithms and experimental insights.

Appendix: MIP formulation for \(\textsc {dec{-}location{-}LEF} \)

We describe the MIP formulation used to solve the $\textsc {dec{-}location{-}LEF}$ problem. We are given a set of objects O, a set of agents N equipped with preferences over those objects (for the ease of exposure we refer here to $r_{i,o}$ as the rank of object o in the preference order of agent i), and a graph $G = (V,E)$.

Together with the real valued decision variable e, which will be used to express the envy bound we try to minimize, we make use of the following (binary) decision variables:

• $x_{i,o}$: agent i holds object o

• $l_{i,p}$: agent i is located on node p

• $s_{i,j,o}$: agent i sees that agent j holds object o

We first express that each agent must receive exactly one object, and that each object must be assigned to exactly one agent (constraints (1) and (2)). Similarly, each agent must be assigned to a single node of the network, and each node must have a single agent assigned (constraints (3) and (4)).

$$\begin{aligned} \forall i\in N :&\sum \limits _{o\in O}{x_{i,j}}=1 \end{aligned}$$

(1)

$$\begin{aligned} \forall o\in O :&\sum \limits _{i\in N}{x_{i,o}}=1 \end{aligned}$$

(2)

$$\begin{aligned} \forall i\in N :&\sum \limits _{p\in V}{l_{i,p}}=1 \end{aligned}$$

(3)

$$\begin{aligned} \forall p\in V :&\sum \limits _{i\in N}{l_{i,p}}=1 \end{aligned}$$

(4)

When agent i is located on a node p connected to a node q where agent j holds o, i sees that j holds o:

$$\begin{aligned} \forall i, j \in N, \forall \{p, q\} \in E :&l_{i,p} + l_{j,q} + x_{j,o} - 2\le s_{i,j,o} \end{aligned}$$

(5)

Finally, we try to minimize the amount of envy between any pair of agents (MMPE), which is expressed by setting, together with the objective function $\min e$, constraint (6):

$$\begin{aligned} \forall i, j \in N: \sum _{o \in O} r_{i,o} \times s_{i,j,o} - \sum _{o \in O} r_{i,o} \times x_{i,o} \le e \end{aligned}$$

(6)

Note that in the case of $\textsc {dec{-}location{-}LEF}$, we are only interested in whether we can find a solution which sets the envy bound e at 0, i.e., whether an LEF allocation exists.