Absence of Envy among “Neighbors” Can Be Enough

Abstract

We assume that the set of agents is decomposed into several classes containing individuals related each other in some way, for example groups of neighbors. We propose a new definition of fairness by requiring efficiency and envy-freeness only within each group. We identify conditions under which absence of envy among “neighbors” is enough to ensure fairness in the entire society. We also show that equal-income Walrasian equilibria are the only fair allocations according to our notion, deriving as corollaries the equivalence theorems of Zhou (1992) and Cato (2010). The analysis is conducted in atomless economies as well as in mixed markets.

Fairness and formation rules of coalitions

“…In atomless economies with connected coverings, local and global strict fairness coincide. This is shown in Donnini and Pesce (2021a). However, we prove here that imposing an inclusion or exclusion restriction on the set of potentially envied coalitions enlarges the set of strictly fair allocations.…”

“…of possible coalitions whose union gives back T . In Donnini and Pesce (2021a) we allow each agent to compare her own bundle only with the average bundle of any coalition contained in a group she belongs to and we impose the absence of envy just within each C i of the covering . We identify a class of coverings-the connected coverings 2 -for which equity within each C i is sufficient to characterize equal-income competitive allocations.…”

“…Consider the following feasible allocation → x T : in the case of a connected covering  is proved in Donnini and Pesce (2021a). Consider now, in the economy defined in Example A.2, the connected covering .…”

Fairness and fuzzy coalitions

“…In Donnini and Pesce (2021) we define two elements C i and C j of R to be connected if μ(C i ∩ C j ) > 0 and a covering R to be connected if for every pair of its elements there is a path linking up them, where a path is a sequence of elements of R connected to each other. With similar arguments used in Donnini and Pesce (2021), it can be proved that if R is connected, absence of envy among members of the same class C i of R is enough to ensure fairness in the whole economy E, and thereby this local notion of fuzzy strict fairness coincides with F S F. It is worthwhile noting that, thanks to the fuzzy approach, this equivalence holds in mixed markets with no assumption on the space of agents as differently needed in Donnini and Pesce (2021).…”

“…The literature of fair division is huge and rich of various equity notions proposed in decades of research. This paper focuses on the notion of strict fairness due to Zhou (1992) which has been recently studied by Cato (2010), who proposed a local version of S F by imposing absence of envy only among neighbors (see also Donnini and Pesce 2021), by Basile et al (2018) who compared S F with a coalitional fairness notion in economies with infinitely many goods, by Donnini and Pesce (2020) who got the equivalence W ei = S F in mixed markets under stronger assumptions than those used to prove the Core-Walras equivalence theorem (Shitovitz 1973). In this paper, we keep the model of mixed markets but, weakening the notion of strict fairness, we avoid additional conditions on the space of agents as well as restrictions on the set of potentially envied agents.…”