# Minimal Envy and Popular Matchings

## TL;DR

This paper explores the relationship between fairness and popularity in object allocation, showing that minimal envy matchings are equivalent to popular matchings when they exist, and introduces algorithms for finding such matchings.

## Contribution

It establishes the equivalence between minimal envy and popular matchings, and provides polynomial-time algorithms for computing them in various scenarios.

## Key findings

- Popular matchings are equivalent to minimal envy matchings when they exist.
- A path-based algorithm can transform any matching into a popular matching.
- A market with majority-based exchanges converges to a popular matching if it exists.

## Abstract

We study ex-post fairness in the object allocation problem where objects are valuable and commonly owned. A matching is fair from individual perspective if it has only inevitable envy towards agents who received most preferred objects -- minimal envy matching. A matching is fair from social perspective if it is supported by majority against any other matching -- popular matching. Surprisingly, the two perspectives give the same outcome: when a popular matching exists it is equivalent to a minimal envy matching. We show the equivalence between global and local popularity: a matching is popular if and only if there does not exist a group of size up to 3 agents that decides to exchange their objects by majority, keeping the remaining matching fixed. We algorithmically show that an arbitrary matching is path-connected to a popular matching where along the path groups of up to 3 agents exchange their objects by majority. A market where random groups exchange objects by majority converges to a popular matching given such matching exists. When popular matching might not exist we define most popular matching as a matching that is popular among the largest subset of agents. We show that each minimal envy matching is a most popular matching and propose a polynomial-time algorithm to find them.