# Heterogeneity and Instability in the Stable Marriage Problem

## TL;DR

This paper extends the stable marriage problem by introducing heterogeneity in group activity levels, using simulations to explore how these variations affect social optimality and stability.

## Contribution

It proposes a generalized SMP model with varying active message sender percentages and analyzes the impact on stability and social optimality through simulations.

## Key findings

- Homogeneous groups achieve social optimality.
- Mixed activity levels lead to less stable matchings.
- Active messaging from both groups yields more realistic outcomes.

## Abstract

The Stable Marriage Problem (SMP) has been extremely discussed in the literature and it is useful to a number of real-world applications. We propose a generalized version of the SMP in which numbers of the matching groups are different as in [9]. However, we go further to make a percentage of each group behave as active message senders. As such, the special case in which all Males are active messengers (beta = 1) and all Females are not active (alpha = 0) replicates the results in [9]. Moreover, we use numerical simulation to present three cases (and their extremes) in which we vary the percentage of active messengers in each group. Whereas we are able to replicate previous work, our numerical simulations also suggest that socially optimal comes only when the groups are homogeneous. More real-world like results are presented when members from both groups are active message senders.