The Marriage Problem and the Fate of Bachelors

TL;DR

This paper models the marriage problem using statistical mechanics, analyzing the effects of bachelors and polygamy on societal matching efficiency through exact solutions and entropy analysis.

Contribution

It introduces a generalized model including bachelors and polygamy, providing exact solutions and entropy behavior at zero temperature, and discusses the conditions for bachelor occurrence.

Findings

Exact solutions for the optimal matching problem.

Entropy vanishes quadratically with temperature.

Bachelor occurrence depends on societal pairing probabilities.

Abstract

In the marriage problem, a variant of the bi-parted matching problem, each member has a `wish-list' expressing his/her preference for all possible partners; this list consists of random, positive real numbers drawn from a certain distribution. One searches the lowest cost for the society, at the risk of breaking up pairs in the course of time. Minimization of a global cost function (Hamiltonian) is performed with statistical mechanics techniques at a finite fictitious temperature. The problem is generalized to include bachelors, needed in particular when the groups have different size, and polygamy. Exact solutions are found for the optimal solution (T=0). The entropy is found to vanish quadratically in $T$. Also other evidence is found that the replica symmetric solution is exact, implying at most a polynomial degeneracy of the optimal solution. Whether bachelors occur or not, depends not only on their intrinsic qualities, or lack thereof, but also on global aspects of the chance for pair formation in society.