Mathematically modeling marriage

Summary: Sociologists and others have made many demographic studies of marriage, even modeling it.

Many kinds of arrangements have existed throughout history under the umbrella of marriage, with the expectations and responsibilities of married partners and their rights both to enter into marriage and within the marriage changing considerably over time and across (or within) cultures. It has always included legal and economic dimensions, which have played into the changing demographics of the married.

History of Marriage

The modern concept of “marriage for love” is a relatively recent development in the history of marriage; for several millennia, marriage was an important societal convention fulfilling critical economic, legal, and political functions. Among elite people, marriage was a tool for the control and consolidation of wealth and power by forming strategic alliances between families. Political and military agreements were sometimes forged in the context of a marriage. In middle and lower classes, marriage played a similarly important societal role, especially economically. Marriage’s economic role was further reflected in conventions such as illegitimacy, the dowry, and large families of children, which proved a vital source of labor and economic gain for the family. Marriage was also the societal device for conferring a host of legal rights.

The sexual marriage, a marriage that is freely arranged between two people on the basis of love, is a newer development that evolved from cultural changes that occurred during the Enlightenment and were further developed by the Industrial Revolution. The economic and legal changes that grew from this period gradually eroded the historical reasons behind arranged marriages. This gradual change in marriage perhaps culminated with the 1950s concept of the “Leave It To Beaver family”; however, this short-lived paradigm of marriage experienced dramatic shifts in the socially turbulent decades to come.

The legal and political advances for women in the early twentieth century, coupled with important economic and demographic advances in the latter half of that century, paved the way for important changes in the way people approach marriage. Women made significant strides economically and socially that allowed them the possibility of viable, independent lives apart from marriage. The innovation of birth control also played an important role in the evolution of marriage by allowing women to effectively separate sex and child rearing.

Statistically Analyzing Marriage

Marriage in the United States has undergone critical demographic changes that are closely allied with education level and socioeconomic status. Data spanning five decades of the latter twentieth century and early twenty-first century demonstrate a steady decline in marriage rates. In 1970, 84% of adults aged 30–44 years were married compared with only 60% in 2007. The decline in marriage rates mirrors corresponding rises in the divorce rate and a greater tendency of couples to find alternate arrangements, such as short-term relationships and cohabitation.

Table 1. Percentage of married adults 30–44.

Marriage is associated with well-established economic benefits. Most obvious is the economy of scale realized when a couple can share major assets, like a house, a car, or furniture, that they would otherwise each need to purchase individually. This economy of scale is still a significant advantage even when the additional economic cost of raising children is factored in.

However, economic benefits are also realized when a spouse marries someone with a higher income. In 2007, individual income for married men was an average of 12% higher than for single men. Married women outearned their single counterparts even more substantially, with a 53% higher average income. However, this statistic is not a simple causal relationship between being married and accruing greater wealth; these economic gains are closely tied to education level and earning power. Essentially, people with a higher educational level are more likely to be married, more likely to be married to a spouse of a similar educational level, and more likely to realize and compound the economic benefits of marriage. Interestingly, this is a trend not present in the 1970 data, where the marriages rates across the socioeconomic spectrum were nearly identical. The period since the 1970s has seen significant changes in the number of women attending college and their choices in forming relationships.

Research literature also indicates important health and emotional benefits associated with marriage. These benefits stem not only from lifestyle changes (for example, the healthier diet of a married couple or the shared division of household labor); contemporary studies suggest an even more important factor is the mitigation of stress and its effects on health. Married people live longer, experience less illness, and are less prone to many diseases. Importantly, studies clearly indicate that the quality of the marriage is an important factor; poor marriages have been shown to be even unhealthier than being single. There are also clear gender differences in the extent and the way in which spouses realize the health benefits of marriage.

Mathematically Modeling Marriage

Marriage statistics are extensively tabulated like many other social statistics, but researchers also use mathematical modeling to study marriage. The 2003 book The Mathematics of Marriage: Dynamic Nonlinear Models was authored by an interdisciplinary team including mathematicians. It used mathematical ideas, such as difference equations, phase space, null clines, influence functions, inertia, and stable steady states (attractors), to model marriage, with applications to other psychological phenomena. In 2009, a team of mathematicians from the United Kingdom and the United States analyzed the behaviors of 700 couples over the course of 12 years to develop a probabilistic model that accurately predicted which marriages would last. It was based on classifying couples into one of five types using behavioral variables. Only one type suggested a long-lasting marriage. In 2010, Spanish economist José-Manuel Rey developed an equation based on optimal control models and the “second thermodynamic law for sentimental interaction,” which states a relationship will disintegrate unless it receives input “energy” or effort.

As with the Birthday and Cocktail Party Problems, mathematicians have identified a similar social puzzle in the Stable Marriage Problem. First introduced as a matching problem by D. Gale and L. S. Shapley in 1962, the stable marriage problem consists of equal numbers of single men and women. Every man creates a preference ranking of each woman as a potential match; similarly, every woman ranks each of the men. The goal is to pair the men and women in couples so as to create stable, happy marriages.

The technical challenge is to avoid an “unstable matching,” which arises when a man and woman who are not paired under the matching would each prefer to be with each other over their paired spouse. The immediately interesting question—whether there always exists a stable matching given a set of preference rankings for each individual—was answered in the same seminal work. The Stable Matching Algorithm provides a solution to this problem and, furthermore, is guaranteed to always produce a stable matching. Curiously, this algorithm maximizes one gender’s happiness while minimizing the other’s, depending upon which gender does the proposing and which does the accepting. This same algorithm has other applications, for example, in matching medical school applicants with schools and in pairing roommates for college residence halls.

Bibliography

Coontz, Stephanie. Marriage: A History. New York: Penguin. 2005.

Fry, Richard and D’Vera Cohn. “Women, Men and the New Economics of Marriage.” Pew Research Center (January 19, 2010). http://pewsocialtrends.org/010/01/19/women-men-and-the-new-economics-of-marriage.

Gale, D., and L. S. Shapley. “College Admissions and the Stability of Marriage.” American Mathematical Monthly 69 (1962).

Gottman, John, James Murray, Catherine Swanson, Rebecca Tyson, and Kristin Swanson. The Mathematics of Marriage: Dynamic Nonlinear Models. Cambridge, MA: MIT Press, 2003.

Knuth, Donald. Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms. Providence, RI: American Mathematical Society. 1997.

Parker-Pope, Tara. For Better: The Science of a Good Marriage. New York: Dutton, 2010.