How does a couple break up? Mathematical anatomy of a fall


Anatomy of a Fall , the film by French director Justine Triet, winner of the Oscar for best original screenplay, uses the reconstruction of a body’s fall as a MacGuffin to carefully dissect the fall of the couple’s romantic relationship. protagonist, Sandra Voyter and Samuel Maleski.

The process of breaking up the relationship that the film describes is not an exception, it is a prevalent phenomenon in the world we live in.

The data present high levels of marriage failure globally, with a general upward trend since the last third of the last century.

For some cohorts of Western marriages, the percentage that ends in divorce after 25 years is 50%, popularizing the claim that “one in two marriages ends in divorce.”

The all or nothing couple

In Triet’s opinion, “the rare thing is that a relationship works, most of them are hell, and the film aims to delve into that hell.”

Certainly, negative divorce statistics underestimate the number of relationships that are unhappy. Maybe most of it is hell. However, other relationships are not only lasting successes: they are better than relationships have ever been. That dichotomy – majority failure or exceptional success – seems to summarize the state of contemporary marriage in the West, which has been called all or nothing .

Scientific studies establish that couple relationships decline , that is, the quality or satisfaction of the relationship declines over time on average. Some couples – the successful ones – stop their decline and stabilize at satisfactory levels forever. But the majority of relationships fall slowly until they enter a state of such discomfort that the breakup is a matter of time.

Understanding these disparate falls, how and why they occur, is the objective of the science of relationships .

The second law of the thermodynamics of feelings

The psychology of relationships says that maintaining a relationship over time requires effort. This is what has been called the second law of thermodynamics of romantic relationships.

The all-or-nothing theory suggests that successful couples require, in addition to compatibility, a significant investment in time and energy. These couples manage to reach a high level of satisfaction, compared to those who are not capable of making the required effort and fall, like that of Samuel and Sandra in Triet’s film.

Why do some couples stop the decline and stay happily ever after? Like that of Samuel and Sandra, all couples initially belong to the same class: that of lovers who want to be together and happy forever. If it is also assumed that they are highly compatible and are willing to make joint efforts – they are called homogamous couples –, they form a relationship that can be called Adam and Eve type.

The Adam and Eve model

Mathematical analysis of the Adam and Eve model confirms the all or nothing theory.

Dynamic systems are the mathematical tool to understand the evolution of a variable over time, in the case of romantic relationships, that of the feeling of love or feeling in the couple. The fact that effort is necessary to sustain the relationship turns the sentimental dynamic into a dynamic system controlled by effort, with the goal of lasting forever.

The theory of controlled dynamic systems demonstrates that a successful couple requires a demanding effort, above the favorite level (the one preferred a priori ), which is also difficult to sustain over time. Those who exceptionally manage to maintain that effort gap also achieve a fulfilling happy relationship. It happens, however, that it is easy for them to fail in the attempt.

Mathematics of an asymmetric fall

There are times when a relationship is chaos, sometimes you fight alone, sometimes accompanied by the other, and sometimes against the other. This is how Sandra Voyter expresses it at one point in Triet’s film. That passage gives an idea of ​​how difficult it can be to govern the chaos of every relationship.

The relationship between Samuel and Sandra has ingredients in common with any other relationship: the starting point is very high – the feeling is at its peak – there is a common approach that the relationship will never end, both are willing to contribute. to the happiness of the relationship with an individual effort that must be managed separately, and both know that an external shock or disturbance is likely to occur that alters the state of affairs.

It is known that homogamous couples, avatars of Adam and Eve, are more stable than those who are not. In general, couples are made up of members who are different with respect to some trait, for example, socioeconomic, cultural or religious. In this case they are called heterogamous.

The most basic heterogamy consists of the different efficiency of the members to transform effort into feeling or happiness for the couple. This disparity can mean an asymmetry in the respective levels of efforts to contribute in a successful relationship, which also turn out to be higher than the favorites , as in the case of homogamous couples.

This is what happens in the relationship between Samuel and Sandra: Samuel expresses at one point in the film that things are unbalanced between them, and Sandra replies, convinced that, to begin with, she does not believe in the idea of ​​equal reciprocity in a couple, which she frankly considers depressing. She seems convinced that the couple’s effort levels should not be equal. An idea that is likely the subject of much debate in all relationships.

Who puts more?

Our recent computational models for controlling the dynamics of asymmetrically efficient couples allow us to simulate the evolution of happiness in the relationship, both in less uncertain environments and with different levels of uncertainty . The simulations suggest that the two members must exert themselves asymmetrically, with Sandra right in her reply.

In the film, in a typical scene of negative couple dynamics, Sandra and Samuel reproach each other for the efforts they make or do not make to sustain the relationship. They both have things to say, as in many real couples. Also in some passage of the script it is implied that Samuel has made or makes more effort than Sandra for her relationship with her partner and family. Our analysis shows, perhaps surprisingly, that the partner most efficient at transforming her effort into mutual happiness must exert greater extra effort to sustain the relationship. In the movie, that’s Samuel.

An external impact

The analysis also suggests that when the couple is subjected to a stressful episode due to an unexpected cause, both members must increase their effort gap. But, in addition, the gap between the most efficient should again increase relatively more. Precisely, Sandra and Samuel’s relationship is subject to tremendous misfortune whose impact affects the crux of the story for a long time. That’s why Samuel feels much more stressed than Sandra.

The mathematics offers an outcome in line with the plot of the film: the continuous gap of greater overexertion that the most efficient member must make, increased even more in a prolonged period of sentimental crisis, can lead to a sensation so unbearable that it can only be resolved with the fall of the relationship until it breaks. In the case of the film, also, with Samuel’s fall into the void.

Author Bios: José-Manuel Rey is Professor at the Faculty of Economics and Business Sciences at Complutense University of Madrid and Jorge Herrera de la Cruz is a Hired Professor Doctor Department of Mathematics and Data Science at CEU San Pablo University