# Get PDF Its Marriage, Do The Math!

The study began a decade ago in Seattle, where Prof. Murray and psychologist John Gottman examined couples who were planning to get married or had recently wed.

## The Math of Marriage

While the couples talked with each other for 15 minutes, every piece of their conversation was given a score. Jokes, smiles and affectionate gestures earned points; rolled eyes, criticism, and mocking subtracted them. Murray told British Broadcasting Corp. It quantifies one person's effect on the other, and it is not difficult.

It's the largest in a series of similar studies that the Seattle researchers have been publishing since the late s, and the results have divided psychologists and therapists. Some say it's revolutionary, while others contend that humans are too complex for such formulas to ever become a useful counselling tool. Gottman, the lead psychologist on the project.

Governments should consider making this kind of test mandatory before granting marriage licenses, Dr. Guerney said, as a way of mitigating the high economic and social costs of divorce. But others contend that the formulas have never been field-tested, suggesting that the studies so far lack validity because the couples involved volunteered to be analyzed. Sollee said other psychologists have also complained the tests look only at a couple's communication skills, and ignore other factors that affect divorce rates. This is a space where subscribers can engage with each other and Globe staff.

Non-subscribers can read and sort comments but will not be able to engage with them in any way. Click here to subscribe. Ronnie brings a refreshing, straightforward and sensitive approach to working with couples. She is known for her humor.

## Math predicts marriage outcome - The Globe and Mail

In her spare time, Ronnie loves hanging out with her husband of 24 years, and enjoys traveling, writing, and trying to keep up with her four children! The Psychology of Toxic Relationships. Every relationship takes work, even the healthiest one, so how do we know if our problems are just a normal phase or a sign of a toxic relationship? Romantic relationships are important for our happiness and well-being. Relationships seem to be more and more transitory and complex. Couples considering marriage would often like to access their marriage compatibility.

Being compatible certainly increases the chance of having a Each man knows exactly k women and each woman knows exactly k men.

Is it possible to marry off everyone so that in each resulting couple the man and woman know each other? Perhaps only a mathematician would use the term marriage for such a situation! Yet, the theorem that gives insight into this situation is as wondrous as the experiences that lead to marriage in the real world. Here is a mathematical model which tries to frame the problem above, though it does not quite take the path followed by Hall about which more later.

Suppose we represent the men by a collection of dots, and the women by a different collection of dots, as shown in Figure 1 below. Note that, for the moment, we need to have the same number of men and women. In the diagram shown, each man is connected to 3 women and each woman is connected to three men, there being five men top set of dots and five women bottom set of dots.

A more general situation occurs if we have equal numbers of men and women but we do not necessarily have the same number of women known by each man and by each woman. Can you see why it is impossible to marry off all the men to women they know in the diagram above? Look at the men at the far left and the far right!

By contrast, in the diagram below, all the men can be married off to women they know. Can you see how to pair off the men and women so that they know each other? Not a total triviality. Here is a solution: So, we have several issues to address here: When can one marry off the men to the women, and is there an algorithm, a simple set of rules, that can be followed, that will find such a pairing when one exists in a reasonable amount of time, even for large numbers of men and women?

The ideas that lie behind this marriage theorem were discovered independently in several different contexts. Hall had come across his theorem in as a result about cosets in group theory ; in recent years it is typically stated as a theorem about sets. The basic idea is to define a system of distinct representatives SDR.

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Think of having a list of committees which have potentially overlapping membership. The goal is to elect a different chairperson for each committee. Abstracting, we have a collection of unordered sets of objects whose members have been chosen from some list M of objects. The connection with marriage theorems is that one can think of the sets as being the names that each woman lists for the men who are the elements of the set M, which explains the use of the letter M whom she would accept as a mate.

Assigning each woman a unique man means finding an SDR for the women's sets. Hall's Theorem states that an SDR exists if and only if for any number i of the women for i equal to one up to the total number of women the lumping together of the men on the lists of the i women contains at least i men. If, for example, 5 of the women only have 4 men on their lists of potential mates, no marriage assignment is possible. The statement of Hall's Theorem , though very elegant, does not lead immediately to finding an SDR in an efficient way, it only gives a test to see if such an SDR exists.

The role of an SDR in the set formulation is related to the notion of a matching in a graph. In this setting, finding an SDR becomes more practical. A matching in a graph is a collection of edges line segments which have no vertex in common. Given a graph one can look for matchings which have the property that no edge can be added to the matching. Alternatively, one can look for a matching with the largest possible number of edges. A graph is said to have a perfect matching when every vertex of the graph is the endpoint of some edge in the matching. A necessary condition for this to happen is that the graph have an even number of vertices.

However, graphs which have an even number of vertices need not have a perfect matching.

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The red lines in Figure 2 above represent a perfect matching. Interpreting the vertices as women and men shows how to marry off the women to men they know. Extensions of the idea of matchings allow the graph to have weights assigned to the edges. For example, one might have a collection of people and jobs for which these individuals are qualified.

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If person i is qualified for job j there is an edge between the vertices representing i and j. This edge can be assigned a weight which indicates: The amount of time that it will take i to complete j , or b. The efficiency with which i will be able to carry out j , or c. The cost of completing j if i does j. This leads to problems of matching workers to jobs to maximize something perhaps the efficiency with which all the jobs can be completed or minimize something the cost of completing the jobs with a particular assignment of workers.

Researchers have been successful in obtaining many theoretical and applied results for problems of this sort, including, in many cases, polynomial time i. Beyond the Hall Marriage Theorem The Hall Marriage theorem aims to examine when it is possible to marry a collection of men to a collection of women who know each other. However, one can imagine that this might not be a very satisfactory situation because the people who are paired are not happy with the partners that they are assigned. A natural way to try to do better is to ask the men how they rank the women and the women how they rank the men.

It is this straightforward idea that leads to a remarkable story of the interplay of theoretical and applicable ideas. In fact, David Gale and Lloyd Shapley were not the first to either think of the model or apply it to a practical situation. None the less Gale and Shapley showed that there was a truly remarkable mathematical theorem that governed what happened in this situation. Let us consider the simplest version of the Gale-Shapley framework, which has been extended in many directions, some of which I will mention.

Suppose that we have two sets of equal size: Each man produces a ranking for each woman where indifference is not allowed. Each woman produces a ranking for each man where indifference is not allowed. For example, if the men's names are Man 1, Man 2, and Man 3; and the women's names are Woman 1, Woman 2, and Woman 3, then the rankings produced might look like: Men's rankings of the women: For example, here are some potential pairings: Woman 1, however, finds herself paired with her least favorite.

Man 2 in this arrangement gets his second choice, he preferring Woman 1, and we have already noted that Woman 1 would prefer any of the other men to the one she was assigned.

Thus, this arrangement tempts Man 2 and Woman 1 to "have an affair" because each prefers the other to the mate they were assigned. If a pairing of men and women has no blocking pair, then the assignment or matching is called stable. Unfortunately, Arrangement 2 is also unstable. Man 1 and Woman 2 would prefer each other to the mates they have been assigned. Note that it may seem to be hard to find a stable matching, but given a proposed matching we can check for a given man whether or not he is a member of a blocking pair.

Take Arrangement 2 above, and Man 2. Is he a member of a blocking pair? Since he was assigned Woman 1, his first choice, he can not be a member of any blocking pair. What about Man 3? He is mated with Woman 2, his second choice. Hence, we need only check if he forms a blocking pair with the woman he ranked first, namely, Woman 3.

Since Woman 3 prefers Man 3 to the person she was paired with Man 1 , the two form a blocking pair. In this example we have 6 potential ways to marry off the men to women and we can check them all to see what happens.

Can you find an assignment where the stability condition holds? The obvious question is whether for any n at least 2 , given arbitrary ranking tables for n men and n women, there is a way of marrying the men to the women in such a way that the assignment is stable.