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A Mathematical Model of Sentimental Dynamics

Accounting for Marital Dissolution

Jose ´-Manuel Rey*

Departamento de Ana ´lisis Econo ´mico, Universidad Complutense, Madrid, Spain

Abstract

Background: Marital dissolution is ubiquitous in western societies. It poses major scientific and sociological problems both

in theoretical and therapeutic terms. Scholars and therapists agree on the existence of a sort of second law of

thermodynamics for sentimental relationships. Effort is required to sustain them. Love is not enough.

Methodology/Principal Findings: Building on a simple version of the second law we use optimal control theory as a novel

approach to model sentimental dynamics. Our analysis is consistent with sociological data. We show that, when both

partners have similar emotional attributes, there is an optimal effort policy yielding a durable happy union. This policy is

prey to structural destabilization resulting from a combination of two factors: there is an effort gap because the optimal

policy always entails discomfort and there is a tendency to lower effort to non-sustaining levels due to the instability of the

dynamics.

Conclusions/Significance: These mathematical facts implied by the model unveil an underlying mechanism that may

explain couple disruption in real scenarios. Within this framework the apparent paradox that a union consistently planned

to last forever will probably break up is explained as a mechanistic consequence of the second law.

Citation: Rey J-M (2010) A Mathematical Model of Sentimental Dynamics Accounting for Marital Dissolution. PLoS ONE 5(3): e9881. doi:10.1371/

journal.pone.0009881

Editor: Jeremy Miles, RAND Corporation, United States of America

Received September 17, 2009; Accepted February 14, 2010; Published March 31, 2010

Copyright: ? 2010 Jose ´-Manuel Rey. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This work was partially supported by Ministerio de Ciencia e Innovacio ´n (Spain) through MTM2006-02372 and MTM2009-12672. The funders had no

role in research design, decision to publish, or preparation of the manuscript.

Competing Interests: The author has declared that no competing interests exist.

* E-mail: j-man@ccee.ucm.es

Introduction

Sentimental relationships of a romantic nature are typically

considered a fundamental component of a balanced happy life in

western societies [1]. When people are asked what they believe

necessary for happiness they usually give priority to ‘love’ or to ‘a

closerelationship’[2],[3],[4].Itishardtothinkofanotheraspectof

human life involving so many cultural, sociological, psychological or

economic issues. Whereas the initial stage of romantic relationships

seems to be controlled by chemical processes (see [5] and references

therein), the issue of maintaining a sentimental relationship may

rather belong in the realm of rational decisions. People usually

engage in long-term relationships –typically marriage– only after

due consideration. Even in the prevalent western scenario of

sequential monogamy, couples generally assert their intention to

make their relationship last and be happy together (see data

reported in section 2). But the high divorce rates massively reported

across Europe and in the United States show a resounding failure in

their program implementation. The phenomenon of couple

disruption is considered epidemic in the US where the statistic

‘oneintwo couples end indivorce’isquotedrepeatedlyinthe media

and in academic reports. The average rate in EU27 is not far below

that figure and some countries in Europe show higher rates of

divorce. Furthermore, data on unmarried couples tell an even worse

tale of sentimental break ups (see section 2.)

There is general agreement among scholars from different fields

on mainly attributing the rise in marital instability in the twentieth

century to the economic forces unleashed by the change in sexual

division of labour [6], [7]. However, that reason cannot account

for the ongoing and pervasive marital disruption observed in the

last decades [8]. Indeed, it is not understood at this juncture why

so many couples end in divorce while some others do not (see [9],

pg. xi). That understanding is of paramount importance since the

social change induced by marital disruption deeply affects the

social structure of contemporary western societies as well as the

well being of their members.

The fact that, for most couples, both partners plan enduring

relationships and commit to work for them, poses a contradiction

with the reportedly high divorce rates. This contradiction is

referred to in this article as the failure paradox. According to

Gottman et al [9], the field of marriage research is in desperate

need of (a mathematical) theory. This paper aims to alleviate the

need. In particular, it offers a consistent explanation for the failure

paradox.

The work by Gottman et al –collected in [9]– seems to be the

only mathematical contribution to the study of couple relation-

ships so far. They used a pair of nonlinear difference equations

estimated from the short-term interaction between two partners

when observed in the lab. A simple dynamical system modelling

for couple interaction was first suggested by Strogatz [10]. We

adopt here a different dynamical approach: the couple is taken as a

unit –no inside interaction is considered– and their sentimental

dynamics is rationally prescribed by their intention to be happy

together forever.

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In view of the ubiquity of the phenomenon of couple break-up,

it seems sensible to look beyond specific flaws in relationships and

search instead for an underlying basic deterministic mechanism

accounting for break-ups. Building on sociological data, we

propose a mathematical model based on optimal control theory

accounting for the rational planning by a homogamous couple of a

long term relationship. A couple is said to be homogamous when

the individual partners have similar characteristics. Homogamous

mating is the most common type of sentimental partnership in

western societies [11], [12]. Our model actually requires a weak

form of homogamy (see section 2). We describe the evolution of

this form of relationship by a dynamical equation based on the

second thermodynamic law for sentimental interaction (second law for the

sequel), as it has been called by Gottman et al [9]. The second law

asserts that a sentimental relationship will deteriorate unless

‘energy’ is fed into it. This generally accepted fact allows us to

model sentimental relationships as a control problem, with energy

in the form of effort playing the role of the control variable.

Optimal control theory has been used extensively in applied

sciences, e.g. in engineering or economics. Our optimal control

modelling brings a novel mathematical approach to the analysis of

marriage and close relationships.

Given some feasibility conditions, our analysis of the model

shows that long-term successful relationships are possible and

correspond to equilibrium paths of the dynamics. While it may

appear obvious that long-term relationships are not possible

without some effort, a remarkable finding of the model is that the

level of effort which keeps a happy relationship going is always

greater than the effort level that would be chosen optimally a

priori (if only the present counted.) Relationships are viable

provided that the effort gap between the two levels is tolerable. The

main result of the mathematical analysis is that sentimental

dynamics subject to the second law are intrinsically unstable. This

implies that when effort is relaxed, gradual sentimental deterio-

ration may easily occur. The analysis identifies a plausible

mechanism accounting for progressive degradation leading either

to rupture or to unsatisfactory sentimental lives.

The results in the paper contribute to the resolution of the

failure paradox: under the second law, the optimal design of a

durable happy relationship is compatible with its dynamic

instability and in turn with its probable break-up. This striking

finding dismantles the failure paradox, since real relationships are

expected to be subject to further sources of instability and

uncertainty. Also, the results may indicate how to keep a long term

relationship alive and well.

In section 2 key evidences supported by sociological data are

presented that will serve as a framework to test the consistency of

the model findings. The issue of the failure paradox is derived here

from sociological evidence. The elements of the model are

introduced along with a thorough discussion of the underlying

assumptions. The main predictions of the model analysis are

gathered in section 3 and some of them are shown to be consistent

with facts presented in section 2. Aiming at a more fluent

discussion, the mathematical technicalities are relegated to an

appendix.

Methods

Stylized Facts

Martin and Bumpass [13] used 1985 data to show that, within a

span of 40 years, two out of three marriages in the US will end in

separation or divorce. This proportion may not have been reached

yet but the data for 2002 show that we are not far below. About

50% of people in their early forties have already divorced at least

once [14]. The much publicized figure of 50% turns out to be only

slightly higher than the average divorce rate (44%) in the EU27 in

2005, and in some European countries this proportion is as high as

71% [15].

The figures go up when unmarried cohabitations are included,

although data sets on cohabitation status are notably difficult to

obtain. A recent study [16] confirmed that non-marital cohabi-

tations are overall less stable than marriages. They report that

49% of premarital cohabitations break up within 5 years (62%

after 10 years), whereas 20% of marriages end up in separation or

divorce within 5 years (33% after 10 years). A first stylized fact of

the phenomenon we are looking at may thus be formulated as

follows:

Claim #1: There is an epidemic failure in love relationships.

This notorious instability of sentimental relationships is not

correlated with a significant loss of belief in the formulae of

marriage or cohabitation as the main ingredient for happiness. On

the contrary, people massively declare that a satisfactory

sentimental relationship is the first element on which to build a

happy life [1]. Moreover they also claim to want their partner to

last them for life:

Claim #2: Couples typically conceive a relationship that lasts

to be the main element in their pursuit of happiness.

Moreover, most of them think that their own relationship

will not collapse.

The available data supports claim #2. When asked to select the

item that would make them happiest, 78% of college students in

the US picked the one called: ‘falling and staying in love with your

ideal mate’ [17]. In a national survey in the US [18], 93.9% of

interviewed married couples thought their chances of a divorce or

separation low (19.9%) or very low (74%), while 81.1% of

unmarried respondents answered in the same way (32.4% low

versus 47.7% very low).

It is intriguing that, in spite of the acknowledged high

probability of breaking up, the vast majority of people think that

their own relationship will not break down. Indeed, claims #1 and

#2 together pose an apparent paradox. According to the data

quoted above, a newly formed couple claims to be 90% certain

that its own relationship will last. However the chances of breaking

up after 5 years of cohabitation are 50%; and after 10 years it is

definitely more probable than not that they will not be staying

together. This fact could be stated as follows:

The failure paradox: how is it that a sentimental relationship

planned to last will very probably break down?

The model proposed below shows that, under plausible

assumptions, claims #1 and #2 are compatible. In order to test

further the consistency of the model, we will consider two more

stylized facts.

Claim #3: Couple disruption is the outcome of a gradual

deterioration process.

The available data support this fact. According to 80% of all

men and women interviewed in the California Divorce Mediation

Project [19], the major reason given for their divorce was the

‘gradually growing apart and losing a sense of closeness, maybe

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staying together but emotionally detached until their loneliness is

not longer bearable’.

Claim #4: The subjective well-being of partners decreases

after marriage.

Although it is accepted that marriage goes with higher levels of

happiness than singleness [1], [20], the average self-perceived

satisfaction with life among those married is reported to peak

around the time of marriage. This fact is supported by recent

findings [21] –see also [22]. The pattern they find implies that,

after marriage, the average reported satisfaction with life decreases

(see figure number 2 in [21]).

The Model

A simple dynamical model is formulated next that accounts for

the scenario described above.

The core of the model lies in two key assumptions, namely the

second law –to be discussed in A2 below– and the long-term

planning of a couple’s relationship –plausibly sustained by claim

#2 above. These assumptions –along with weak homogamy (see

A1 below) and a natural cost–benefit evaluation of the relationship

state (assumption A3 below) –permit us to see the couple’s

sentimental relationship as an optimal control problem.

Modelling starts (time t=0) when the romantic period is over

and the feelings of partners about their relationship are at their

peak (probably at the moment of commitment). At the initial time,

the two partners, having an intense feeling for one another, agree

on becoming a couple and undertake to do whatever is required to

ensure a long future together. We assume:

A1 (Weak Homogamy) Both partners share the same traits

according to the model specifications below. Equivalently,

the couple is the decision unit for the planning problem.

This assumption implies that the parameters, variables and

utility structure defined in the model will all refer to the couple, as

formed by two similar individuals. The fact that most people tend

to feel attracted to individuals sharing the same traits they

themselves posses has long been recognized in the literature [5],

[11], [23], [24], [25]. Ample evidence in western societies supports

this fact [12]. Thus assumption A1 stands as the rule, rather than

the exception. In strict terms our theory only requires similarity in

emotion rather than in personality between partners (see A3

below) although the two are shown to go together in dating and

married couples [25].

As mentioned above, the following assumption is critical for our

model.

A2 (Second law of thermodynamics for sentimental relationships.)

There is tendency for the initial feeling for one another to

fade away. This kind of inertia must be counteracted by

conscious practices.

There is general consensus in the literature about this fact [5],

[9], [26], [27]. There seems to be a natural law that unattended

love erodes as time goes by. Jacobson and Margolin [26] identified

this fact as a major cause for marital instability. They write:

‘Marriages start off happy, but over time reinforcement erosion

occurs that is the source of marital dysfunction’. The popular

motto ‘love is not enough’ reflects this fact and implicitly suggests

that erosion can be prevented somehow. The formulation of A2 as

a law is taken from Gottman et al [9] (page 143), where the

sentimental wearing out is suggestively explained as ‘something

like a second law of thermodynamics for marital relationships:

things fall apart unless energy is supplied to keep the relationship

alive and well.’

In order to turn A2 into mathematics, a non-negative variable

x(t) is defined to represent the state of the relationship at time t$0.

This is the feeling variable and it can be understood as the (common)

sentiment that the partners have about one another. The variable

x(t) serves as an ordinal variable probing the qualitative level of the

relationship. Specific values of x(t) are uninformative, but the

sentiment level at different times t1, t2can be compared according

to whether x(t1)$x(t2) or x(t1)#x(t2). At t=0 the common feeling

x(0)=x0 is assumed very large. We assume the relationship

becomes unsatisfactory when x(t) falls below a certain threshold

value xmin.0, which varies with the couple in question.

According to A2, the fading inertia can be counteracted by

working on the relationship. This working will be represented by a

non-negative and ordinal variable c(t) –called the effort variable–

assumed piecewise continuous (see Appendix S1 about this). The

scope of c(t) includes any everyday life practice serving as a

reinforcement for the relationship. For instance, therapist suggest

constructive actions (asking questions, listening actively, making

plans together), and tolerant attitudes (accepting partners short-

comings, giving her/him privacy, respecting differences in tastes

and habits), to name only a few among the recommended

practices [5], [27]. The importance of effort/sacrifice, either

passive or active, and its benefits on the relationship persistence

have been widely recognized in the literature (see [28] for a

review.)

A simple version of the second law can be written in terms of

feeling and effort variables as the differential equation

dx

dt(t)~{rx(t)zac(t), for t§0,

ð1Þ

with r.0 and a.0. Without intervention (i.e. c(t)=0), Eq. (1)

implies that x(t) fades at a constant rate r, specific to each

relationship, which is a measure of the strength of feeling fading.

This simple linear law is well-known to steer many natural and

social phenomena. In fact, its discrete version was used in [9] to

describe the baseline evolution of uninfluenced partner behaviour

in short-term marital interaction. At any rate, Eq. (1) with c(t)=0 is

the first obvious working hypothesis for the decaying law of feeling.

Effort enters as a recovery term in Eq. (1) counteracting the

weakening of feeling. The parameter a obviously indicates effort

efficiency. Selecting an effort plan c(t) determines the evolution of the

feeling by solving Eq. (1) for x(t). Eq. (1) implicitly entails that x(t)

changes smoothly, except at effort discontinuities.

The intensity of c(t) can be decided by the partners involved, in

contrast to the level of the (non-rational) variable x(t), that cannot.

The rational nature of the effort variable c(t) allows one to interpret

it as a control variable in the scenario of optimal control theory [29].

In this setting, the controlled variable –the state variable– is x(t) and

Eq. (1) is the state equation linking both variables.

Our next and last assumption refers to the cost-benefit valuation

of effort and feeling levels. A standard utilitarian approach is

considered. A mathematical representation of the emotional

evaluation of feeling is rather straightforward (see A3 below).

However formalization of effort valuation requires some consid-

erations. The typical form of effort is sacrifice –forgetting one’s

self–interest for the sake of a close relationship–, whose potential

benefits and costs have repeatedly been considered in the literature

(see [30] and references therein.) Empirical research on sacrifice

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and related practices has evidenced that effort making may entail

both emotional cost and benefits. This apparent contradiction is

reconciled in [30] by means of a motivational analysis of sacrifice

based on attitudes of approach and avoidance. While seeking to

please one’s partner wishes may lead to positive emotions,

avoiding conflict may induce tension and distress. Our interpre-

tation of the emotional differences in effort making is related to the

intensity of effort since we consider effort to be emotionally

rewarding up to a certain level but costly (distressing) beyond then.

This is formalized as follows.

A3 (Utility structure) There are two independent sources of

utility. One comes from the level of feeling of attachment

and the other is the consequence of the intensity of effort.

i)

function U(x) such that U’(x)w0, U’’(x)v0, and

U’(x)?0 as xR‘. In words, for any feeling level, its

marginal utility is positive and decreasing but it vanishes

when feeling is large.

Disutility of effort c$0 is given by a differentiable

function D(c) satisfying D’’(c)w0, D’(c?)~0 for some

c*$0, and D’(c)?? as cR‘. That is, effort dissatisfac-

tion reaches its absolute minimum level at c* and

marginal dissatisfaction goes up without bound as the

effort level increases.

Utility from feeling is described by a differentiable

ii)

Notice that specific mathematical expressions for U and D are

not required. The theory is valid for general functions as long as

they satisfy the qualitative properties above.

The term utility may be interchanged with happiness, well-

being or life satisfaction. The assumptions in part i) above are

standard when utility depends on the consumption of some

good.Utilitydefinedonfeeling

superstructure: while x (how one feels) is directly linked to the

(unprocessed) sentiment towards the relationship, U(x) produces

a valuation of the feeling level x based on individual judgement

and probably depends on past experiences or personality traits.

For example, two different couples may attach quite different

values to similar feeling levels, so that their valuations will

isnotan unnecessary

be represented by different utility functions. The assumption

on the existence of a utility function of feeling can be argued

to be as sensible as it is in the case of utility dependent on

consumption.

The function D represents disutility, on the basis that making

extra effort entails a cost in terms of utility. Its negative (2D) can

thus be thought of as utility. The typical graphs of both functions

are represented in Figure 1.

In the dynamic setting of the model, U and D mean to measure

instantaneous utility and disutility, that is, of current levels of

feeling and effort. The assumption that D may be non-monotonic

leaves room for the fact that effort making may be felt as

rewarding on its own within a certain range of low levels. To

illustrate this, think of planning some recreational activity with

your partner: it entails low effort and may certainly be enjoyable

rather than distressing. Although future benefits of (current) effort

making are implicitly taken into account via feeling utility –since

current effort serves to enhance future feeling through equation

(1)– the current benefits of effort making would not be admitted if

D is always non-decreasing.

While making a small effort may plausibly be pleasant if the

effort level is low, it is surely emotionally costly for sufficiently high

effort levels. It is thus assumed in A3ii) above that making an

additional effort increases utility until a level c* is reached, but

decreases utility when the effort level goes beyond c*. The

parameter c* thus corresponds to the a priori preferred effort level

for the couple, and it plays a key role in the analysis. The theory

admits D monotonic as a particular case, when c*=0. This is the

situation in which (current) effort generates (current) dissatisfaction

from the very first effort unit. The proposed structure for D

permits a more plausible situation.

The problem for a couple is how to design an effort policy that

guarantees their relationship will endure and provide both

partners with as much satisfaction as possible. The effort evolution

is thus determined using an ideal criterion of pursuing maximal

happiness. This is an optimality problem that can be formulated as

follows.

(P) The effort control problem for sentimental dynamics: Assume

feeling evolution given by Eq. (1), a utility structure as

described in A3, initial feeling level x(0)=x0&1, and denote

Figure 1. Utility structure: typical shapes of utility and disutility functions. The shape of feeling utility U is the standard picture assumed in

the social sciences. Utility from effort, represented by 2D, increases till it reaches c* but decreases beyond this point. Marginal effort utility is

decreasing and vanishes at c*. Thus 2D is concave in shape reaching an absolute maximum at c*.

doi:10.1371/journal.pone.0009881.g001

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the impatience factor by r.0. Under these conditions

find the effort plan c#(t)$0, for t$0, that maximizes

total discounted net utility and such that the associated

evolution of both feeling and effort are sustainable in the

long run.

Total satisfaction is obtained by aggregating discounted net

instantaneous utilities for t$0, which can be expressed –in a

standard way– as

W~

ð?

0

e{rtU(x(t)){D(c(t))

ðÞdt:

(the exponential term accounts for the discounted valuation of

future utilities.) Problem (P) is a standard infinite horizon optimal

control problem [29]. Because of claim #2, the planning period of

the problem is considered unbounded. The issue of sustainability,

a key requirement in the couple’s problem, is concerned with two

issues: admissibility and viability. Not only long term levels of both

feeling and effort must be admissible (i.e. feeling must be kept

above xmin,), but also the transition to those asymptotic levels must

be viable (see below.)

Results and Discussion

The main implications of the model are derived and discussed

next. Remarkably, the empirical evidence stated as claims #3 and

#4 are derived theoretically from the model analysis. Also, claims

#1 and #2 are shown to be compatible within the model

framework, which somehow solves the failure paradox. The

mathematical details of the analysis are placed in Appendix S1.

The optimal (when positive) effort at time t must satisfy:

dc

dt(t)~

1

D’’(c)(rzr)D’(c){aU’(x)

ðÞ,

t§0:

ð2Þ

Equation (2) gives the law of variation for optimal effort. Equations

(1) and (2) form a system of differential equations for the optimal

levels of feeling cum effort trajectories. These are denoted by

(x#(t),c#(t)).

Sentimental equilibrium

Stationary solutions of (1)–(2), if viable, guarantee a sustained

happy sentimental life that is achieved on the basis of an invariant

effort routine. Enjoying a permanent rewarding feeling, without

turbulences in effort making, is obviously an attractive feature of a

lasting sentimental dynamics. This makes equilibrium the desired

configuration for a long term relationship.

Existence and viability.

Equilibria are characterized by

setting time derivatives equal to zero in (1)–(2). Under the

specifications of the model, it is proved (Appendix S1) that there

exists a unique well-defined sentimental equilibrium E=(xs

which is depicted in Figure 2.

This is an admissible solution provided that xs

crucial finding of the analysis is that the stationary effort level cs

lies above c* (see Appendix S1), as shown in Figure 2. This has the

important implication that the extra effort c#

sustain the relationship dynamics in equilibrium. An equilibrium

solution is viable provided the effort gap c#

too costly by the couple. A relationship is in equilibrium when

#,cs

#),

#lies above xmin. A

#

s{c?w0 is needed to

s{c?w0 is not seen as

Figure 2. Sentimental equilibrium. Under the specifications of the model, there is always a unique feeling-effort equilibrium E of the optimal

sentimental flow defined by Eqs. (1)–(2). This is a viable solution if xs

sentimental flow is vertical, that isdx

and its graph is always decreasing and located above the line c=c*. The graph represented above corresponds to the case that U9(0),+‘, in turn

implying cH(0),+‘.

doi:10.1371/journal.pone.0009881.g002

#.xminand the effort gap cs

#2c* is tolerable. The vertical nullcline (where the

dt~0) is the line ac=rx The horizontal nullcline is the curve cH(x) where the sentimental dynamics is flat, i.e.dc

dt~0,

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