A Mathematical Model of Sentimental Dynamics
Accounting for Marital Dissolution
Jose ´-Manuel Rey*
Departamento de Ana ´lisis Econo ´mico, Universidad Complutense, Madrid, Spain
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
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/
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: firstname.lastname@example.org
Sentimental relationships of a romantic nature are typically
considered a fundamental component of a balanced happy life in
western societies . When people are asked what they believe
necessary for happiness they usually give priority to ‘love’ or to ‘a
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  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 , . However, that reason cannot account
for the ongoing and pervasive marital disruption observed in the
last decades . Indeed, it is not understood at this juncture why
so many couples end in divorce while some others do not (see ,
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 , 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
The work by Gottman et al –collected in – 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 . 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
PLoS ONE | www.plosone.org1 March 2010 | Volume 5 | Issue 3 | e9881
disruption occurring massively in sentimental relationships that
were initially planned to last forever. Two forces work together to
ease the appearance of the deterioration process. First, it happens
that since an extra effort must always be put in to sustain a
relationship on the successful path, partners may relax and lower
the effort level if the gap is uncomfortable. Then instability enters
the scene, driving the feeling-effort state out of the lasting
A further significant finding is the fact that partners construct
and perceive their relationships as definitive projects is compatible
with the evidence that their union may probably fall apart –which
is typical in the model dynamics. This dismantles the failure
paradox, accounting for probable couple disruption as a
gravitational consequence of the second law under optimality.
The model analysis may offer advice to partners about how to
keep a long term relationship afloat. Lasting relationships are
possible only if the effort gap is tolerable and the optimal effort
making is continuously watched over to stay on the target
dynamics. A realistic lasting relationship, when the effort gap is
satisfactory, may be described by a trajectory travelling near the
stable branch for a while and then wandering near equilibrium
alert at keeping effort at the right level. These kinds of
relationships are seen often enough although they may appear
exceptional. This is consistent with the exceptionality of durable
successful relationships within the model.
Two apparent facts serve as a first test to validate the theory
proposed in this paper: (i) the model formulation builds on
accepted evidence (namely, the second law and the intention of
couples to design their relationships to last forever) and (ii) the
mathematics of the model shows consistency with further
empirical facts on divorce and separation, namely the typical
progressive deterioration of failing relationships (which is claim #3
in section 2) and the decrease of well-being after marriage (claim
#4 in section 2). Further research to validate the model should
address testing –in a lab experiment or a field survey– the two
main findings of the theory, i.e. the existence of the effort gap and
the unstable nature of feeling-effort dynamics.
The pessimistic conclusions for couple durability should remain
valid in a less ideal scenario as long as the formulation of the
second law is considered valid. More realistic assumptions like
(weak) heterogamy, presence of external shocks or sub-optimal
behaviour, probably enter the scene as contributing factors
enforcing instability. The effort gap plus the unveiled instability
identify an essential intrinsic mechanism for probable sentimental
ical derivations for the analysis in the main manuscript.
Found at: doi:10.1371/journal.pone.0009881.s001 (0.16 MB
Supporting document containing the mathemat-
The author thanks Carmen Carrera for many language style suggestions
after a careful reading of a first version of the manuscript.
This paper is dedicated to the unique long-standing sentimental
equilibrium of Pepe Rey and Ana Simo ´.
Wrote the paper: JMR. Conceived and developed the research: JMR.
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