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Wrote the paper: JMR. Conceived and developed the research: JMR.

The author has declared that no competing interests exist.

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

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.

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.

Sentimental relationships of a romantic nature are typically considered a fundamental component of a balanced happy life in western societies

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

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

The work by Gottman et al –collected in

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

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

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.

Martin and Bumpass

The figures go up when unmarried cohabitations are included, although data sets on cohabitation status are notably difficult to obtain. A recent study

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

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’

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 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.

The available data support this fact. According to 80% of all men and women interviewed in the California Divorce Mediation Project

Although it is accepted that marriage goes with higher levels of happiness than singleness

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

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

As mentioned above, the following assumption is critical for our model.

There is general consensus in the literature about this fact

In order to turn A2 into mathematics, a non-negative variable _{1}, _{2} can be compared according to whether _{1})≥_{2}) or _{1})≤_{2}). At _{0} is assumed very large. We assume the relationship becomes unsatisfactory when _{min} >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

A simple version of the second law can be written in terms of feeling and effort variables as the differential equation

The intensity of

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 considerations. 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

i) Utility from feeling is described by a differentiable function

ii) Disutility of effort

Notice that specific mathematical expressions for

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. Utility defined on feeling is not an unnecessary superstructure: while

The function

The shape of feeling utility

In the dynamic setting of the model,

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

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.

_{0}≫1, and denote the impatience factor by ^{♥}(

Total satisfaction is obtained by aggregating discounted net instantaneous utilities for _{min},), but also the transition to those asymptotic levels must be viable (see below.)

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 ^{♥}(^{♥}(

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.

Equilibria are characterized by setting time derivatives equal to zero in (1)–(2). Under the specifications of the model, it is proved (_{s}^{♥},_{s}^{♥}), which is depicted in

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 _{s}^{♥}>_{min} and the effort gap _{s}^{♥}−_{H}_{H}

This is an admissible solution provided that _{s}^{♥} lies above _{min}. A crucial finding of the analysis is that the stationary effort level _{s}^{♥} lies above _{s}^{♥}. Since that level is the unique solution of _{s}^{♥},_{s}^{♥}), maximal well-being is achieved. However, the existence of the effort gap is a possible source of non-viability for the equilibrium solution.

A fundamental issue is whether or not perturbations will vanish or expand as time passes. If perturbations are amplified, the system is unstable. While stability contributes to a solid long life for the relationship, instability may be a serious drawback. In the unstable case, small shocks –typically due to lowering effort– will drive the feeling-effort configuration far from the equilibrium state. With no intervention, the final fate of the perturbed configurations is the dismantling of the relationship. This will be made clear in the analysis of the global sentimental dynamics. It is proved that the sentimental equilibrium defined by (1)–(2) is unstable (see

The initial state of the relationship is not generally placed at the equilibrium point because the initial feeling for each other is typically much higher than the stationary level _{s}^{♥}. Therefore the discussion must proceed by looking at the dynamics (1)–(2) for an initial feeling _{0}≫_{s}^{♥}. We need to look at the global configuration of the phase space to explain the transitory dynamics towards equilibrium.

Under the assumptions of the model, there is a unique effort policy that takes the initial feeling _{0} to the unique equilibrium E. This is achieved by setting the initial effort at point A to get onto the stable manifold _{s}^{+} and then following path AE to approach equilibrium. Trajectories starting above _{s}^{+} (e.g. at point A′) are not acceptable. The target trajectory AE always lies above the line ^{♥}(_{s}^{♥}>_{min}. Furthermore, since the target trajectory AE is unstable, trajectories starting at lower effort levels (e.g. at point A″) depart from AE and eventually lead to abandon effort (setting

The stable and unstable manifolds –composed of points (_{s} and _{u}^{♥}(

The key issue is whether or not, given an initial feeling _{0}, there exists an effort policy that leads to equilibrium and if it does what it is that characterizes the effort strategy. The stable manifold is the only curve supporting trajectories leading to equilibrium. Any other trajectory is either non acceptable or corresponds to a non-lasting relationship. Indeed, trajectories lying in regions I and II above _{s} (see _{0}, there is a suitable level _{0}^{♥} for which A = (_{0},_{0}^{♥}) lies in _{s}^{+} and evolves towards E (_{s}^{♥} is greater than _{min}. Since the target trajectory AE embedded in _{s}^{+} lies entirely above the line

Since the target path lies in region I, ^{♥}(^{♥}(

As explained above, typical dynamics occur within the shaded region in _{0} is large and trajectories leading to increasing levels of effort are not plausible. Along trajectories in the shaded area, the effort eventually decreases until the ^{♥}(_{s}^{+} leads the state of the system into the shaded region, where optimal trajectories diverge from the target curve. This critical feature is the main source of sentimental instability.

A possible mechanism –via _{s}^{+}. If at a certain point effort inattention occurs, that is if the effort level is lowered, the state is driven out of _{s}^{+}. If effort is not returned to the correct level and if the system follows the optimal dynamics (1)–(2), the new deviated state finds itself at an initial condition of a trajectory moving away from the target trajectory. This new trajectory may be followed for a while until new effort inattention occurs, expelling the state to a new position with lower effort level, in turn following a new decaying trajectory moving further away from _{s}^{+}. Through a sequence of effort inattentions, instability causes the decaying trajectories to cross the threshold level _{min} (

The model produces a plausible scenario, through a sequence of effort inattentions, for the deterioration of a relationship in a gradual form, which seems to be typical according to data. Because of the effort gap, there is a tendency to lower the right effort level. Then the intrinsic instability of sentimental dynamics obeying the second law causes the piecewise decaying trajectories to move further and further away from the target trajectory and eventually to cross the threshold level _{min}. This is considered a point of

If the system is following a decaying trajectory, the target path dynamics can be restored by increasing the effort level. However, the longer it takes to react and correct deviations, the farther the state is from the target path, and the more difficult it is to restore the system to the lasting path. If effort is neglected for too long, it may become irreversible. A considerable amount of reported unhappy marriages seem to fit this diagnosis

The mathematical theory introduced in this paper unveils an underlying mechanism that may explain the deterioration and 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 successful dynamics.

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 failure.

Supporting document containing the mathematical derivations for the analysis in the main manuscript.

(0.16 MB DOC)

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 Simó.

^{nd}reprint. North-Holland