Martin and Bumpass 
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 
. 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% 
The figures go up when unmarried cohabitations are included, although data sets on cohabitation status are notably difficult to obtain. A recent study 
confirmed that non-marital cohabitations 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 
. 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’ 
. In a national survey in the US 
, 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 
, the major reason given for their divorce was the ‘gradually growing apart and losing a sense of closeness, maybe 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 
, 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 
–see also 
. The pattern they find implies that, after marriage, the average reported satisfaction with life decreases (see figure number 2 in 
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 
. Ample evidence in western societies supports this fact 
. 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 
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 
. There seems to be a natural law that unattended love erodes as time goes by. Jacobson and Margolin 
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 
(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
) 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
) serves as an ordinal variable probing the qualitative level of the relationship. Specific values of x
) are uninformative, but the sentiment level at different times t1
can be compared according to whether x
) or x
). At t
0 the common feeling x
is assumed very large. We assume the relationship becomes unsatisfactory when x
) 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
) –called the effort variable
– assumed piecewise continuous (see Appendix S1
about this). The scope of c
) 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 shortcomings, giving her/him privacy, respecting differences in tastes and habits), to name only a few among the recommended practices 
. The importance of effort/sacrifice, either passive or active, and its benefits on the relationship persistence have been widely recognized in the literature (see 
for a review.)
A simple version of the second law can be written in terms of feeling and effort variables as the differential equation
>0 and a
>0. Without intervention (i.e. c
0), Eq. (1) implies that x
) 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 
to describe the baseline evolution of uninfluenced partner behaviour in short-term marital interaction. At any rate, Eq. (1) with c
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
) determines the evolution of the feeling by solving Eq. (1) for x
). Eq. (1) implicitly entails that x
) changes smoothly, except at effort discontinuities.
The intensity of c
) can be decided by the partners involved, in contrast to the level of the (non-rational) variable x
), that cannot. The rational nature of the effort variable c
) allows one to interpret it as a control variable
in the scenario of optimal control theory 
. In this setting, the controlled variable –the state variable
– is x
) 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 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 
and references therein.) Empirical research on sacrifice and related practices has evidenced that effort making may entail both emotional cost and benefits. This apparent contradiction is reconciled in 
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 interpretation 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.
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. Utility defined on feeling is not an unnecessary 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 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 (−D) can thus be thought of as utility. The typical graphs of both functions are represented in .
Utility structure: typical shapes of utility and disutility functions.
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
1, and denote the impatience factor by ρ
>0. Under these conditions find the effort plan c
)≥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
(the exponential term accounts for the discounted valuation of future utilities.) Problem (P)
is a standard infinite horizon optimal control problem 
. 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.)