On Fights and Break-ups Between Couples

Abstract

We present a new economic model of romantic relationships by theorizing breakup as an intertemporal process, unlike the traditional static approach in terms of outside options. Contrary to Becker (2009), our approach underlines the possibility of divorces even when outside options are less than gains from marriage. This allows us to provide a rationale for the anti-Benthamite legally mandated separation periods during divorce proceedings, within the Benthamite paradigm. Further, we provide a rational explanation for the curious tendency of partners of being in long periods of acrimonious fight before making up by undertaking actions that they could have undertaken right at the start of their fight. Finally, in line with a UK government report (1989), our paper posits that unions without legal recognition are likely to be more unstable in comparison to legally recognized marriages.

JEL codes: K36, C72,D90

Keywords

Divorce Legal separation Bentham intertemporal fights

1 Introduction

Sacred scriptures of most religions in the world have invariably frowned upon break up of union between man and woman. However, it is only the classical Islamic law which decrees that divorces should only be granted after disagreeing partners spend a period of separation which is typically three or four months from the first pronouncement of repudiation (Awad & Mawla, 2013).¹

1
Those who swear to keep away from wives (with intent to divorce) have four months of grace; then they reconcile (during this period), surely God is forgiving and kind. And if they are bent on divorce, God hears all and knows everything. (Qur�an 2: 226�227 in Peters, 1993, p. 249)

Now, such an arrangement may be seen as imprisoning of partners, who have agreed to seek relief from their incompatibility, in a pleasure-less union for this so-called separation period. And so, it may be viewed as a contravention of the founding principle of Benthamite utilitarianism which unequivocally states,

Nature has placed mankind under the governance of two sovereign masters, pain and pleasure. It is for them alone to point out what we ought to do, as well as to determine what we shall do. (Bentham, 1996)

Therefore, it is only natural to expect the modern societies that emphatically champion Bentham's utilitarian decision-making to do away with such an archaic practice. Yet, in several countries around the world such as Chile, France, Ireland, India, Italy, Norway, Portugal, Sweden and some states in USA, the process of legal divorce by mutual consent requires partners to spend a duration of trial/legal separation from the date of moving the application for divorce, which is usually more than six months long.²

See https://en.wikipedia.org/wiki/Divorce_law_by_country

This paper attempts to resolve this apparent contradiction without abandoning Bentham's utilitarian perspective and the accompanying rational decision-making paradigm. In particular, we consider two partners with identical socio-economic characteristics involved in a sufficiently ‘healthy’ relationship where both partners, a priori, do not find quitting the relationship to be an attractive option. Our model focusses on their ability to resolve conflicts in a dynamic setting while being unabashed pleasure seekers.

We find that even such healthy relationships may not survive a conflict among partners. Moreover, even if such relationships survive the conflict, it would do so—either under a severely unfair norm where one partner has a hegemonic power over the dispute outcome, or when partners decide probabilistically to make up or fight in each period. In the latter case, a resolution of dispute is likely to be preceded by several periods of fighting. And so, this equilibrium explains the curious, but commonplace, behaviour where partners engage in conflict for long periods of time before arriving at a resolution, by undertaking actions that they could have agreed to in the first period itself and saved themselves from the painful periods of acrimony.

Now, it is difficult to imagine that an unequal hegemonic relationship equilibrium will pervade a liberal society. And so, it can be argued that the cultural change arising out of social emancipation would manifest itself in the shift of society from the ‘unequal’ equilibrium to the ‘unstable’ or ‘acrimonious’ equilibrium. Indeed, an instance of this shift can be seen while comparing societies at different levels of gender equality. For example, Guatemala is roughly ten times more gender unequal than Sweden, but Sweden has a ten times higher divorce rate than Guatemala.³

Requisite references appear later in the paper.

Also, to the extent that social emancipation facilitates greater economic empowerment for women, our theoretical assertion finds support in the empirical observations of positive relationship between divorce rates and increased participation of married female work force in labour markets (Becker, 2009; Jimenez-Rubio, Garoupa & Rosales, 2016; Nock, 2001; Trent & South, 1989; White, 1990). In fact, Sayer and Bianchi (2008) argued that it is actually the ideological disequilibrium with respect to gender expectations in a marriage, generated by gradual women empowerment, that contributed to rise in divorce rates in USA.

Note that, while no case can ever be made for slowing down on the path of greater woman empowerment, both rising acrimony among partners and weakening of the institution of marriage are, arguably, not in the best interests of the modern society.⁴

For example, continued decay in the institution of marriage may lead to societal situations like in Japan, where fertility rates have drastically fallen to the extent that the population is actually shrinking. This phenomenon has been accompanied by emergence of a high end market for emotional services ranging from intimacy to grief (https://www.economist.com/news/asia/21706321-most-japanese-want-be-married-are-finding-it-hard-i-dont and https://www.economist.com/news/asia/21706321-most-japanese-want-be-married-are-finding-it-hard-i-donthttps://www.youtube.com/watch?v=_aLEu1lx8bg).

Therefore, the critical question that arises is: What can be done to rescue healthy relationships from being so unequal or unstable or acrimonious?

It is in the pursuit of an answer to this question that the idea of a mandatory legal separation period becomes instrumental. We find that the restriction of legal separation mitigates any adverse impact of gender equality on stability of healthy marriages. Of course, one can ascribe many other benefits of such a legal device that appeals to the emotional nature of human decision-making. However, we show that even when partners are completely rational (and behave as the classical Homo economicus), the legal directive mandating a trial separation period serves a crucial role in limiting the friction between equality and stability of marriages. And this would explain, within the rational paradigm, why modern societies have continued with the concept of ‘legal separation’ in their divorce laws under mutual consent category, even when it goes against the essence of Bentham's ideals.

Finally, note that break ups can, typically, be legally regulated only if unions are legally recognized. Hence, in light of our findings, one may argue that giving legal status to a union of partners, as a marriage or otherwise, is likely to increase the chances of the union being stable.

2 Relation to Literature

We are unaware of any other paper rationalizing this device of ‘legal separation’ in modern divorce laws. Also, to the best of our knowledge, there is no other paper that looks through a ‘formal’ model at the dynamics within a marriage consequent to a dispute that may or may not cause irreversible breakdown in relationship. This is quite in contrast to the importance accorded to these issues by Kitson, Babri, and Roach (1985) and White (1990); who argued that divorce must be studied as intertemporal processes in addition to the conventional demographic approach. Also, unlike other papers, we adopt a gender neutral perspective by considering partners to be identical in all conceivable socio-economic dimensions. This allows us to isolate and focus on the internal dynamics of a marriage without any interference from mismatch in outside options available to the partners.

The majority of related literature has analyzed stability of marriages in terms of trade off between gains from staying in marriage and outside options available to partners. In fact, Becker (2009, p. 331) states unequivocally,

A husband and wife would both consent to a divorce if and only if they both expected to be better off divorced.

In contrast, our model shows that couples may divorce even when outside options are less than gains from marriage. Further, instability of marriages in our model manifests itself in break ups that occur in the very first period of conflict among partners. This is consistent with the observation of Becker (2009), which states that majority of separation and divorces in USA occur within the early years of a marriage (see the last paragraph of Becker, 2009, p. 327). Note that while Becker (2009) explains this observation by invoking imperfect information among partners, our assertion is derived in a complete information setting.

Also, as mentioned earlier, our results are consistent with widespread empirical observations of greater economic independence of women being associated with greater divorce rates. Further, our results nicely relate to the empirical findings of Nock (2001), which state that equally dependent spouses are more likely to get divorced than their counterparts. We argue that equally dependent partners are likely to be observed in emancipated communities that will never accept a hegemonic equilibrium and so, partners would coordinate on the unstable or acrimonious equilibrium. Finally, in line with the UK government report (HM Government, 2010), our results imply that marriages are likely to be more stable than unregistered live-in unions, as the latter cannot be subjected to legal separation period while breaking up.

3 Model

Consider two identical partners ${1, 2}$ that are in a relationship and have got embroiled in a dispute at time period $t = 0$ . At each period $t \geq 1$ , both of them must make a simultaneous choice to either make up (m) or fight (f) or quit (q). If any partner, at any period $t$ , chooses to make up or quit, the game ends.⁵

5
This model represents couple interaction consequent to a single dispute. So, an end to the game by some partner making up does not preclude possibility of future disputes arising and similar game unfolding again.

If both partners choose to fight, then the game moves on to the next period

t + 1

when both partners again make simultaneous choices from

{m, f, q}

. We assume that partners do not have an agreed, a priori, deadline by which they would end the conflict and so, model it as an infinite horizon interaction. To maintain comparability among the probable infinite streams of payoffs that arise out this infiniteness of horizon, we assume that partners discount future at the rate

δ \in (0, 1)

. That is, any payoff amount

x

obtained at period

t + τ

, where

t \geq 1, τ \geq 0

, is worth the amount

δ^{τ}

in period

t

The period payoffs in our game, at any period $t$ , are as follows: If both partners play make up simultaneously, then both get a period payoff $v - ε$ , where $v$ is the value of relationship and $ε \in (0, v)$ is some arbitrarily small cost associated with admitting any faults when the other partner also apologizes. If any partner $i$ makes-up while the other partner $j$ fights, then $i$ gets $v - c$ and $j$ gets $v$ . This cost $c$ denotes the loss of utility that ensues from being the first person to swallow pride and unilaterally make up, and is assumed to be greater than $ε$ . This assumption of $c > ε$ embodies the idea that displeasure associated with making up is lesser when other partner, also, apologizes, than when making an unilateral apology.

As mentioned earlier, if any partner $i$ quits on the relationship, the game ends. If the other partner $j$ has not quited simultaneously in the same period then partner $i$ gets $θ > 0$ which is her outside option with respect to the relationship. This value $θ$ could be interpreted as the best possible quality of relationship that she can build with another partner, upon quitting the present one.⁶

Note that the value $θ$ can be suitably modified to model any extra displeasure associated with the act of quitting when the couple has a child. And since, as will be argued later, the assumption of identical partners can easily be relaxed without qualitatively altering our results, the validity of our analysis would continue to hold even if the impact of a child on the outside options are asymmetric across partners.

The partner

j

who was quit upon, gets payoff

\frac{θ}{2}

which signifies that he is now forced to face his outside option with a sense of abandonment that comes from being judged to be unacceptable as a partner.⁷

Anecdotal instances can be found where such sense of abandonment leads to a downward spiral of self-loathing that culminates into acute depression.

Accordingly, if both partners decide to quit upon each other at the same period

t

, no partner ends up feeling the despair of being deserted and so, both get a payoff of

θ

each. Here, we presume that partners are less pained in being unwanted, when they themselves do not want the other. Finally, if both partners fight, they incur a cost

γ > 0

which signifies an aversion to acrimony in relationship.

In this paper, we focus on relationships that are not fractured beyond repair. Hence, we impose the following regularity condition on the parameters that ensures that both partners attach a substantial amount of value to their relationship:

θ + c < v

We denote this infinite horizon relationship game by $G$ . A strategy in this game is a function from $ℕ \to {m, f, q}$ , where $ℕ$ is the index set of periods. We restrict our attention to stationary strategies that depend on the state of game, irrespective of the time period, and look for subgame perfect Nash equilibrium (SPNE).

4 Results

The first result states that any pure strategy equilibrium must either involve quitting at the first period by both partners, or making up by only one partner in the first period. In either case, the game ends in the first period.

Theorem 1. There are only two possible pure strategy equilibria: (a) where one partner always fights while the other always makes-up, and (b) where both partners quit in the first period.

Proof. We begin by noting that there can be no SPNE in pure strategies where the game ends with a partner making up at some period $t > 1$ . To see this, suppose that there exists an SPNE $σ$ such that $σ_{i}$ requires some partner $i \in 1, 2$ to fight for all periods $1 \leq t < t^{i}$ and then make up at period $t^{i}$ . Then partner $j$ ’s equilibrium strategy, $σ_{j}$ , must require her to fight for all period $1 \leq t \leq t^{i}$ and make up at some period $t^{j} \geq t^{i} + 1$ (or never make up with $t^{j} = \infty$ ). But then, given any such $σ_{j}$ , $i$ has a profitable unilateral deviation from $σ_{i}$ where she makes-up right at the first period at $t = 1$ to get payoff $v - c$ , which is greater than her equilibrium payoff $δ^{t^{i} - 1} (v - c)$ under our supposed equilibrium $σ$ . This implies that $σ$ cannot be a Nash equilibrium (NE) and hence, contradiction.

Hence, in any pure strategy, SPNE of $G$ that leads to the partners making up, for any partner $i \in {1, 2}$ , either $i$ makes-up in the first period or she fights always without making up ever. Further, it is easy to see that both partners making up at the same period $t = 1$ cannot be an SPNE outcome, since each partner would have a profitable unilateral deviation in fighting in the first period and making up some time later. Finally at any period $t \geq 1$ , if one partner is going to fight at all times in future, then the best response of the other partner is to make up at the current period $t$ (and vice-versa). Therefore, there exists an equilibrium of type (i), where any one partner always fights and the other always makes up.

Now, we consider any possible SPNE of $G$ that leads to break up of the relationship at some period $t \geq 1$ . Clearly, this requires at least one partner $i$ to choose to quit at some time period $t^{i}$ . However, if $t^{i} > 1$ , then partner $j \neq i$ has a profitable unilateral deviation where she quits in period $t^{i} - 1$ and thus, generates no less stage payoff without the delay and cost of fighting for one extra period. Therefore, for any SPNE where the relationship does not survive conflict, break up must occur in the very first period $t = 1$ with both partners quitting simultaneously. Arguing, in the same manner, we can show that in such an SPNE, at any period $t > 1$ , both agents must choose to quit at the current period simultaneously. Finally, for any $t \geq 1$ , if one partner is supposed to quit at the current period, then the best response of the other partner is to quit in the current period as well. Thus, the equilibrium of type (ii) follows. $□$

We interpret equilibrium (i) of Proposition 1 as an unequal (no-break up) equilibrium, because one partner remains absolutely inflexible, while the other remains absolutely accommodating. Similarly, we interpret the equilibrium (ii) as an unstable (immediate-break up) equilibrium, because both partners are unwilling to work through their dispute and quit at the earliest opportunity leading to a break up by ‘mutual consent’. Thus, Proposition 1 underlines a dichotomy between equality and stability in a relationship, even when both partners value it high enough (so that $θ + c < v$ ). A reason behind absence of a more stable, but equal, dispute resolution equilibrium in such healthy relationships is that both partners are afraid that the other will quit unilaterally and so, they try to pre-empt by quitting first. Therefore, in our model, both end up quitting as soon as possible leading to a bilateral break up, ostensibly by mutual consent, in the first period itself. Such a sociological phenomenon is not difficult to see, particularly among urban youth where both partners have similar levels of economic and educational emancipation.⁸

8
We feel that the problem gets accentuated as one's own notion of self-worth expands. That is, bigger the ego of a person, greater is the hurt, pain and fear of being quited upon. Perhaps this is why many show-business celebrity spectacle marriages dissolve soon after.

As mentioned earlier, such a phenomenon has also been documented by Nock (2001).

In contrast, it is in the patriarchy-stricken societies, generally in relatively rural areas, that one gets to see the unequal equilibrium of Theorem 1 playing out. Such backward societies, typically, have long lasting heterogeneous marriages that are ‘domineering at best and abusive at worst’ in nature. A common feature of such marriages is the stark male domination that manifests in social practices and sartorial choices, irrespective of income or the educational levels of the partners. Unsurprisingly, divorced women in these societies usually face a precarious and insecure future, not to mention, a severe social stigma that leads to an almost negligible outside option in terms of our model. Quite disappointingly, this equilibrium continues to survive even when we allow it for partners having ‘different’ outside options $θ_{1} \neq θ_{2}$ that are sufficiently small (that is, $0 \leq θ_{1}, θ_{2} < v - c$ ).

In the following result, we show that there is a mixed strategy SPNE in this game where partners mix by putting positive probability only on both making up and fighting.⁹

In the context of human relationships, we find it difficult to believe that people in real life take decisions to break up in probabilistic manner. Hence, we ignore any mixed strategy equilibrium where partners put positive probability on quitting.

Proposition 1: There exists a mixed strategy equilibrium where partners mix only between making up and fighting with probabilities $p_{m}^{*}$ and $1 - p_{m}^{*}$ , respectively, with

p_{m}^{*} = \frac{B - \sqrt{B^{2} - 4 C δ (c - ε)}}{2 δ (c - ε)}

where $B : = (v + γ) - (c - ε) - δ (v - 2 c + ε)$ , and $C : = (v - c) (1 - δ) + γ$ .

Proof. Suppose there exists a mixed-strategy SPNE where each partner $i \in {1, 2}$ mixes between making up and fighting with probabilities $p_{m}^{i}$ and $1 - p_{m}^{i}$ , respectively. Therefore, expected equilibrium payoff of partner $i$ by making up, ${EP}_{i} (m)$ , and by fighting, ${EP}_{i} (f)$ , must be equal. Further, our supposition implies that ${EP}_{i} (f) = p_{m}^{j} v + (1 - p_{m}^{j}) (δ {EP}_{i} (m) - γ)$ . Therefore, ${EP}_{i} (m) = p_{m}^{j} v + (1 - p_{m}^{j}) (δ {EP}_{i} (m) - γ)$ implies ${EP}_{i} (m) = \frac{p_{m}^{j} v - γ (1 - p_{m}^{j})}{1 - δ (1 - p_{m}^{j})} = (v - c) + p_{m}^{j} (c - ε)$ . After some tedious algebra simplifying this inequality, we get that

δ (c - ε) {p_{m}^{j}}^{2} - B p_{m}^{j} + C = 0,

which is satisfied if

p_{m}^{j} = p_{m}^{*}

. Arguing similarly for partner

j

, we can show that

p_{m}^{i} = p_{m}^{*}

Therefore, if we show that $p_{m}^{*} \in (0, 1)$ then the result would follow. Now, since $δ \in (0, 1)$ , it is easy to see that $B > c + γ > 0$ and so, $p_{m}^{*} > 0$ . Further, $p_{m}^{*}$ can be inferred to be a fraction because $p_{m}^{*} < 1$ is equivalent to the inequality $δ (c - ε) < B - C = v - (v - c) (1 - δ) - (c - ε) - δ (v - c)$ , which follows from $ε$ being positive. $□$

Remark 1. Note that Proposition 1 implies that on equilibrium path, partners may indulge in periods of painful fighting for long periods of time. This provides some sense to the perplexing tendency of partners to resolve disputes after long periods of painful acrimony by taking actions that they could have taken right at the beginning and saved themselves from the painful periods of acrimony.

Note that a partner may make up unilaterally under the mixed-strategy SPNE specified in Proposition 1 but the probability of this happening in any period could be very small for a large range of parameter values. And so, if one is prepared to accept the idea that no relationship between human beings can exist without disputes and conflicts then Theorem 1 and Proposition 1 paint a grim picture. They posit that even when the relationship is sufficiently valuable (so that $θ + c < v$ ) to both partners, it may not survive. And even if it does, it is likely to do so—either under an unfair equilibrium, or under possibly long periods of acrimony.

Now, it is highly unlikely for the unequal equilibrium to frequent a society that celebrates human equality. Then, from Theorem 1 and Proposition 1, it follows that most relationships in a gender egalitarian society are likely to be brittle or acrimonious. Moreover, long periods of acrimony is may be expected to decay relationships to a level beyond repair where $v < θ + ε$ . It is easy to see that in context of our model, this parameter restriction would imply that making up is a dominated strategy in the stage game and so, no partner would ever make up in equilibrium. Thus, the unique equilibrium outcome under this parameter restriction would be one where both partners quit simultaneously at the first period.

In contrast, unstable equilibrium is likely to be observed in more parochial (and almost always, patriarchal) societies, where gender inequality manifests itself in play of the hegemonic relationship equilibrium. Anecdotal evidence confirms this by reflecting the proliferation of legal break ups via mutual consent divorces in advanced liberal societies all over the world, with very small rates of divorce in more primitive patriarchal ones. A striking example of this pattern holds for two countries: Guatemala and Sweden. While the Gender Inequality Index for Sweden is $0.055$ , for Guatemala it is almost ‘ten’ times more at $0.523$ .¹⁰

See https://en.wikipedia.org/wiki/Gender_Inequality_Index#Top_ten_countries and http://lac.unwomen.org/en/donde-estamos/guatemala.

In line with our findings, the divorce to marriage rates show similar divergence in opposite direction, where Sweden has almost ‘ten’ times greater rate at

47

than Guatemala which has meagre rate of

5

.¹¹

See https://en.wikipedia.org/wiki/Divorce_demography

Since the computation of Gender Inequality Index does not involve divorce rates in any manner, this data pattern across Guatemala and Sweden endorses our theoretical observations.

However, this bleak scenario is not beyond repair. If we alter the game $G$ slightly, we can establish an equilibrium where there is no break up and no partner dominates the other, leading to equality as well as stability of the relationship. The proposed alteration is depicted by Figure 1, which describes the altered game $G^{'}$ , where no partner can quit in the first period when $t = 1$ but from $t = 2$ onwards, they play according to $G$ . Note that the idea of ‘legal separation’ period effectively imposes this exact restriction.¹²

Such arrangement may also be specified in a legal contracts such as ‘prenuptial agreements’ that are increasingly used in marriages involving partners with huge value of assets.

Figure 1:

The Modified Relationship Game $G^{'}$ Where $x_{i}^{*} (δ)$ Is The Equilibrium Payoff to Partner $i$ In $G$

The following corollary states this idea formally:

Corollary 1. There exists a pure strategy equilibrium in $G^{'}$ , where partners makes-up in the first period.

Proof. From Theorem 1 it follows that both partners quitting at all periods, constitutes an equilibrium in $G$ , yielding the equilibrium payoff $θ$ to each partner at any period. Therefore, it can easily be seen from Figure 1 that, given $v > θ + c$ , there exists a pure strategy equilibrium $σ^{'}$ in $G^{'}$ , such that any one partner plays a strategy to make up in the first period, and quit at all periods in future, while the other partner plays a strategy to fight at all periods. That is because (a) for both partners ( $i, j$ ), $σ_{i}^{'}$ is best response to $σ_{j}^{'}$ in $G^{'}$ and (b) at any time $t > 1$ , both partners quit at all periods in future, as shown earlier in Theorem 1, constitutes a best response to each other. $□$

Thus, Corollary 1 shows that the absence of any opportunity to quit over a period of time allows any one partner of a healthy enough relationship to make at least one more effort to patch up things, even when he behaves as the narrowly self-interested Homo economicus. And so, if there is enough of a relationship left to retrieve and restore (as captured by the parameter restriction $θ + c < v$ ) then couples who have approached courts for divorce by mutual consent may well make up during a trial separation period ‘imposed’ on them.

In fact, there is much anecdotal evidence across the world where sparring partners found it possible to make up amicably during such periods. And so, the legal directive of trial separation can be seen as a mitigating influence upon the dichotomy between equality and stability of relationships in our model and thus, it is a very effective instrument in strengthening the institution of marriage. Further, in recent times, some societies like the states of Arizona, Arkansas and Louisiana in USA have come up with novel codes of marriage law, especially designed to reduce the chances of grant of quick divorce by mutual consent.

Remark 2. Note that legal restrictions apply only to legal unions and in most countries, these are marriages. However, live-in relationships that have no legal recognition cannot be subjected to these restrictions. Therefore, it is possible to argue that chances of survival of a relationship goes up when it is legally recognized. That is, the institution of legal union or marriage strengthens chances of survival of a relationship. The UK government report (HM Government, 2010) confirms this reasoning.

5 Conclusion

Our paper provides a rationale for the extensive prevalence of the non-utilitarian construct of mandatory legal separation in utilitarian societies. It does so by theorizing divorce as an intertemporal process, unlike the traditional methods of analyzing divorce in terms of outside options or demographics. Contrary to the assertions of Becker (2009), this approach underlines the possibility of divorces even when outside options are less than gains from marriage. We also present a justification for the curious tendency of partners to spend long periods of acrimonious fight before making up, even when they could have done so at the first period of fight. Finally, our paper posits that unions that are not legally recognized are may be more unstable in comparison to a legally recognized marriage.

It would be very interesting to empirically substantiate these findings. We leave this exercise for future research.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

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