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Mistreatment as a Coordination Problem

This post combines and translates two notes I wrote in Spanish in 2019, Acostarse con boludos and Maltrato.

Magdalena López wrote this piece in response to an interview I don’t want to discuss. I don’t want to comment on Magui’s piece either; I want to show a different way of thinking about these issues. In the interview, the person interviewed says:

What is patriarchal is to assume that women are submissive, passive. That a woman can’t say “hey, dude, you’re an idiot, bye.” Because the question is why you keep waiting for someone who left you on read.

At one point Magui says the following, which I will take as a reply:

To blame a woman for continuing to desire the man who doesn’t answer her messages, doesn’t call her, who is careless and cruel, is to reproduce the short-skirt argument.

And later:

to attribute salvation to a woman telling a jerk “get lost, jerk” is, once again, to go for the individual way out.

My answer is: if there were an individual way out, women would take it, and that would be that. The problem is precisely that there is no individual way out. The only way out is collective.

The argument

One can think about this as follows. A woman and a man are a couple. The man mistreats the woman. The psychoanalyst says the woman should simply leave him. Magui replies: if the woman keeps desiring the man, we can’t expect her to leave him. I understand that desire can be shaped (evidence: different people have different tastes, and the distribution of tastes changes over time), but one can’t change one’s desire at will, at least not in the short run. So women have to live with the desire they “received.” Here is one story: women are socialized so that their desire for men doesn’t decline when they are mistreated. This is, of course, good for men, since it lets them mistreat women without consequences. A story with a similar result is that men are socialized to desire submissive women. This gives women an incentive not to respond to mistreatment, so as not to become less desirable, and gives men an incentive to mistreat, because it reveals whether the woman is submissive.

Now suppose the woman’s desire for the man does fall with mistreatment (reasonable, no?). Why doesn’t she leave the man who mistreats her? The explanation above no longer works. Suppose men prefer to mistreat women.¹ (Again, we can think of preferences as malleable and partly received.) Suppose also that the woman prefers being in a couple to being alone, and likewise for the man. If the woman leaves the man, she spends some time alone until she finds another partner. The psychoanalyst’s advice is “leave him.” Even ignoring the cost of leaving, of waiting, and the expected time until finding another partner, there is a very simple reason not to: if all men mistreat, it is not rational to pay the cost of leaving an abuser, because the result is being alone for a while and then being with another abuser.

I make hyper-simplifying assumptions (everyone is identical, couples are heterosexual and monogamous, couples form at random, etc.) because they let us think of this as a game. At each moment there are some couples; the man may mistreat or not; the woman may end the relationship or stay; if the relationship ends, both are single for a period, and then each is matched with a random partner. This game has (at least) two equilibria:

1. All men mistreat. Given that, women accept the mistreatment, because the alternative (leaving) is not better: it is costly, and then they end up with another abuser.
2. All women leave a man who mistreats them. Anticipating this, men don’t mistreat, because they know that if they did, their partner would leave, they would pay a cost, and they would end up with another woman who wouldn’t accept mistreatment either.

The setup is a bit silly, but it delivers the following insight. In a patriarchal society (equilibrium 1), women are trapped, and they cannot solve their problem individually. If women coordinate, if they act collectively, perhaps paying some costs during the transition, they can move to equilibrium 2 and free themselves from mistreatment.

If one woman decides to put an end to mistreatment on her own, she can threaten to leave her partner, but the threat is not credible, because the other women do accept mistreatment and the other men do mistreat. If all women threaten to leave their partner if mistreated, the threat becomes credible, and men accept not mistreating to avoid being rejected.

Put differently: if a man expects the women he can date to accept mistreatment, he will mistreat. If he sees that enough women around him don’t accept it, it pays him to be an “ally.” In this model men are allies out of convenience, because they expect a sanction otherwise; if they could, they would go back to their old ways. We don’t need a change in values, men becoming good against their own interest. What we need is a change in expectations: that both men and women expect women to sanction mistreatment. The way out is not a “cultural change” but a change in collective behavior.

Some implications

The model can be enriched as follows. In reality, when a woman leaves her partner she doesn’t get a random partner; she gets one from her social circle (Tinder makes this less true, but let’s leave that aside). We live fairly segregated by class, education, and so on, so patriarchal “regions” can coexist with non-patriarchal ones. If the men you know don’t mistreat, you know you can leave the jerk who mistreats you, because you’ll find someone better. But this is not true for all women. The advice “leave him, he’s a jerk” may work in some social circles and not in others.

Also, “feminist” women, defined as the ones who don’t accept mistreatment, perhaps at a personal cost, generate a positive externality on the other women they are connected to. Non-feminists benefit from what feminists do. Another implication: it matters that women talk to each other and share WhatsApp screenshots, because even if that is not enough to eradicate patriarchy from society as a whole, it can reduce it in their own circles. A very strong signal (a massive march) is excellent for coordination, but local networks help too.

One could add inequality, or the fact that for some women leaving is especially costly. And surely there are far better ways to model this. My methodological point is that analytical individualism can help us think about these things.

The model

There is an infinite set of women \(W\) and an infinite set of men \(M\) of the same cardinality, both uncountable. A matching is a set \(R\) of pairs \((w,m)\) with \(w \in W\) and \(m \in M\) such that if \((w,m), (w',m') \in R\), then \(w = w'\) if and only if \(m = m'\). Time is discrete and infinite, \(t = 0, 1, \ldots\). The game starts with an arbitrary matching \(R_0\), and at each time \(t\) there is a matching \(R_t\). If \((w,m) \in R_t\), they simultaneously choose actions \(x\) and \(y\), respectively, each \(0\) or \(1\). The woman gets utility \(xr - yc\) and the man gets \(xr + yc\). The woman chooses whether to stay in the relationship (\(x = 1\)) or not (\(x = 0\)), and the man chooses whether to mistreat (\(y = 1\)) or not (\(y = 0\)). If the relationship goes on, both get \(r > 0\), the benefit of the relationship. If the man mistreats, he gets an extra benefit \(c\) and imposes a cost \(c\) on the woman. I assume \(c > 0\) (the man benefits from mistreatment and the woman suffers it) and \(r > c\) (even with mistreatment, the woman prefers to be in a couple). This last assumption is questionable, but if the woman preferred to be alone we would need to bring coercion into the model, and everything changes. The model only says something about the case in which women can leave freely.

If the woman stays, \((w,m)\) remain matched in the next period. If not, \(w\) is matched with someone at random other than \(m\), and likewise for \(m\). A technical detail: if only one couple breaks up, its members have no one else to match with next period. So I assume that in every period one of the continuing couples is chosen at random and broken up, so there is always at least one man and one woman available. The idea is that some couples split for exogenous reasons. Since there are infinitely many people and only one couple breaks up, the probability that this affects any given couple is \(0\), so I can ignore it in the calculations.

Everyone discounts the future by a factor \(\delta\) with \(0 < \delta < 1\). I assume \(\delta > \frac{c}{r-c}\) and \(\delta > \frac{r}{r+c}\), i.e., agents are patient enough. For this to be possible I need \(c < \frac{r}{2}\), a bit more than \(c < r\). Strictly speaking this is only needed for the man’s benefit from mistreatment, not the woman’s cost, but making that explicit requires more notation.

The equilibrium concept is subgame perfect equilibrium. Each player has a strategy, a function from histories of the game up to time \(t\) to actions at time \(t\). The strategies form an equilibrium if, for every time \(t\) and every history up to \(t\), each agent’s actions from then on are optimal given that everyone else follows their strategies from then on.

First equilibrium. All men always mistreat, and all women always stay. Would a man gain from not mistreating at some point? No: he loses \(c\), and it has no effect on the future. Would a woman gain from leaving? No: if she leaves she gets \(-c\) this period, gets a new partner next period, and is mistreated again, so she is worse off.

Second equilibrium. Men don’t mistreat their partner unless they mistreated her in the previous period, in which case they mistreat again. Women leave their partner if and only if he mistreated them in the past.

Consider first a man who has not mistreated his partner. If he doesn’t mistreat, he gets \(r\) this period, she doesn’t leave, and so on, so he gets \(r\) forever. If he mistreats, he gets \(r + c\) today; next period she leaves, so he gets only \(c\); then he has a new partner whom he doesn’t mistreat, so he gets \(r\) from then on. The difference is in the first two periods: \(r + \delta r\) if he doesn’t mistreat versus \(r + c + \delta c\) if he does. Since \(\delta > \frac{c}{r-c}\), the first is larger, so he prefers not to mistreat.

If the man mistreated his current partner in the previous period, he must prefer to mistreat her again. Given the women’s strategy, he expects her to leave, and then he starts a new relationship in which he won’t mistreat. Mistreating now yields \(c\) and has no consequence, so he does it.

Now consider a woman whose partner has not mistreated her. She expects him not to, so staying yields \(r\) from now on. Leaving yields \(0\) today and \(r\) from tomorrow on, which is worse.

Finally, consider a woman who was mistreated by her partner in the past. She expects to be mistreated again today. If she leaves, she gets \(-c\) today, and next period she has a new partner who doesn’t mistreat her, so she gets \(r\) from then on. If she stays, she gets \(r - c\) today; tomorrow her partner mistreats her again and she leaves, getting \(-c\); then she gets a new partner who doesn’t mistreat her, and she gets \(r\) from then on. From the third period on the payoffs coincide, so we compare the first two: \(-c + \delta r\) if she leaves versus \((r - c) - \delta c\) if she stays. Since \(\delta > \frac{r}{r+c}\), leaving is better. This completes the check.

Are these the only equilibria? No. For example, if women can observe all men’s past actions, they can call out and “cancel” men who mistreat. But this produces the same outcome as the second equilibrium; it does not make women better off.

Footnotes

1. This is an ugly assumption, but, first, it is realistic, at least if we allow for some heterogeneity. The obvious evidence is that mistreatment happens in the real world, although men may mistreat women even if they don’t prefer it as an isolated action. For example, mistreatment can signal that one is not too interested in the other person (which increases one’s bargaining power in the relationship), or a man may mistreat because he thinks it is “the right thing to do” (there is a norm, and one would have to explain why it is there). There might also be a biological story. Second, I think we are all a bit of a jerk when we believe there are no consequences. We are altruistic, we follow norms, but beyond that we are not always nice to others. With a bit of honest introspection, I think the reader will grant me this.