I wrote a reflection on Game Theory and partner abuse. Curious about this community’s thoughts.

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I confess: I’ve been getting sucked into the Johnny Depp/Amber Heard trial. Last week devolved into a blur of articles, Twitter threads, and subreddits, culminating in the testimony of the two protagonists themselves.

It got to the point where I was posting comments.

There’s a problem at the center of the case: when both sides accuse one another, how do we know who to believe?

Game Theory can be helpful here: it’s a field which uses math to model strategic interactions and make sense of human behavior. Once I started thinking about Game Theory, the Depp-Heard trial lost some of its obsessive pull, and I was able to return to my life.

Let’s look at Game Theory, and see if we can figure out how to react—as bystanders, fans, or friends—when two people make bilateral accusations of abuse (we’ll keep the math contained, I promise). By the end of this article, we’ll come up with a Game Theory informed algorithm for analyzing cases like this, and I’ll apply it to the Johnny Depp/Amber Heard case.

Part 1: Definitions and Axioms

Definitions (for edification and clarity)

Abuse: A pattern of behavior involving control and manipulation, where one person treats another as a resource (for sex, emotional release, stimulation, housework, etc). Can include physical violence, intimidation, gaslighting, outbursts of anger, displays of jealousy, verbal attacks, and isolation.

Partner abuse: Abuse in the context of a romantic relationship (also called “intimate partner abuse,” but we’ll keep it simple).

Psychological abuse: For the purposes of this article, a catch-all for abuse which doesn’t involve violence.

Unilateral abuse: In the context of a relationship, a situation where only one partner is abusive to the other.

Bilateral abuse: In the context of a relationship, a situation where both partners are abusive to the other.

Abuser: One carrying out abuse.

Target: One on whom abuse is being carried out. I’ll use it interchangeably with “victim.”

Violence: Use of physical force on another person. As above, violence can be unilateral or bilateral (or multilateral, but that’s outside the scope of this article).

Toxic relationship: A relationship where one partner is spiraling to a lower level of consciousness/functioning, and pulling the other person down with them. As above, toxicity can be unilateral or bilateral.

Axioms

1. All the above behaviors/relationship-types exist.

2. Abuse can involve violence, but doesn’t have to.

3. Abuse is a form of toxicity, but there are forms of toxicity besides abuse.

(An example to clarify point 3: Let’s say your partner is addicted to heroin and robs liquor stores to fund their habit; they may be totally unabusive, but they sound pretty toxic.)

4. Abusive relationships can include other forms of toxicity.

5. One person in a relationship can be abusive, and the other toxic; toxicity doesn’t justify abuse.

(An example to clarify point 5: Let’s say you try to stop your heroin-addicted partner from robbing liquor stores by screaming insults and beating them; this makes you abusive, while your partner remains toxic. Your behavior is not justified.)

6. Severe violence is a worse form of abuse than minor violence or psychological abuse.

Part 2: Abuse and Game Theory

The main “game” we’re trying to play is how we, as third parties, approach situations where two people are making accusations of abuse against each other. But before we get to that, we gotta dial in on that game-within-a-game: what’s going on with those two people making accusations, from their perspective?

There are a few possibilities:

1. They’re both abusive, and both telling the truth.

2. They’re both toxic and neither is abusive; both misperceive the other’s toxicity as abuse.

3. Neither was abusive during the relationship, but post-relationship, they’re bitterly trying to destroy each others’ reputations.

4. One is abusive, one is not. One is making a true accusation, and the other is lying.

I’m sure we can think of a lot more, but let’s leave it there. Those first three collapse into the same thing, game-theory wise: they’re symmetrical situations, where both people are making accusations to achieve their aims.

Number 4 is interesting, and, I suspect, extremely common. Many people think that something like this is happening in the Depp-Heard case. But why would it be common?

To answer that question, let’s make a game:

Don’t worry, I’ll explain everything.

This game represents a unilateral abusive relationship post-breakup. A game has three elements: players, strategies, and rewards. The players in this game are abuser and victim. The victim’s strategies are along the left: stay silent, or make an accusation. The Abuser’s strategies are across the top: make a false counter-accusation, or don’t accuse. The numbers in the middle are the rewards for each set of actions: the first number is the victim’s reward, the second is the abuser’s.

Let’s walk through an example.

At the beginning, no one’s made accusations, so we start off in the lower-right corner. Players with a co-operative strategy might choose to stay here: less drama, easier on the kids, fewer lawsuits. The spoils of silence.

In this scenario we assume that both parties are equally credible, and that if they make an accusation, each will be partially believed (some people will believe them, some won’t, and neither has enough evidence to prosecute).

The victim made an accusation! Maybe the abuser’s stalking them, and they want them excluded from group social events. Maybe they want to warn other people in the community who might date the abuser. Maybe they’re just super honest.

(Older versions of Game Theory assumed everyone only acted in their own rational self-interest, but newer versions ditch that assumption).

Now things look great for the victim: they have support, and the abuser is being held accountable. Look at that disparity: victim 3, abuser 0.

The abuser doesn’t like this one bit. They’re being trashed by their ex and people are giving them the cold shoulder. Sure, some stuff went down, but their ex deserved it. And now they’re the one being blamed??

Screw that. Lucky for the abuser, there’s a way out: make a false counter-accusation.

In the past, the abuser’s counter-accusation would be something like “she’s an insane bitch”: a general accusation of toxicity. But these days, with abuse awareness on the rise, the savvy abuser makes a false counter-claim of abuse. It’s very du jour, allowing the abuser to play to popular tropes: the lying man can say ‘men get abused too,’ while the lying woman can claim she’s the victim of gendered violence (we’re not focusing on gay relationships in this article).

So now we’re in the top-left corner. Could be that mutual accusations are better than silence for either victim or abuser, but I’ve set it pretty low to demonstrate that we’re likely to end up here, even if it sucks. The top-right corner is the Nash Equilibrium: a point where no one can change their strategy to increase their reward, assuming the other person’s strategy doesn’t change.

We can also get to this corner through a sequence where the abuser makes the first accusation, but I’ll let you work that one out yourself.

This example illustrates something important: once the accusations start, we’re almost certain to end up in the top-left. This explains why accusers almost always make a counter-accusation in response to an accusation.

Part 2: What to believe?

Strategy #1: Believe all Women

Before we dive in, let’s take a digression from Game Theory into statistics:

The research on partner abuse is weird. There’s a major controversy as to whether it’s symmetrical by gender, or whether it’s perpetrated at a (much) higher rate by men. The “it’s symmetrical” side mainly relies on something called the Conflict Tactics Scale: essentially a survey where researchers ask people questions like “Have you ever thrown something at partner that could hurt? Has your partner ever done that to you?”

The results of these studies are shocking: turns out men and women perpetuate partner violence at a roughly equal rate. Some studies even find that women are more likely to be perpetrators!

Case closed. The old feminist argument that women are the primary victims has been debunked. Men are as likely to be victimized as women. Time to convert some women’s shelters into men’s shelters and pack up that outdated hashtag.

Except, maybe not. Turns out, the Conflict Tactics Scale has more issues than Johnny Depp and Amber Heard’s relationship. The CTS looks at individual acts of violence outside of any context (like who initiated the violence, or whether it’s in self-defense). It also ignores violence after separation (a time when the risk of violence is dramatically elevated). The Conflict Tactics Scale has low intra-couple reliability: when husbands and wives are both surveyed, their level of agreement about the violence that took place in their relationship is lower than chance.

If the Depp-Heard trial teaches us anything, it’s that we can’t take people’s claims of abuse at face value; unfortunately, the Conflict Tactics Scale does just that.

Other methodologies paint a different picture. A table from this study says it all:

A few statistics, from methodologies other than the Conflict Tactics Scale:

- This study asked children about their parents’ violence (likely more accurate than asking people about their own violence). It found that 78% of intimate partner violence was carried out by men, and 88% of severe violence (kicking, choking, beating) were carried out by men.

- This UN study found that of people murdered by a romantic partner, 77.4% were female, 22.6% male. Women are more likely to be killed by a romantic partner than by all other types of assailants combined.

- This study found that 25% of women and 7.6% of men reported being raped or physically assaulted by a partner; that 5% of woman and 0.6% of men reported being stalked by a partner; and that 41.5% of women and 19.9% of men assaulted by a partner were seriously injured.

It’s time to trash the CTS and acknowledge that abuse is an asymmetrical issue: while men can be abused, and women can be abusive, male-on-female abuse is more common and tends to be more severe.

Whew. We got through statistics. Back to Game Theory.

So we’re currently evaluating strategy #1 for dealing with a Depp-Heard-like situation: a man and a woman each accuse the other of partner abuse. They each deny the counter-allegation. We know the abuser has a strong incentive to lie.

If we’re playing the numbers game, #BelieveAllWomen isn’t the worst approach (the hashtag applies to many forms of assault, but we’ll focus here on partner abuse). As we’ve seen, women are much more likely to be victims than perpetrators; with a blanket policy of #BelieveAllWomen, we’re beating the house. Further, research shows that abuse is more damaging to women than it is to men, not only physically, but economically and psychologically. #BelieveAllWomen might win the utilitarian calculus.

Unfortunately, #BelieveAllWomen has some drawbacks. It’s a form of “guilty until proven innocent:” with #BelieveAllWomen, some innocent male victims will unjust punishment. The principle of “innocent until proven guilty” prefers false negatives to false positives, and there’s something nice about that. It’s charitable.

#BelieveAllWomen also truncates the full humanity of both women and men. #BelieveAllWomen assumes women don’t lie and manipulate, which we are perfectly capable of, thank you very much.

We gotta keep in mind that crucial distinction: universal statements are different from tendencies. Abuse TENDS to be a male-on-female problems, but women can DEFINITELY be abusers. Abusers TEND to be male, but NOT ALL abusers are male. We are complex, we contain multitudes.

Another issue with #BelieveAllWomen is it turns the female abuser into a queen. It also allows any women to take down any man with impunity, even if she wouldn’t have been an abuser otherwise—#BelieveAllWomen is a moral hazard. The female-abuser game becomes something like this:

In a world of #BelieveAllWomen, the abusive woman gets to act with impunity, and no possibility of accountability. Great for her, but not so much for the rest of us.

Do we retire #BelieveAllWomen? At the level of collective punishment, we must. But individually, it may still have a place: I might, if I was so inclined, avoid dating guys who’ve been accused of partner abuse. Again, I’d risk false positives: maybe I miss out on the love of my life, due to an unfortunate false accusation. But I’d also reduce my chance of ending up with an abuser. This doesn’t actually need to be #BelieveAllWomen: I can withhold judgement on whether the accusations are true, and use the presence of an accusation as a (better than chance) indicator. Conversely, a single man might take a #BelieveAllMen stance, and not date any women he’s been warned against by other men.

Beyond that, #BelieveAllWomen has to go.

Strategy #2: Formal Neutrality

Johnny Depp says Amber Heard was abusive. Amber Heard says Johnny Depp was abusive. Who’s to say, really? In cases like this, one strategy is to refuse all judgement. Remain neutral, be Switzerland.

In a situation which really does seem like a toxic bilateral soup, where no is more or less at fault, neutrality may be the best approach. But neutrality isn’t the best policy to take in a situation of abuse, for a few reasons:

1. The abuser gets to avoid accountability.

2. The victim might, post-breakup, need to avoid the abuser for safety reasons. A de jure ‘policy of neutrality’—which continues to include the abuser in social contexts—might de facto isolate the victim.

3. The victim might still be in danger, and need support beyond that of neutral non-involvement; the most likely time for relationship violence to occur, once again, is post-separation.

A true Switzerland Strategy is careful to provide equal access, divide resources, and withhold judgement. In practice, this is challenging—it’s hard to stay close to two people in the bitter throes of mutual accusation. The policy of neutrality often devolves into:

Strategy #2b: Pick the person I like

This approach is fairly natural: after a relationship ends, it makes sense for everyone to return to their corners and lick their wounds. After all, abusive people need support too: if you suspect a friend or relative is abusive, there’s no moral imperative to cut them off or vilify them. Abusive people aren’t monsters, they’re people with a serious problem, one they’re usually only partially aware of. They need other people in their lives who understand the situation and can help them change (if you find yourself supporting an abusive person, I’d recommend Lundy Bancroft’s book, Why Does He Do That).

Here’s the issue: for a lot of people, “supporting” their friend (or their Johnny Depp) means uncritically accepting their side of the story. This isn’t support, it’s enablement.

All that aside: the point of these strategies isn’t to figure out who to remain friends with, it’s deciding who to believe. And epistemologically, believing the person we happen to like is as effective as flipping a coin.

If not worse. Abusers isolate their the victims and mess with their head. The abuser has problems (possibly a personality disorder), but those problems can make them charming and likable; in the immediate aftermath of abuse, the victim might be anxious, paranoid, hostile, or out of touch with reality—in other words, acting pretty weird. The abuser may very well be in a better position to win a popularity contest.

Strategy #3: Atomic Individualism

Men abuse. Women abuse. What else is there to say?

Well, I’ve been saying a whole bunch of other stuff for 2000+ words, but many people think those four are enough. We know both sexes are capable of abuse, so why not judge each case gender-blind? Look at the evidence, weight things neutrally, don’t let all this gender controversy get in the way.

The nice thing about this approach is it’s fair. This is a good approach for, say, the legal system. The legal system knows their bias: they’re trying to reduce false positives AKA wrongful convictions. They’re not trying to understand the deepest truth of every situation, to probe the complexities of human nature, or figure out who to be friends with.

You and I, however, are humans, not legal systems. And if we want to understand what’s happening around us, going at things gender-blind is… well, it’s putting on blinders. Reality isn’t gender-blind, as gender is rooted in history, biology, and (if you’re willing to get spiritual with me) maybe even ontology. If we’re going to relate to each other truthfully, we need to take account of one another as embodied, gendered beings.

Here’s an example. Given the research discussed above, a situation of a male-on-female abuse is likely a more urgent situation than one of female-on-male abuse. I’m not saying one of those situations is more important than the other. Both are important. But only one person in those two situations has a massively elevated risk of getting murdered.

Strategy #4: Pattern Matching

With pattern matching, we research both male-on-female and female-on-male abuse, and identify patterns (the two are quite different). If we’ve experienced or witnessed partner abuse in our lives, we’re at an epistemological advantage: we can think back to those situations and generate patterns. Better yet, if we’ve witnessed multiple such situations, we can triangulate.

One guy who’s done this is Lundy Bancroft, who worked with thousands of abusive men, and has their number. Happily for us, he wrote a book. If anyone knows of a similar book about female abusers, please share in the comments.

The best way to pattern match is also the hardest: it’s identifying patterns of abuse within ourselves.

Abuse comes from the impulse to treat another person as a means for getting what we want. We all do this to some degree, but in abuse, we employ tactics of control, manipulation, or physical violence.

I’ve had times where a romantic partner brought up a topic I didn’t want to talk about. Instead of setting a boundary, I allowed myself to get colder, to telegraph anger and annoyance at a level below which I could be accused of overreacting, but above the level of perceptibility. The conscious goal of my emotional expression was to “teach” the other person not to bring up that topic anymore.

This kind of low-level manipulation can easily escalate into full-blown abuse. I could indulge in anger every time my partner does something I don’t like. If the anger doesn’t work, I could escalate into violence.

Pattern-matching is indispensable, but it has some drawbacks:

1) We pattern-match too tightly to our own experiences. Our experiences furnish us with patterns we use as templates, but it’s easy to overfit to those patterns, and get attached to them.

2) We pattern-match to TV. This is happening very literally in the Depp-Heard case (a lot of people are identifying Depp with his public persona and characters), but it can also be subtle. TV is a superstimulus, and hits our brains hard. At the same time, the way people act on TV is not how people act in real life—the medium has a tendency to exaggerate, elide, and pander. It’s all too easy to pick up the patterns of TV people, and start thinking that’s how Real People are. This can dangerously warp our understanding of reality (for a monumentally profound take on this, check out “E Unibus Pluram” by David Foster Wallace).

3) We pattern-match to grand societal narratives. This is what’s so pernicious about #BelieveAllWomen: it urges us to adopt a narrative instead of critically thinking. The rallying point of “Men can be abused too!” is similar: yes, men can be abused, but if that’s the main lens we’re applying to the world, it’s easy to overfit. When we adopt narratives instead of thinking for ourselves, the consequences can be devastating for everyone and everything.

Given these drawbacks, our final, winning strategy looks something like this:

Strategy #4b: Game Theory Informed Pattern Matching Plus

We start with our competing narratives:

Narrative 1: Person A is abusive, Person B is the target.

Narrative 2: Person B is abusive, Person A is the target.

Narrative 3: It’s some kind of bilateral situation.

Then, we carry out the following steps, sequentially:

1. Bracket gender and examine the evidence of the case, gender-neutrally.

2. Reintroduce gender and see if it alters our analysis. Remain wary of pattern-matching too closely to grand societal narratives.

3. Introduce personal experience and try pattern-matching to it.

4. Identify personal bias. Incorporate perspectives with counter-biases to balance that.

5. Weight steps 1 through 4 equally, assigning each a value of one point. Each step “votes” for the narrative it best supports. Whichever narrative has the most votes wins, and is considered most likely to be true. If it’s a tie, default to Formal Neutrality.

This approach allows us to retain the benefits of Atomic Individualism, incorporate the humility of Formal Neutrality, and address the risks of Pattern Matching. Based on everything we’ve looked at so far, this is our strongest strategy, and it’s the one we’ll apply to the Depp-Heard case.

Part 3: Johnny Depp and Amber Heard

With Johnny Depp and Amber Heard, there are three main perspectives:

Narrative #1: Amber as abuser, Johnny as victim

In this narrative, Amber was abusive towards Johnny during their relationship, and made a false claim of abuse during the divorce to get a better settlement. She made a false public accusation to benefit her career, thinking that in the #MeToo era, Johnny would have no recourse. However, he made a true counter-accusation, bringing her abuse to light and suing her. And now we’re in the upper-left corner.

Narrative #2: Johnny as abuser, Amber as victim

In this narrative, Johnny was abusive towards Amber during their relationship. She accused him during the divorce, and tried to keep it quiet. However, after he sued the UK Sun for calling him a batterer, she came went public with an accusation. In retaliation, Johnny made a false counter-accusation and sued her.

Once again, we’re in that upper-left corner.

Narrative #3: The whole thing is a toxic bilateral mess

This includes all our symmetrical options: they’re equally abusive, they’re trying to ruin each other, they were both toxic and neither is abusive, they’ve been brainwashed by an evil magician to remember the past inaccurately.

When considering this possibility, let’s remember axiom 5: if one person was toxic and the other abusive, that’s not a bilateral situation, and the toxicity does not justify the abuse. A situation of one toxic person and one abusive person falls into narrative #1 or #2, not this one.

Now let’s apply the steps.

Step #1: Bracket gender and examine the evidence of the case, gender-neutrally.

While I love a bewildering miasma of memes as much as anyone else, I believe the UK trial is the best source of evidence. Johnny Depp sued the Sun, a UK-based gossip mag, for calling him a batterer. He lost the case after the court established 12 counts of credible battery against Heard. Depp appealed the case, and lost after two more judges reviewed the evidence.

The twelve credible instances include Depp kicking Heard, head-butting her, and throwing objects at her. The evidence includes contemporaneous text messages and journal entries, medical evidence, and witness testimony.

On the other side of it, Heard admitted to hitting Depp, and several of his staff have testified witnessing her being verbally abusive. There’s an audio-recording of her admitting to hitting Depp, saying there’s a “difference” between them, and saying he won’t be believed if he comes forward. Depp also accused Heard of throwing a vodka bottle which severed his finger, and of crapping in their bed as an act of abusive terrorism.

The latter two are likely false. And at the time of the incident, Depp told people he cut off his own finger (342), more than once, and there’s no evidence she did it besides him saying so (in contradiction to his original story). And the bed-crapping story is full of holes: it happened at a time when Depp was away, and wouldn’t be back before Heard needed to use the bed and the cleaning lady came; they had a dog with a history of crapping the bed; and Depp had texted about taking a shit in the bedroom to prank Heard (i.e.: the idea was already in his head before the incident) (see 479 and 480).

We could go on all day. Depp wrote text messages about killing Heard and having sex with her burnt corpse. One of their therapists called the situation one of “mutual abuse.” That same therapist said she small bruises on Heard’s face. That same therapist said that Heard sometimes initiated fights, and that Heard “gave as good as she got.”

There’s a ton of evidence that Depp was violently abusive towards Heard and initiated the most severe incidents. There’s also evidence that Heard hit Johnny and initiated some incidents.

0.5 vote for narrative 2, Johnny guilty, Amber innocent.

0.5 vote for narrative 3, bilateral mess.

Step #2. Reintroduce gender and see if it alters our analysis. Remain wary of pattern-matching too closely to grand societal narratives.

As a society, we need to get to a place where consider a man hitting a woman worse than a woman hitting a man (in the absence of weapons). Men have greater physical strength than women, particularly in their upper bodies: a woman is unlikely to seriously harm a man through hitting, while a man can easily beat a woman to death. This is a knife fight/gun fight scenario, and here the “gender blind” approach fails: a man punching a woman is more serious than a woman punching a man. That said, if a woman uses a weapon, she’s escalated to dangerous violence.

The only credible violence Heard carried out against Depp was hitting. Not-that-it’s-okay or anything, but the violence Depp is credibly accused of is far worse. Even if they were both accused only of hitting each other, Depp’s act would be much more serious. But he probably did way more than that. Further, there’s evidence he was instigator much of the time: even the Heard-critical therapist mentioned above acknowledges that. Even if Heard was the instigator some of the time, in the context of his much more serious physical abuse and credible psychological abuse, this case can’t be considered bilateral.

Vote for narrative 2, Johnny guilty, Amber innocent.

3. Introduce personal experience and try pattern-matching to it.

(The following will sound subjective; it’s meant to.)

I’ve seen many male-on-female abusive relationships, and I lived in an abusive household as a teenager. A lot of what Heard recounted in her testimony rang true: Depp’s jealousy, attempts to control her career, “romantic” boundary violations, fits of jealousy, extreme judgement over how she dresses. The way she applied them to a Hollywood context also made sense (he forbade her from taking roles with sex scenes and romantic subplots, etc).

When I listened to her testimony, I believed her. And I don’t believe Johnny Depp: he’s either lying or bullshitting about a lot. The man strikes me as someone who’s walked a long way down a very dark path.

Vote for narrative 2, Johnny guilty, Amber innocent.

4. Identify personal bias. Incorporate perspectives with counter-biases to balance that.

I’m extremely, highly, embarrassingly biased when it comes to this issue.

Abuse was the first serious-adult problem I had to deal with, and the version I witnessed was male-on-female abuse. This situation began when I was twelve, so this experience is baked into my personality: I tend to side with women over men, and have a history of regarding female friends’ male partners with suspicion.

It’s worth noting that before I researched this case, I accepted the guilty-Heard/innocent-Depp narrative, so it seems my bias isn’t totalizing. That being said, when I became skeptical of that narrative, I had a feeling like, “Yes!! Of course!”

Okay, I’m biased. And most people are pro-Depp. The social media support for Johnny Depp has been overwhelming, and the internet hates Amber Heard. The herd (no pun intended) isn’t always right, but it’s a signal.

Okay, let’s bring in someone else’s perspective to counter my bias. Let’s go with Daniel Pinchbeck, who’s written a thoughtful Johnny-sympathetic analysis. He believes Heard was the primary abuser, though Depp carried out some violence and possibly initiated on certain occasions. He identifies with Depp in several ways, and for him, Heard pattern matches closely with what he’s seen of female-on-male abuse.

In a nutshell: I’m biased, a lot of people disagree with me, and they aren’t crazy (not all of them, anyway). It’s possible Heard planted the evidence for Johnny’s abuse (in particular, contemporaneous text messages, emails, and diary entries), as she was planning to falsely accuse him years ahead of time. This doesn’t seem likely to me, but it’s possible and I have to take that into account.

Let’s call it:

Vote for narrative 1, Amber guilty, Johnny innocent.

Verdict

Narrative #1, Amber guilty, Johnny innocent: 1

Narrative #2, Johnny guilty, Amber innocent: 2.5

Narrative #3, bilateral mess: 0.5

We get the picture of a deeply toxic relationship, where both parties assaulted the other and engaged in abusive behavior, but Depp was the primary abuser.

In conclusion: you won’t find me outside the courthouse with Alpacas. I wish these guys the best, and anyone else who’s tangled up in a situation of abuse.

Game Theory and Partner Abuse