Tuesday, March 22, 2005

Fidelity Games

A man and a woman in a committed relationship are discussing the issue of cheating. At some point, the man asks, “So, if your husband ever cheated on you, would you forgive him or leave him?” What should her answer be?

Part 1: The Non-Credible Threat. For incentive purposes, her best answer would seem to be “leave the bastard.” That way she warns her man that cheating will be met with dire consequences, thus giving him reason not to cheat in the first place. The problem is that this claim could easily be a non-credible threat. Lots of wives stay with husbands who have cheated, presumably because they stand to lose a lot (in companionship, in money, in time) if they leave. And advice columnists regularly encourage couples to try to stay together, with the help of marriage counseling if necessary, rather than “throwing it all away” over one infidelity.

People with preferences given by
{faithful mate} pref {no mate at all} pref {mildly unfaithful mate}
are in a position to make credible threats. They can promise to dump their cheating partners and mean it, because they’d rather have no mate at all than continue living with an unfaithful one. Let us call such people Type 1. Contrast them with Type 2, people whose preferences are given by
{faithful mate} pref {mildly unfaithful mate} pref {no mate at all}
Members of Type 2 cannot plausibly threaten to dump the cheaters, given their preference for an unfaithful mate over no mate at all. (I realize, of course, that leaving one’s mate doesn’t mean being mateless forever. For convenience, I’m using “no mate at all” to stand for “going back on the market and enduring at least a temporary mateless phase.”)

Moreover, the very existence of Type 2’s weakens the credibility of promises to dump made by Type 1’s. Why? Potential cheaters often lack knowledge of their partners’ true types, which means the likely outcome is a “pooling equilibrium”* (a situation where players of different types send the same signals). The potential cheaters will perceive a single group, all of whose members may claim they’ll dump unfaithful mates, but only some of whom (the actual Type 1’s) will go through with it. Potential cheaters will have a mild incentive not to cheat (since some mates will actually dump them for it), but not the strong incentive they’d have if the threat of dumping were fully credible.

Type 1’s therefore have an incentive to distinguish themselves from Type 2’s, so their partners will believe their (real) threats to dump cheaters. But how? To create a “separating equilibrium”* (a situation in which players of different types send different signals), there must be some costly signal that only Type 1’s would find it worthwhile to send. They might, for instance, attend a church that is associated with strong support for marital fidelity. If the cost of attending church is (for some reason) higher for Type 2’s than Type 1’s, then church-going might be taken by potential cheaters as a signal of one’s true type.

(* I was surprised that I couldn’t find a good online reference on signaling games with pooling and separating equilibria. Contact me if you know of one.)

Part 2: The Binding Commitment. I’ve assumed thus far that a credible threat of dumping will in fact deter cheating. Given this assumption, it follows that Type 2’s would actually like to make their dumping threats credible by committing to them. By binding themselves in advance (Odysseus-like) to dump cheaters, Type 2’s could assure their most preferred outcome of a faithful mates. But what if some amount of cheating will happen no matter what? Then a dumping commitment might not be desirable after all.

The likelihood of a potential cheater actually cheating can be represented as the sum of fixed and variable components, like so:
Prob(cheating) = p + q
The first element, p, is the fixed likelihood that one’s partner will cheat despite any relationship consequences, perhaps when confronted with an irresistible cheating opportunity. The second element, q, is the additional likelihood of cheating if there’s no threat of being dumped; q vanishes in the presence of a credible dumping commitment. The larger is p relative to q, the more dangerous it is for Type 2’s to make such a commitment. In the limiting case, where q = 0 regardless, the commitment is all cost and no benefit: it doesn’t deter any cheating, and it assures the worst outcome (no mate at all) will occur some fraction p of the time. At the other extreme, when there’s no cheating opportunity that cannot be deterred (p = 0), the commitment is all benefit and no cost. The reality is likely to be somewhere in between, which means Type 2’s face a trade-off in deciding whether to make binding commitments to dump cheaters. On the one hand, they can deter some cheating, but on the other, they potentially force themselves to dump partners they’d rather keep.

In any case, even if a Type 2 decides making a binding commitment is worth the risk, she may be unable to do so. I can’t think of any way to guarantee you’ll dump someone in the future against your own wishes at the time. But less ironclad commitments are possible. Bold claims in front of friends (“No man will ever cheat on me and get away with it! I swear that I will kick his sorry ass to the curb!”) can tie your reputation to your verbal commitments. If the loss of reputation exceeds the marginal gain from staying with an unfaithful partner, the commitment can be effective. But given the possibility of partners who will stray regardless, only some will find the commitment worth making.


Anonymous said...

If the threat is highly credible, then any cheating that does occur is likely to be unnecessarily destructive. Once he's crossed the line of "worst possible penalty" he has little or no incentive to use a condom/fess up/get an STD test, and he can't easily back out.

Anonymous said...

eeew, i don't like this kind of calculating application of game theory in relationships, but i guess it is inevitable to a certain degree.
for everyone's sake i hope they're all type 1s. i'd rather be by myself than be miserable with someone else (even if there was no cheating). that relationship would suck and drive me nuts because of lack of trust. i'd think only people with low self-esteem would remain. but i've never been in that situation so i guess i can't really be judgemental; especially because women tend to get attached. and i guess if it was workable, staying together might (i say might b/c it's probably a rare case) make the relationship stronger depending on the parties involved. some people should just permanently just end it. but i would never stay if 1)a man cheated 2)abused me verbally or physically 3)molested my children or did some other illegal stuff.

I'd hope that credible threat of dumping will in fact deter cheating. Moreso, I'd hope that the cheating will be curbed hopefully by the parties involved because they care about the partner. But wouldn't cheating signal that there is something wrong with the relationship? Cheating is often never really about the sex; well, with the exception of some philandering men.


Will Wilkinson said...

For men, I think agressive jealousy helps play this function. If one's jealousy comes off as sufficiently involuntary, then it need not signal too much a lack of trust. And NOT threatening to dump the partner, but instead threatening to physically brutalize anyone they cheat with (or even are flirting too seriously with), significantly raises the cost of flirting and cheating. Also, a credible threat to viciously slander any cheating partner as a backstabbing unfaithful slut who is not to be trusted under any circumstances also significantly raises the costs of cheating. This latter tack is important because the weight of threat to dump depends on the availability of alternatives. If I threaten to dump my girlfriend if she cheats on me, but she knows that she can easily acquire a subsitute, then it has little weight. But if she believes that I will ruin her reputation among the pool of potential substitute,the threat to dump has more weight. Likewise if the pool of potential substitutes believe that I will try to kick the living shit out of them.

The Ev Psych literature implies that assymetrical information about paternity/maternity should create adaptive pressure for men to become better cheater detectors, and to develop a suite of behaviors to discourage cheating (intense suspicion, agressive jealousy, altruistic punishment) and that women should therefore have developed more sophisticated strategies for evading cheater detection.

Make of that what you will.

Anonymous said...

i'm dying laughing. Will: your man's man general outlook on life is revealed ever so clearer with each post. just remembering your 'women should be women and men should me men otherwise world would be hard for you' post.

i was thinking about all the couples i know in my life driving to work and contrary to the popular stereotype, about 70% of the cheaters are the women. And all of them are still with their men. I have to say, i'm guilty of adhering to a double standard b/c i remember thinking that it was sweet of their bfs to take them back and work their problems out. Because I remember telling my other gf who was cheated on (and verbally abused regularly) to take that bastard to the cleaners. so maybe if the tables were turned i'd think differently. And all of them are childless, i think that factors into cheating for women especially (it's just easier to).

on another note, some people are just more likely to be affected by a fixed p where q equaling null doesn't make a difference. you can sometimes avoid this situation when choosing a mate. my indy film director friend is constantly bombarded with hot actresses who wants to sleep with the director. although he calls them all whores, i've made a mental note of "this guy is never to be a bf." it's not that directors can't be faithful (there were other red flags like wanting to cheat on his current gf); but it's what you can handle and can't handle in a relationship.

all i have to say is 1) don't neglect your women emotionally. they are most likely to be vulnerable when their emotional needs are not met. (unless she's an emotional drain; that person can't be helped) 2) if you want to date a friend, don't tell her all your relationship problems with the current gf (like thoughts about cheating); they will certainly make mental notes of you. and you will be blacklisted. 3) women get bored and unsatisfied too.

Blar said...

I'm with Julian on this one. Her answer should be something like this:

"I don't know; I haven't really thought about it much. And I don't plan on thinking about it much. I wouldn't be involved with a guy if I thought he would do something like that."

I think that this response is good for the relationship before any affair occurs, good at dissuading cheating, and good for mending the relationship after any infidelity that might occur.

Making an explicit threat to someone is not a good way to establish the kind of union that makes for a good relationship. It's better to be more positive, to be on the same side, and to establish trust. She's indicating that she trusts him not to cheat. The expectations that she places on him may make it harder for him to be unfaithful ("I wouldn't do that to her"), and her expectations for the relationship make it clear that it might not survive infidelity. If he went on to cheat anyways, it's clear that the burden would be on him to repair the damage to the relationship and regain her trust. It would be his cheating and failure to maintain the relationship, not her decision, that would "throw it all away."

In summary, the positive approach, which emphasizes the kind of relationship that you are striving for, can dissuade cheating in a way that is more consistent with other relationship-relevant goals.

Anonymous said...

My guy and I have been together for seven years. He's an artist and works around a lot of young hot chicks with a conspicuous absence of traditional morals. He has plenty of opportunity to cheat if he wants, but I don't think he has ever done so. I've never done so, either, and I work with a bunch of male engineers... the kind Mom (who married an engineer) warned me about. :)

Sometimes the gals at the office talk about our men over lunch. The topic comes up, how do you deal with threats to your marriage? My answer, which seems flippant at first blush, is that I didn't marry him in the first place. We are really radical-libertarian in a lot of ways, my guy and I, and we decided that if we were going to be a couple, it was going to be because we chose to be, not because we went through some social ordeal or jumped through some largely meaningless government hoops.

I guess we've lowered the cost of leaving the relationship to its absolute minimum. When you know that your relationship's permanence depends on your actions, you make sure those actions show that you want the relationship to be permanent. If you do want that.

Now, that means that either of us could leave at any time without the formality of a divorce. All we would have to do in the case of infidelity would be to say, "I don't choose to be in a relationship with someone who thinks so little of me that they have to lie and cheat."

I guess what I'm saying is that I don't see the particular use of "binding commitments." You mean them when you make them, and you may not mean them later. They're a good way to get in trouble if one spouse (say) relies on the commitment to protect the marriage, while the cheating spouse uses that false security as a cover for their activities. Our way works because we have no false security.

(Now, on this blog you've discussed polyamory, and that's an option that my guy and I have discussed. But it's a very, very unlikely option that would involve all three people being as compatible as the two people were, plus total honesty and openness throughout the process of recasting the relationship as a threesome. The chances of our finding "that perfect person" are what, exactly?)