MAKE ME A MATCH MADE IN HEAVEN
Reproduction and the Mathematics of Romance
IT IS QUITE POSSIBLE THAT WE OWE THOSE WONDERFUL THINGS THAT we call love and sex to viruses. If viruses did not exist, all animals, including humans, would apparently reproduce asexually.
A large percentage of plant species and some animals do in fact reproduce asexually, meaning they do so without involving a second organism. But most complex species would be unable to survive viral infections if it were not for sexual reproduction. There is an unceasing war constantly taking place between animals and their virus enemies. The most effective weapon that animals have developed for winning this war is genetic variation.
Viruses attacking animals, including humans, are forever striving to adapt themselves to the genetic structures of their victims. Our genetic structures are analogous to locks, with the viruses looking for the corresponding keys to open those locks. Once they have those keys, they can attack every animal whose locks are similar enough to be opened by the keys. If there is sufficiently broad genetic variation in a given population, the viruses need to carry a very large bundle of keys in order to attack every individual. If, in contrast, a population is genetically identical, a virus with a single key can wipe it out entirely. Sexual reproduction enables two individuals with different genetic structures to mate and produce an offspring whose genetic structure is different from that of both parents. In essence, sexual reproduction is an insurance policy guaranteeing the genetic future of the parents.
This is also the source of the evolutionary taboo against sexual relations involving family relatives. It might seem that if the goal of evolution is to create a hereditary chain of animals sharing as much genetic similarity as possible, natural selection would favor reproduction within the family unit, with brothers and sisters as optimal mates for producing children. In fact, incestuous reproduction is a sweeping evolutionary liability.
There are genetically transmitted diseases whose incidence is greatly amplified in children produced by incestuous relations. In addition to the social taboo against such relations, we have also developed an efficient psychological mechanism that prevents us from being attracted to close family relatives. All of this protects the genetic variation in our species, even at the price of reducing the genetic nearness between ourselves and our offspring. A tight genetic similarity between us and our offspring means less variation in the human population, making our species more vulnerable to extinction at the hands of a viral plague.
This is, of course, another example of emotions being a hugely valuable mechanism for preventing a bad outcome: the logic required to understand the evolutionary risks of incest is far more abstract than the immediacy of our revulsion to it. Nearly all of us recoil at the thought of conducting sexual relations with family relatives such as siblings or cousins, but many studies show that most of us are in fact sexually attracted to those who resemble us both in appearance and personality. Psychologists who have conducted studies of the phenomenon have found that siblings and cousins who are not aware of the familial ties between them (as can happen in cases of adoption, separation of parents, or very large families) report significantly greater sexual attraction between them relative to most couples. It is reasonable to suppose that this attraction stems from the fact that if the viral threat were not present, there would be considerable evolutionary advantage to marrying relatives.
It is interesting to imagine a science fictional account of a world without viruses in which humans reproduce asexually. If there were no need to fight viruses by confusing them with genetic variation, there would definitely be an evolutionary advantage to asexual reproduction. Sexual reproduction is genetically inefficient: it is complicated, leaves too much to fate, and above all it produces offspring who are not genetically identical to their parents. Asexual reproduction, in contrast, would have enabled us all to clone ourselves and form perfect genetic copies of ourselves. It is reasonable to assume, based purely on evolutionary considerations, that if we lived in an asexually reproducing world, asexual reproduction would be as pleasurable to us as we find sexual relations to be. If reproduction were not pleasurable we would not do it, and then we would cease to exist as a species.
Evolutionarily, we could probably get along fine with asexual reproduction, but what about human society in such a world? What would play the roles of courtship, love, romance, and flirting? What would art and music be like without the subject of romantic love? Narcissism and egocentrism would doubtless be primary personality characteristics in an asexually reproducing world. We would all be focused on ourselves, rarely interacting with others. In sum total, our lives would probably be emotionally poorer and far more boring.
But if two are better than one when it comes to sex and reproduction, why aren’t three better than two? This question was posed and studied by Motty Perry and two colleagues in a very interesting research article.1 Given that sexual coupling is a way to ensure genetic variation in a population as protection against viruses, why didn’t natural selection go on to create threesome sexual reproduction? After all, by combining the genetic material of three individuals, even more variation would be created.
I should point out that by threesome sexual reproduction I do not mean ménage a trois scenarios as one might find in certain French genre films from the 1970s—two men and a woman sharing a bed, or two women and a man. A world with threesome reproduction means a world with three different sexes: male, female, and a third sex for which we obviously do not even have a name in our languages. In such a world, each successful sexual encounter would require the involvement of one member from each of the three sexes, each contributing genetic material for the creation of offspring. There is no species on earth that reproduces in such a way. There is good reason for this: the advantages that threesome sexual reproduction has relative to the sexual coupling reproductive method we are so familiar with are overwhelmed by the disadvantages it brings.
From a technical perspective, there is no difficulty at all in imagining threesome reproduction occurring. There are cases in which DNA tests for the purposes of establishing paternity have mysteriously failed to show a genetic connection to either the father or the mother. In some of those cases, further investigation revealed that the child in question actually had three parents, as the result of an ovum that was fertilized by the spermatozoa of two different men, with the child then bearing the genetic heritage of both men and the mother. The child’s mother, it turned out, had indeed engaged in sexual relations with two different men in a short space of time and her ovulating egg had been penetrated by the sperm of both men.
Perry and his coauthors showed that adding more distinct sexes in the act of procreation does indeed lead to greater genetic diversity in a population, but the increase in diversity gained by moving from two sexes to three sexes is marginal. On the other hand, reproduction requiring three (or more) distinct sexes reduces fertility significantly, because it requires three individuals desirous of reproduction to find each other and meet, which is much more complicated than the analogous chance meeting of two individuals. The conclusion is that reproduction in pairs is the optimal form of reproduction for organisms seeking to avoid being wiped out by viruses. It is comforting to know that the human approach to sexuality through a romantic attachment to one other individual, as opposed to groups of three or more, is not arbitrary but the result of mathematically well-founded evolutionary considerations.
Previous chapters of this book pointed out that human sexuality differs from the sexuality of most other animals because it is based on a mix of emotions and commitment. But the emotions that so move us in romance and sexuality are not arbitrary either. In contrast to common perceptions, we do not suddenly fall in love or get swept away by romantic emotions; love develops at the right time and with the right person. In fact, love is in most cases the result of decisions we make.
When I completed my university studies and went to the United States as a young researcher, I was astonished to discover that some of my colleagues, researchers from India, had married their spouses by way of arranged marriages. These colleagues were young, modern, and liberal in their views and were extremely well educated and intelligent. They had lived in the United States for many years, but when it came to marriage, they accepted the traditions of their culture and entered into matrimony in matches arranged by their parents.
When I spoke at length with my Indian friends on the subject of love and relations between the sexes, they described their experiences of growing to love their spouses as the result of rational and deliberate decisions that they took. When they had met their spouses-to-be for the first time, a wedding date had already been set, as had their future living arrangements and the amount of the dowry that the father of the bride was to pay the father of the groom. The question of whether or not the bride and groom were appropriate for each other as a couple remained almost entirely in the hands of the parents.
In the Indian arranged marriage tradition, during negotiations on the amount of the dowry both the virtues and the negative aspects of the bride and groom are openly discussed. If the gap in the “qualities” of the members of the proposed couple is regarded as too large by the parents, negotiations are ended and the respective parents will seek new matches for their children. Small gaps are “bridged” by appropriate tweaks to the amount of the dowry, reflecting the relative qualities of the children whose marriage is being arranged.
A colleague of mine named Ragavan went to India to get married in the middle of his doctoral studies at the University of Oxford. He got to meet his future wife for half an hour before the marriage agreement was concluded between their parents. Within two days the newlyweds were on their way to Oxford. Ragavan and others have often told me that the process by which their marriages were arranged has not reduced in the least the love that they feel for their spouses. In fact, they claim that quite the opposite effect is achieved. They were able to concentrate on building a loving relationship after all the other details regarding their marriage had been determined, arranged, and completed. Some of my Indian friends have even told me that they have difficulty understanding me and my wife. How could both of us start dealing with such an emotional matter as a loving relationship, they ask, when there was still so much uncertainty surrounding it?
For all its positive aspects, the system of arranged marriages as practiced in India (and many other countries), and especially the custom of paying dowries for marriages, can also be the source of several social ills, most prominently inequality between the sexes. The dowries that the father of the bride is required to pay the father of the groom can be beyond what the bride’s family can bear. There are Web sites in India containing dowry price lists. The price is mainly determined by variables such as the groom’s occupation, his caste, and the caste of the bride. Large gaps between the standing of the bride and groom can raise the price of a dowry to upwards of $130,000. It is, therefore, not surprising that the birth of a daughter is regarded as a burden in many Indian families, while the birth of a son is treated as a treasure. In recent decades, with the development of technologies that can identify the sex of a fetus in the early stages of gestation, a trend toward aborting female fetuses has gained momentum in India (and in China as well). Prior to the trend of female fetus abortion, biology gave the general human population an identical proportion of women and men. The abortion of female fetuses has changed that, with men now outnumbering women in the world population by 2 percent. In India the gap between men and women in the population stands at 4 percent more men, while in China that gap is now 6 percent. In some provinces of India the gap is even wider. Interestingly it is in wealthy areas that some of the widest gaps have emerged, because wealthy women carrying female fetuses are more able to bear the cost of having an abortion than poor women.
These imbalances have inevitably set in motion correcting market forces. A shortage of women has brought about significant reductions in dowry prices. In some places, in a reversal of tradition the parents of brides are now demanding that the groom’s parents pay them a dowry for the right to marry their daughter. In areas with especially large gaps between the numbers of men and women, the shortage of nubile women is so great that another perturbing economic phenomenon has emerged: two brothers marrying the same woman to enable their family to pay for the large dowry demanded by the bride’s parents.
The dating market in Western societies is freer and more spontaneous, but a careful look at the rational and economic considerations operating in it shows that it is not that different from the traditional arranged marriage system customary in India. The phrase “love is blind” sounds poetic, but reality is usually much more prosaic. In most cases we fall in love with individuals with whom we expect we can form mutual bonds, while avoiding developing feelings of love for those whom we believe are “unattainable.” Romantic attachments often form between couples belonging to the same ethnic group and sharing the same social and economic standing.
My colleague and friend Eva Illouz has conducted a thorough study of the way women and men choose romantic mates in modern Western societies.2 Illouz’s study shows the extent to which liberalism with regard to relationships, along with technological advances that now enable romantic meetings to be arranged at the press of a button, have brought capitalist consumer culture into our love lives. When it comes to modern love, it turns out, we refuse to compromise on anything less than the best deal we can obtain, just as we do when we go shopping. To obtain that idealized goal we are willing to endure hundreds of Web-enabled dates that end in frustration and disillusionment. As a result we often avoid investing the commitment that is required for building stable relationships.
Gary Becker, winner of the 1992 Nobel Prize in economics, has also claimed that the decisions we make with regard to relationships and love closely resemble our decisions in market situations. In two articles he published in 1973 and 1974, entitled “A Theory of Marriage,” Becker presented a mathematical model of the marriage market.3,4 He was not the first to do so. A decade earlier David Gale and Lloyd Shapley, mathematicians specializing in game theory (Shapley was awarded the 2012 Nobel Prize in economics), fashioned a similar model for the marriage market.5
Both models describe a two-sided market with women forming one side of the market and men the other. Each man lists the women in order of preference: the women to whom he is more attracted are at the top of the list, while those to whom he is less attracted are toward the bottom of the list. Each woman has a similar ranking of the men, from most attractive in her eyes to least attractive. Each market participant, male or female, retains the right to remain single if the only members of the opposite sex that are available to him or her appear so low in his or her subjective attractiveness ranking that being alone seems preferable to marriage.
The central concept underlying both models is the beautiful idea of a stable set of matches. A set of matches is a monogamous assignment of the men in the market to the women in the market. Each man is matched with no more than one woman (although some men may remain single, i.e., not matched with any woman) and each woman is matched with no more than one man (again, some women may remain single). A set of matches is stable if it is not possible for any couples to divorce, or for the people involved to be better matched than they already are. Man A may desire Woman B, but if she also desires him and they are each married to someone else, it is not a stable set of matches. Similarly, in a stable set of matches each individual who was matched to a member of the opposite sex prefers being with his or her spouse to remaining single.
From these definitions alone, it is not immediately clear that such an ideal stable set of matches can always be constructed given a marriage market of men and women. Using a short and very elegant proof, however, Gale and Shapley proved an optimistic mathematical theorem: a stable set of matches always exists—no matter what the preferences of the men and women in the marriage market are! Gale and Shapley even showed how a simple and easily implementable procedure, that can be run on a computer, can find a stable set of matches, given an input of the preferences of each man and each woman.
Gale and Shapley’s model, more than Becker’s model, has turned out to have very broad applications. In fact it is one of the most influential and practically applied economic models of all times. It has been used, for instance, to find placement for medical interns, a market that got dramatically more efficient as a result. And, after the esteemed Stanford economist Alvin Roth further developed the theory, it has helped school boards in the United States and the United Kingdom improve the placement of children in their preferred schools. In recent years Roth has also been the driving force behind the introduction of the Gale-Shapley algorithm for a novel application that is literally life-saving: kidney transplants.
Successfully implanting a kidney requires a high level of genetic compatibility between donor and recipient. Not unlike marriages, many potential implants never happen because the two are not (genetically) compatible. Alvin Roth, along with several colleagues, had the insight to recognize that many lives could be saved by using matching algorithms to bring together compatible donors and recipients. The idea goes as follows: suppose that Ron wants to donate a kidney to his ailing sister Ruth, but unfortunately their compatibility is insufficient for a successful transplant, while at the same time Maya wants to donate a kidney to her husband Gary, with that transplant also vetoed by doctors due to incompatibility. If Ron’s kidney, however, can be successfully transplanted to Gary, while Maya’s kidney is compatible with Ruth’s body, an organ “swap” between the pairs can be conducted, saving two lives that might otherwise be lost.
Roth correctly understood that the potential “market” between such organ donors and recipients is similar to the marriage market and the interns market mentioned above: it is a two-sided market with donors on one side and patients in need of kidney transplants on the other side. An algorithm can therefore be applied in the transplant market to create long chains of kidney donors and recipients, saving thousands of lives annually across the country.
Yet the Becker model has much value of its own: it illustrates the surprising savvy and self-interest embedded in our decidedly nonrational dating market. Here’s how it works. In Becker’s model, people rank the attractiveness of members of the opposite sex using a set of characteristics such as appearance, education, social standing, wealth, and so on. Each individual gives different relative weights to these characteristics in forming his or her preferences. Each potential match between a man and a woman creates “joint utility,” a jargony term meaning the benefit each partner gains from the match, based on the characteristics of each and the weights that the other assigns to those characteristics. A more successful couple has higher “joint utility,” although it is not necessarily the case that they share it equally—more about that in a moment. In contrast to the Gale-Shapley model, which permits individuals only to agree or refuse any particular match suggested to them, in Becker’s model the members of every couple that is formed also need to agree how to divide between themselves the joint utility that their match creates.
Consider, for example, a woman with many attractive characteristics who is rated highly by many men. It is possible for her to marry a man who is not considered attractive by other women. But in that case the division of the joint utility generated by the marriage will be skewed in the woman’s favor. This may be expressed, for example, by the man having to do more of the household work or forgoing the purchase of a sports car that he particularly desires. This assumption is called “transfer of utility” in Becker’s model and it is the essential difference between Becker’s model and that of Gale and Shapley. The extent to which the assumption of transfer of utility is reasonable is a subject of ongoing dispute between economists, and we shall return to it.
Here is an example of how a stable system of matches and agreements on utility transfers may be accomplished under Becker’s model. To simplify matters, the marriage market in the example is small, containing only two women, Rachel and Miriam, and two men, Sam and David. Each of the four possible matches that can be formed in this marriage market creates joint utility for the couples involved. These joint utilities are illustrated in the following table:
As the table shows, if, for example, David and Rachel are matched together, their joint utility as a couple is 8, representing the benefits that the couple gets from being a couple. This includes both the material and the emotional benefits that both members of the couple get from the fact that they have married.
Note, though, that in our example, although David and Miriam are the most successful potential couple (creating a joint utility of 9, higher than all the other utilities in the table), they cannot be matched in a stable system of matches. To see why, consider the situation in which David-Miriam and Rachel-Sam are the matched couples. Suppose that Sam, David, Rachel, and Miriam individually receive the utilities S (for Sam), D (for David), R (for Rachel) and M (for Miriam), respectively, under the utility division agreements that the couples sign. This means that S + R = 4 and D + M = 9. Put simply, Sam and Rachel are a bad couple: based on our chart, he would be better off with Miriam and she better off with David. Whatever they are getting in their current match, they can get more by being together and dividing the pie of 8 units. Thus, even though David and Miriam are a happy couple, the unhappiness of Sam and Rachel makes this set of matches unstable. They have an incentive to divorce. Another way to look at this is to note that the sum of the couples’ “utility” is greater in a stable set of matches. Sam and Rachel’s joint value is 4, and David and Miriam’s is 9, for a total of 13; on the other hand, David and Rachel’s utility is 8, and Sam and Miriam’s is 7, for a total of 15. The greater number, in this case, reflects stability.
The important insight here is that the stability of a couple depends on more than the direct relationship between the two members of the couple—it also depends on what is possible outside of the partnership, meaning the possibility that each member of the couple can do better by finding an alternative spouse. For the same reason, it is entirely possible that the most successful relationship (meaning the relationship that creates the greatest amount of joint utility) will never be part of a stable system of matches. For a system of matches to be stable it must maximize the sum total of the utilities of all the individuals in the marriage market.
In our example the system of matches composed of David-Rachel and Miriam-Sam is stable. It creates a sum total of 15 units of utility, the greatest amount of total utility possible in this marriage market. The next interesting question is then how will these utilities be divided between the man and the woman in each couple? The answer again depends on the entire market, and it will not necessarily be divided equally between the man and the woman.
Suppose that the utilities were divided equally in our example, so that David and Rachel agree to divide their joint utility of 8 in a 4:4 split while Sam and Miriam agree to a 3.5:3.5 split. This is an unstable arrangement, because Miriam and David could divorce their spouses and form a new couple enabling them to divide more units of utility between themselves (9 units instead of 7.5). There is a division of utilities in this case that does lead to a stable system of matches, as follows: David and Rachel divide their 8 units of utility equally between them at 4 apiece, while Sam and Miriam divide their utility of 7 by giving Sam only 2 units of utility and Miriam 5—Miriam thus receiving three more units than her spouse Sam.
Under such an arrangement, why wouldn’t Sam rebel against the inequality imposed on him and demand, for example, that Miriam handle the laundry in their household, given that he is already doing all the cooking and driving the children to soccer practice? The (cynical) answer given by Becker’s model to this question is that if Sam does receive a larger share of the joint utility at Miriam’s expense, then Miriam will have an incentive to divorce Sam and marry David instead, in a new arrangement that would make both Miriam and David better off.
If you find the materialism, unmitigated self-interest, and selfishness that pervades Becker’s model somewhat unpalatable when the subject matter is relationships and love, I sympathize with you. But let’s be precise in our criticism of the model. Becker’s model isn’t necessarily talking about pure materialism, since as mentioned above the numerical values in the model also represent emotional utilities. But the model is indeed based on self-interest and selfishness, which is one of its weaknesses. The model implies, for example, that if one member of a couple is injured to the extent that the joint utility of the couple is significantly reduced (such as might happen if there is a serious illness or a similar disaster) then his or her spouse should immediately start seeking a new relationship. This is not an accurate portrayal of a loving relationship—not from a moral perspective or an empirical one.
Gary Becker proposed his model of the marriage market in the 1970s, when he was a member of the economics faculty of the University of Chicago. The “Chicago School,” as it was known, was marked by an approach to economics characterized by material self-interest on the part of economic agents and an unequivocal belief in the forces of the free market. Becker was very much a supporter of this approach. Given this, it is not surprising that Becker also controversially proposed that human organs be bought and sold on the free market for the sake of reducing chronic shortages of organs for transplants.
Despite the criticisms that can be leveled against Becker’s model, it is still a very important model because it provides us with several insights into how the marriage market actually works. Some of these insights are well supported empirically. The model, for example, correctly predicts that increasing the participation of women in the labor market can improve the standing women have within their relationships with their partners but can also increase the frequency of divorce. This follows from the fact that a woman’s utility from staying single increases once she has the means of supporting herself.
Consider what would happen in the above example if men and women can remain single. If each person’s utility from staying single is 1, nothing would change in our example (since being part of a couple delivers each person more than that). Suppose now that the opportunity to work and support herself were to increase Rachel’s utility from being single to 4.5, up from 1. Then the system of matches previously detailed would no longer be stable. Rachel would demand at least 4.5 utility units from David in exchange for remaining in a relationship with him, leaving David with only 3.5 (out of 8). But David need not agree to Rachel’s demand. He may instead offer Miriam 5.2 units of utility if she agrees to marry him (which is more than she is getting from Sam), which would make both of them better off. This analysis provides some support for the claim sometimes raised by activists in women’s organizations that many men are opposed to their wives working outside of the house not because of concern that this will reduce the quality of care their children are getting, or because of fears that household work won’t get done, but because they are worried that economic independence for women will increase women’s bargaining position within relationships.
Another weakness of Becker’s model is the assumption that utility can be transferred. By this assumption, just about any negative characteristic that one member of a couple sees in the other can be corrected by dividing the joint resources in the marriage in an appropriate way. This sort of brutal smashing of the entire basis of romantic attachment in relationships is not only outrageous, it is also not an accurate portrayal of reality. Rather than speaking in anyone else’s name, let me relate a personal story. When I was a freshman student, I had a brief romantic relationship with a woman who had just about everything I could ask for. She was good looking and intelligent and had an amazing sense of humor and great sensitivity. But what I could not find in her, despite trying very hard to find it, was that amorphous and ill-defined thing that distinguishes a very good friendship from a sweeping sense of falling in love. I cannot imagine that this exquisite young woman could have given me anything at all to compensate for that missing ingredient.