Ask a Mathematician / Ask a Physicist

Mathematician: The Physicist and I were once asked “how do I find the love of my life?”. Never ones to shy away from applying math to love (or anything else), the Physicist gave his take on this question (noting the connections between finding a mate and the famous “secretary problem”), and here I am, taking a stab at love from a different angle, tackling the problem using math and logic in the hope of producing useful advice.

It’s important to note that the question itself is deeply ambiguous. Part of the reason that math is so powerful is because it relies on applying deduction to precise definitions, so when mathematicians get answers, pretty much everyone who understands them can agree they are correct. Even for the insanely complex and specialized proof of the Poincaré conjecture, a consensus about its correctness was reached in just a few years (a short time by the standards of other fields). Hence, in the spirit of mathematical proof, we proceed by trying to formalize our question by developing a precise definition.

Let the “love of your life” be the currently living person who (if you got to know them) would:

1. Fall romantically in love with you for as long as he or she lives.
2. Cause you to fall romantically in love with her or him for as long as you live.
3. Increase your average happiness at least as much as any other person satisfying both (1.) and (2.), if you were to become life partners.

Unfortunately, due to the limitations of the English language, and  a notable lack of happiness measuring devices, we will be forced to sweep under the rug sticky issues such as:

• how, precisely, to define “romantic love”
• what exactly is meant by “happiness” (e.g. how do we compare the happiness generated by a good shoulder rub with the happiness caused by a great conversation? The appropriate conversion rate between different types of happiness, if such a rate even makes sense to talk about, is very problematic to define)
• whether there could be more than one “love of your life” for a given person (my answer: with high probability each person only has one, due to the fact that our definition involves the person who makes you happiest, and total happiness over a long stretch of time can take on a huge variety of distinct numerical values, so it’s unlikely that the two best people for you give you precisely the same value).

In any event, I am sad to report that when applying the above definition for “the love of your life”, finding “the one” is essentially impossible. I strongly urge you not to try it. The probability that you meet the single person that would make you happiest of all is extremely small. There are about 7 billion people on earth, and (let’s say) more than a billion within a reasonable dating age range of you. That implies that there are at least 500 million people of the appropriate gender (a bit more if you are bisexual). Even if it is the case that you are rather narrow minded, and just 1 in 200 people are culturally similar enough to you for you to even consider a romantic relationship (e.g. you are a Baptist American, who would only ever be willing to date another Baptist American), that still leaves at least 2.5 million people to search through. To meet just half of those people (and therefore have anywhere close to a 50% chance of meeting the “love of your life”) you would have to be meeting, on average, more than 40 potential mates each day over a period of 80 years.

In practice, maybe you could do a bit better by being selective about who you meet (for example, concentrating your efforts on those places where you would suspect “the one” would hang out). But even so, you’re (non-literally) screwed, as the odds of meeting this person are negligible. In fact, the only way that you could have a decent chance of meeting that person is if you had such stringent dating criteria that you would be unhappy with anyone that fell outside of a small, easy to locate group. Unfortunately, while this strategy is great for maximizing the chances of finding the “love of your life”, it is terrible for maximizing your happiness (official mathematician advice: don’t go join a cult where you are not permitted to date non-cult members, on punishment of eternal damnation).

So if trying to find the “love of your life” is a bad idea, what should a person do? Well, when mathematicians don’t like what they can derive from one starting point, they often can just alter their problem or definitions a little (you should feel bad for those poor physicists who are limited by what actually exists. We mathematicians frolic in the realm of pure ideas!).

Looking for love shouldn’t be about finding the best person. As we’ve seen, that is usually close to impossible, and even when possible, a bad idea. It is much smarter to view the search for love as an attempt to maximize your total lifetime romantic happiness over the rest of your life. This viewpoint leads to a very different optimal strategy than one would use to try to find the single best person. The total romantic happiness maximizing approach implies working to increase the moment to moment satisfaction you feel due to your romantic life, added up (or if time is continuous, integrated) over all of your remaining moments.

From now on, we will refer to a person employing such a romantic happiness maximizing strategy as a “Romaximizer”, and will kick the mathematics into high gear, introducing the Romaximizer equation:

where

• L is the number of years you have left to live
• H is the average amount of happiness per relationship year (that is, per year of time spent in relationships) that you will derive from your future relationships
• T is the average number of years that you will spend in each future relationship
• P is the average number of new people that you will meet per year
• F is the fraction of the people that you will meet that you will find sufficiently physically and personally attractive to consider dating
• D is the fraction of those people you will want a relationship with who will actually be willing to have a relationship with you
• M is the fraction of those people that you will find sufficiently attractive to consider dating that you will decide to try to actually begin a romantic relationship with

The Romaximizer equation is only an approximation, and hence the reason the equation uses  “” rather than ““, but it should be plenty accurate for our purposes (for those interested, I’ve included the proof of the equation at the bottom of this article). An important thing to note about it is that increasing the value of any one of the variables will increase Total Romantic Happiness, so long as all of the other variables are simultaneously left unchanged. Hence, all else being equal, L, H, T, P, F, D and M are things that we should try to increase (though as we’ll see, some of them involve tradeoffs where increasing one variable decreases another). This formula leads us directly to a variety of specific strategies for improving our total romantic happiness, which I will now discuss in detail.

1. Increase L, the number of years you have left to live. It is of course the case that the longer you live, the more potential time for romantic companionship you will have. Dying young is an especially bad strategy for the Romaximizer. Fortunately, there are plenty of simple (though not necessarily easy) things we can do to promote long life. According to government statistics, out of all deaths in the United States in 2007, about 25% were caused by diseases of the heart (staying within a healthy weight range, exercising, and not smoking cigarettes is believed to reduce this risk), 23% were caused by malignant cancerous tumors (maintaining a healthy weight, avoiding cigarettes, and avoiding excessive drinking are believed to help), 5.6% were attributable to cerebrovascular diseases which can lead to strokes (high blood pressure is a culprit, and smoking and obesity appear to be risk factors), 5.3% were caused by chronic lower respiratory diseases (you guessed it, smoking is implicated yet again), 5.1% were due to accidents (an easy way to reduce this risk is to always wear your seatbelt, since if you break it down further, about 1.5% of deaths are due to car accidents), 2.2% were due to Influenza or Pneumonia (remember to get your flu shot if you’re a member of an at risk group), and 1.4% were due to suicide (if you feel depressed on a regular basis, get yourself to a CBT therapist or psychiatrist right away, before it gets any worse). With health, there will always be luck involved, but there fortunately are a number of steps we can take to significantly improve our odds.
   
2. Increase H, the average amount of romantic happiness per year spent in future relationships. One simple approach to raising this variable is to increase your pickiness with respect to those traits that really make a big difference to your happiness in a relationship. So long as your beliefs about what in a mate makes you sustainably happy are fairly accurate, greater selectivity should increase the average quality of your relationships. Unfortunately, there is a tradeoff here, as this will decrease M, the fraction of people who you are attracted to that you ultimately decide to enter into a relationship with (and as we know, decreasing any of our variables tends to reduce total romantic happiness). We get a simple rule of thumb for deciding whether increasing H is worth the amount you’d have to decrease M by noting that when the product of variables T P F D M is significantly bigger than 1 (which basically means that you get into relationships easily with little gap between them and they last for a substantial time), then the Romaximizer equation simplifies to (a value of more than 6 for T P F D M will make the approximation accurate to within about 15%). That means that under those conditions, if you can increase H by becoming more picky (i.e. by decreasing M) then you should do so since there is little cost as seen in the simplification of the equation. There are other strategies for increasing H, of course. For instance, if you have had a number of relationships in the past, you could try making a list of problems that arose in them, and a corresponding list of things that you could have done to help prevent or fix those problems. Review these strategies a few times before getting into a new relationship, so that you are primed to use them the next time around!
3. Increase T, the average number of years that you spend in each romantic relationship. Like H, the variable T can be increased by being more picky about who you enter into a relationship with (i.e. by decreasing M). The Romaximizer equation immediately shows us the tradeoff between T and M: as long as we are increasing the product T M, our total romantic happiness will be improving. That means that if you think you can increase the average length of your relationships by 40% by being 20% more selective regarding who you start a relationship with (so causing T to increase by the multiple 1.4 causes M to decrease by the multiple 1-0.2 = 0.8), then the product T M will increase by 1.4*0.8 = 1.12 > 1, so your total romantic happiness will be improved. It’s worth noting that often H and T come into conflict. When your happiness is falling in a relationship, you have the choice of increasing T (staying, even though you aren’t so happy), or trying to find someone else who will make H larger. The right choice for increasing total net happiness is going to depend on how difficult it is for you to find someone else that will make you happier than your current partner. For example, consider a case where your happiness in the relationship has waned substantially, and you have truly exhausted strategies for improving it (and there aren’t religious or children related reasons to stay together). If finding another person who makes you happier is unlikely to take you very long, then a breakup is likely a romantic happiness maximizing strategy.
4. Increase P, the average number of new people you meet per year. There are at least three reasons that trying to meet more people can be a good idea. First, your time between relationships will tend to be lowered since you’ll meet potential mates more often. Second, meeting lots of people can help provide insight into what traits you really value, as you are better able to compare the things you like and don’t like about different people. Third, the more people you meet, the more selective you can afford to be without adding any wasted non-romantic time (i.e. you can afford to decrease M in order to increase H without much cost in terms of increased non-relationship time). For example, if for each person of the appropriate gender and age range that you meet there is a 1 in 100 chance of having romantic interest in them (e.g. suppose that you find 1 in 5 physically attractive, and 1 in 20 of those seemingly compatible enough with you that you’d be willing to date), then for each 100 people you meet there will be about a 37% chance that you don’t have a romantic interest in any of them, and only an 8% chance that you have a romantic interest in three or more of them. On the other hand, if you had met 300 people rather than 100, there would be only about a 5% that you wouldn’t have a romantic interest, and there would be a 58% chance that you would have interest in three or more of them, giving you the flexibility to choose to try to start a relationship with whomever seems the most likely to make you happy, rather than being forced to go with your only reasonably compatible option. The main drawback to meeting more people is that sometimes lots of choices can lead to doubt, indecision, or regret. If you have serious problems with decision making, it’s still good to meet lots of people to minimize your waiting time between relationships, but you may just want to go with the first person you find that you think is substantially likely to bring you significant happiness so as to avoid tough decision making and doubt. It’s worth noting that if you have the goal that during the next three years you will meet the person you will one day marry, and you find less than 1 in 150 people of your desired age range and gender to be sufficiently personally and physically attractive to strongly consider marriage with them, then to have a pretty good chance of achieving your goal you had better be meeting an average of at least two new people of the right gender and age group each week! So you’d better get cracking. There is this potentially very self defeating view out there that love happens when you aren’t looking for it. This has an element of truth in that neediness, desperateness and insecurity are often found to be unattractive. It also has a large element of falsehood, in that if you don’t make a conscious effort to meet lots of people who could be potential mates, then you will be impairing your chances considerably. If you meet zero people you will never find love, I guarantee (except, of course, of the “self” kind). Likewise, if you meet only a small number of people each year, it will be unlikely that you will find love anytime soon. When you go to a party, make that extra effort to speak (for a few minutes, at least) to each and every person that you find potentially attractive. Sign up for online dating, speed dating, and events where singles are likely to be. Ask friends if they know anyone who they think you would like to meet. Your total romantic happiness will thank you for trying these strategies! It is also worth noting that the amount of time you have to wait to find your next relationship will scale based on how many people you meet. So if you start meeting twice as many people of the same quality as you used to, then all else being equal, you will tend to be single for only half as long as usual! Hence, for many people, the simple strategy of substantially boosting the number of people you meet can lead to a big improvement in romantic life!
5. Increase F, the fraction of the people that you meet that you find sufficiently physically and personally attractive to consider dating. While we may not have a great deal of control over who we find attractive, there are still a variety of things we can do to try to increase F. (i) How likely you are to be attracted to a person you meet is going to depend on where you meet that person. Not every place to meet a person is equally good. For example, an intellectual snob shouldn’t be picking up men at WWE matches (not that there is anything wrong with men in spandex pretending to hurt each other), and the wicked witch of the west shouldn’t be meeting people at pool parties. The main things to consider are, “how many people am I likely to meet at place X?” and, “what’s the probability that each person I meet at place X will be someone I have a mutual romantic attraction with?” Maximizing romantic happiness isn’t just about meeting a lot of people, but also about meeting them at places where you are more likely to find people that you like. (ii) The physical attractiveness we feel for a person can rise over time as we get used to them, especially if we appreciate his or her personality a lot. So if we meet someone who we don’t quite think meets our physical attractiveness threshold, but who we like a lot and think we would be compatible with, it may well be worth it to get to know them better and explore whether physical attraction could develop (e.g. by becoming friends). Eliminating someone because you don’t find them super good looking on the first date may not be a winning strategy overall. (iii) Sometimes our ideas of who we find attractive and who we think we should find attractive can get somewhat mixed up. If you think that social pressure of some kind is affecting how attracted you are to someone (e.g. your friends don’t think she’s hot, or he’s not part of the cool crowd), then ask yourself how much happiness you are likely to get from pursuing this person, versus how much unhappiness you are likely to have due to violating the social pressure. Another thought experiment that can be fruitful is to ask yourself how much you would enjoy being with the person if nobody else’s opinion mattered and nobody else cared. (iv) Sometimes we make rules for who we consider ourselves to be compatible with that don’t actually have much bearing on what really matters. People will have a list of traits that they feel a person must (or must not) have in order to be boyfriend/girlfriend/spouse worthy. The appropriate question to ask yourself is: “If a person lacks (or has) this trait, how much will it affect my happiness with them overall?” If the answer is “only a small amount”, it should probably be taken off the deal breaker list. By lowering your standards in the less important areas, you can afford to raise them in the more important ones (i.e. those that will more substantially affect your happiness).
6. Increase D, the fraction of those people you would consider dating who would be willing to have a relationship with you. Since D is basically a measure of your desirability, we can apply some classic strategies to increasing it. (i) Get in better shape (this only applies to non-olympic athletes). (ii) Practice projecting confidence in your speech and body language (you can practice this when talking to friends, and get their feedback). (iii) Learn how to dance (if dancing is relevant at places you hang out). (iv) Try different haircuts, and poll people of the attractive gender as to which looks the best on you. (v) If you feel like you don’t have interesting things to say, then start taking in more information (from books, documentaries, blogs, magazines etc.) and spend time learning about things that people enjoy talking about (popular TV shows, local sports teams, etc.). If every day you have almost the same experiences (go to work, do the same job, come home, watch the same TV shows) it won’t be surprising if you don’t have much to talk about. (vi) Take a poll of your friends to find out if people find your glasses attractive, and if they don’t, switch to contacts. (vii) When meeting people, wear only the clothes that make you look especially good (ask a stylish friend for help figuring out what to wear, if necessary). (viii) If you have a unibrow, pluck your eyebrows. (ix) think about what those people you really want to attract care about in a boyfriend or girlfriend, and seek to be more that way. The list of things you can try to make yourself more attractive goes on and on. Some people are resistant to changes like these because they view them as superficial. Yes, many of them are superficial of course, simply by definition, but that doesn’t make them bad. They will make people more attracted to you (including even non-superficial people), which means that you will, on average, be increasing your total romantic happiness!
7. Increase M, the fraction of those people that you are attracted to that you decide you actually want a relationship with. M basically measures how un-selective you are in who you decide to enter into a relationship with, above and beyond your basic attractiveness and compatibility constraints. While increasing M would be good if you could hold all the other variables constant, in practice there is a tradeoff, because becoming less selective means that you are likely to reduce H, the average happiness per year of dating that your future relationships bring you, and you may also reduce T as well, the average number of years per relationship. On the flip side, by making M smaller (and therefore being more selective) you also pay a price, since you will have a longer waiting time between relationships. Hence, you’ll potentially be missing out on extra years of good romance. There is an important idea related to selectivity that has to do with how quickly you become attached to people. Romantic attachments can form before you know the person you are dating well. This is dangerous because these attachments can be difficult and painful to break, even if it turns out that you and the other person are not compatible or even if the other person doesn’t treat you well. You might end up dating such a person for a long time when you could have waited a bit to find someone who is much better for you in the long term. Another related consideration is that when people first meet, they are usually trying to make the best impression that they can, and signal all their best qualities. Since this process of romantic deception is not flawless, and since the motivation to maintain it generally falls with time, information about the flaws your date possesses will inevitably leak out with time. For instance, if during each one of those first hours you spend with someone you are dating there is only a 10% chance that a (not immediately obvious) flaw becomes apparent, you will need to spend about 6.5 hours with that person to have at least a 50% chance of knowing just one of these flaws! If you have met someone who you believe has no flaws, then you simply haven’t spent enough time with them yet. Disciplined casual dating (e.g. not rushing into being serious fast, and not depending on each other or having high expectations for each other early on) can allow you to get to know what the person is really like before you have formed a strong attachment, so that you can make an informed and clear headed decision about whether the person is likely to make you happy. That implies being more selective about who you get into a relationship with, because you will likely be eliminating people that normally you would have become attached to. If you are the sort of person who meets strong romantic interests fairly often (or, who could do so by applying the techniques mentioned above), casual dating can allow you to defer making a decision about someone you’ve met until you become confident that he or she is more compatible with you than other people you are likely to meet soon. If you’re the type of person that finds someone you really like only once every five years, then by all means jump into things right when you find someone, as the most important thing will be to not let them get away. But on the other hand, if a year of casual dating will tend to lead to 6 significant interests for you, deferring your decision for a while about the first potential person you meet (while at the same time getting to know them) can lead to better decisions. After all, without yet knowing anything else about someone new you are interested in, there is a 50% chance that the next romantic interest you have will actually be more compatible with you than that first person is. New information about a person can make this probability rise or fall. The main drawbacks of casual dating of course are that your potential partner may become frustrated that things aren’t moving along more quickly, and even if he or she doesn’t become frustrated, you may lose a little bit of utility by delaying for a while the enjoyment of a higher intensity relationship. These drawbacks need to be weighed against the potential benefits.

In conclusion, you do have the power to increase your expected total romantic happiness. It is up to you to take steps to live longer, become more selective about who you enter in a relationship with when lots of dating options are available, avoid forming attachments before you know a person well, increase your options by making an effort to meet people (much) more frequently than you are currently meeting them, hang out at places where people you like are more likely to be found, adjust your standards so that you aren’t eliminating dating partners based on criteria that don’t have much bearing on your likely future happiness with them, recognize that attraction can build over time, make that extra effort to talk to every attractive person at each event you go to, and take steps to become more attractive to others.

Derivation of the Romaximizer Equation