{"id":270,"date":"2012-02-14T07:49:09","date_gmt":"2012-02-14T13:49:09","guid":{"rendered":"http:\/\/linesoftangency.wordpress.com\/?p=270"},"modified":"2012-02-14T07:49:09","modified_gmt":"2012-02-14T13:49:09","slug":"the-square-root-of-love","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=270","title":{"rendered":"The (square) Root of Love"},"content":{"rendered":"

All right, fellas, huddle up.\u00a0 We're going to talk about the best way to find true love.\u00a0 I mean, you can't just go running around all willy-nilly hoping to bump into somebody great.\u00a0 The world is a big place.\u00a0 You need a strategy<\/strong>, man.\u00a0 A dating plan of attack.<\/p>\n

First, some ground rules, some general observations about romantic life, and a few restrictions in the interest of mathematical well-behavedness:<\/p>\n

    \n
  1. You are only going to meet a finite number of datable women over the course of your lifetime.\u00a0 It will be a depressingly low number.<\/li>\n
  2. You are going to be an upstanding citizen and date only one woman at a time.<\/li>\n
  3. You will date a woman for some finite period of time, at which point you'll make a decision either to pull the trigger and propose, or cut her loose.\u00a0 Or, more likely, she'll dump you first.<\/li>\n
  4. Once you propose, no takesies-backsies.\u00a0 And once you cut a woman loose, you can't ever reconsider her for marriage; she will hate you forever.<\/li>\n
  5. You are able to perfectly rank the women you have dated according to a strict, unambiguous order of preference.\u00a0 Tie goes to the blonde.<\/li>\n
  6. You will encounter these women in random order.\u00a0 That is, you are completely ignorant of where the next potential wife will stand in the overall rankings.<\/li>\n
  7. You will date a certain number of women <\/em>without really considering any of them for a proposal.\u00a0 In other words, you'll take some time getting a feel for who's out there.\u00a0 Setting the bar.<\/li>\n<\/ol>\n

    In the world of mathematics, this is what's known as an optimal stopping problem<\/a>.\u00a0 You're going to date, and date, and date..., and stop.\u00a0 Hopefully on the woman of your dreams (hence the optimal part).\u00a0 In fact, this is one of those problems that's so famous it goes by several (mildly sexist) names: the secretary problem, the sultan's dowry problem, the fussy suitor problem.\u00a0 Because it's Valentine's Day, we'll call it the marriage problem<\/a>.<\/p>\n

    <\/p>\n

    Strategy One: The Soul Mate<\/h2>\n

    Out of all the <\/em>datable candidates you're ever going to meet, one<\/em> of them is going to be objectively best.\u00a0 For you economists out there, she will uniquely maximize your utility function (hubba hubba).\u00a0 You want her, and only her.\u00a0 No one else will do.\u00a0 How do you maximize the probability you'll end up together?\u00a0 When should you stop<\/strong> dating?<\/p>\n

    Notice that, according to our rules above, you will begin your search with a period of casual, commitment-phobic dating (which doesn't seem particularly unrealistic, actually).\u00a0 After this initial period, you will propose to the first woman who's better than all the ones before.\u00a0 Assuming, of course, she's still out there.<\/p>\n

    Suppose that you will meet n<\/em> datable women in your life, and that you'll see w<\/em> of those women casually as part of your initial feeling-out phase.\u00a0 Since you've already decided to marry the first woman that's better than any of the first w<\/em>, the question really boils down to, What is the best possible value of <\/em>w?\u00a0<\/em> How many women should you casually date before starting to think about marriage?<\/p>\n

    Now if w <\/em>= 0, you're going to propose to the very first woman who comes along, in which case the probability of marrying your soul mate is just the probability that she is the first woman you randomly meet, which is 1\/n<\/em>.\u00a0 That's not much of a strategy.\u00a0 By the same reasoning, if you wait too long and set w<\/em> = n<\/em> - 1, you're still only successful with probability 1\/n <\/em>(your soul mate just happens to be the last\u00a0<\/em>woman you randomly meet).\u00a0 Clearly the optimal value of w<\/em> occurs somewhere in between those two.\u00a0 But where?\u00a0 Make w<\/em> too small, and you risk marrying too early, make <\/em>it too large and you risk passing over your soul mate in the casual phase, which dooms you to a lonely death.<\/p>\n

    It turns out that this is one of those magical places where the number e<\/em><\/a> shows up out of the clear, blue sky.\u00a0 You <\/em><\/em>should (optimally) date n\/e<\/em><\/strong> of your<\/em> potential mates before looking for a wife, which maxes out your probability of successfully soul-mating at a whopping 1\/e<\/em><\/strong>, or about 37%.\u00a0 Not too shabby.<\/p>\n

    There's a problem with this number, though.\u00a0 For one thing, e<\/em> shows up because the harmonic series<\/a> makes an appearance in the probability calculations (which is almost romantic), and e<\/em> and the harmonic series are bound together in an intimate way. \u00a0Actually, e<\/em> shows up (as the base of the natural logarithm) alongside Euler's number<\/a> in an approximation<\/em> to the harmonic series.\u00a0 But the harmonic series diverges.\u00a0 And it diverges verrrrrry...verrrrrry...sloooooowly.\u00a0 In fact, even with n<\/em> as large as 1000, Euler's number is only accurate to two decimal places.\u00a0 Because of this, we need slightly different threshold numbers for small and large values of n<\/em>, and there's no clear line of demarcation.\u00a0 Besides, I'm not entirely convinced there are 1000 datable people in North America.<\/p>\n

    In addition to the mathematical objections, there is an economic one.\u00a0 In the Soul Mate model, the payoff for ending up with anyone besides your soul mate is zero<\/strong>, which is pretty harsh.\u00a0 Even if you don't end up with the absolute best possible wife, certainly you'll derive some nontrivial level of utility from ending up with a good wife, or a great one.\u00a0 With that in mind, we turn our attention to...<\/p>\n

    Strategy Two: Settling Down<\/h2>\n

    In this model, not only does every woman come with a lineal ranking, she also comes equipped with...let's call it a \"happiness value,\" i.e., the amount of happiness you would derive from marrying her.\u00a0 (Let's assume everybody has a positive happiness value, otherwise you wouldn't really consider her to be \"datable,\" would you?)\u00a0 Your soul mate is still out there, and she still has the highest happiness value, but now we acknowledge that other women are perfectly capable of making you happy as well, albeit to a lesser extent.<\/p>\n

    Notice that our focus has shifted a bit.\u00a0 Instead of trying to maximize the chances of meeting your one-and-only, now you're trying to maximize your expected happiness<\/strong>, which may or may not involve any one woman in particular.<\/p>\n

    As before, you're going to date w<\/em> women before the marriage search begins in earnest.\u00a0 And, as before, you'll propose to the first woman who's better than any of the first w<\/em>.\u00a0 If, however you don't run into any (you already passed over your soul mate), then you'll settle for the n<\/em>th woman; after all, in this model you're better off with anybody<\/em> (with a nonzero happiness value) than alone.\u00a0 How high should you set w<\/em> now?<\/p>\n

    The answer this time is \u221an<\/em>, rounded to the nearest integer.\u00a0 Good news if you're impatient: in general, \u221an<\/em> is going to be smaller than n\/e<\/em> (actually, as long as you're going to meet more than seven datable women in your life).\u00a0 Also good news if you're risk-averse: you trade some of the likelihood of finding your soul mate for the certainty that you won't end up alone and totally unhappy, which is closer to actual human behavior.\u00a0 Also<\/em> good news if you're mathematically fastidious: this solution isn't asymptotic; it holds for all n<\/em>, even small values.<\/p>\n

    In Which I Can Hear Your Objections Already<\/h2>\n

    Obviously these are very simplistic models with some glaring shortcomings.\u00a0 For one thing, we've been assuming tha<\/em>t it's possible to know the true value of n<\/em><\/em>, which is ridiculous.\u00a0 It turns out that, for an unknown number of potential mates, the marriage problem becomes fairly difficult and completely resistant to tidy, blogworthy solutions.\u00a0 There are more realistic, time-dependent models that presume, for instance, you have finite time<\/em> to marry one of the women who arrives at your doorstep, according to a Poisson distribution, of course...but that sounds too geeky to be heuristically useful, even for me.<\/p>\n

    For another thing, the rules I laid out in the beginning are pretty rigid and unrealistic.\u00a0 People routinely multi-date, get back together with exes, break engagements\/marriages, etc.\u00a0 And, while I'll bet most men go through an initial dating period without too much thought toward marriage, they probably don't wait to meet someone who's better than all<\/em> of the women they've dated before.\u00a0 Better than the last couple might be enough.<\/p>\n

    So I won't make any guarantees that this post will score you true love, but it makes for an interesting exercise.\u00a0 Try to estimate how many single, demographically and geographically appropriate women you're likely to meet during your lifetime, punch it into your calculator, hit the square root button, and pony up for some flowers.\u00a0 Oh, and iron your shirt.\u00a0 You look like a slob.<\/p>\n","protected":false},"excerpt":{"rendered":"

    All right, fellas, huddle up.\u00a0 We're going to talk about the best way to find true love.\u00a0 I mean, you can't just go running around all willy-nilly hoping to bump into somebody great.\u00a0 The world is a big place.\u00a0 You need a strategy, man.\u00a0 A dating plan of attack. First, some ground rules, some general […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[28,32,41],"class_list":["post-270","post","type-post","status-publish","format-standard","hentry","category-math-musing","tag-love","tag-math","tag-optimal-stopping"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/270","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=270"}],"version-history":[{"count":0,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/270\/revisions"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=270"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=270"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=270"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}