[MINI] The Girlfriend Equation

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[MINI] The Girlfriend Equation

Economist Peter Backus put forward “The Girlfriend Equation” while working on his PhD - a probabilistic model attempting to estimate the likelihood of him finding a girlfriend. In this mini episode we explore the soundness of his model and also share some stories about how Linhda and Kyle met.

Backus’s equation is an homage to The Drake Equation and takes the form:

\(N\) = \(N_* \cdot f_w \cdot f_L \cdot f_A \cdot f_U \cdot f_B\)
= \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\)


\(N_*\) is the number of people living in the United Kingdoms as of the 2007 census
\(f_w\) is the percentage that are female.
\(f_L\) is the percentage living in London
\(f_A\) is the percentage of those between ages 24 and 34
\(f_U\) is the percentage that have a college degree
\(f_B\) is the percentage of women he’s attracted to

Ok, so 10k possible girlfriends sounds like plenty, right? Backus then adds some additional scalars as follows:

\(\bar{N}\) = \(N \cdot f_{B'} \cdot f_S \cdot f_C\)
= \(\cdot\) \(\cdot\) \(\cdot\)


\(f_{B'}\) is the percentage of matches that are attracted back to him.
\(f_S\) is the proportion of matches that are single.
\(f_C\) is the probability that the matching couple will get along.

Now we’re down to only 26 possible matches in the entire dense city in which the author lives. And that’s just few enough to seem depressingly small, and generate a fun analysis paper.

But is the good statistics?

We discuss that in this episode. The primary objection Kyle raises is mostly about the logic, and only partially about the values of the parameters choosen. Kyle raises some concerns about the lack of consideration of independence. For example, Backus assumes \(f_B\) and \(f_{B'}\) (the probabilities of each party being attracted to the other) are independent. This is not necessarily true. People tend to be attracted to one another at least in part, based on some common ground. Thus, given A is attracted to B, or…

\[Pr(B \textit{attractedTo} A|A attractedTo B) > Pr(B attractedTo A)\]

My calculation instead look like this:

\(\hat{N}\) = \(N_* \cdot f_w \cdot f_L \cdot f_A \cdot f_U \cdot f_{BB'} \cdot \hat{f_S} \cdot f_{C|BB'}\)
= \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\)


\(f_{BB'}\) is my estimate of the the percentage of successful random pairings.
\(\hat{f_S}\) is my estimate of the percentage of females matching previous criteria that will be single. I suspect it’s actually higher than this, but I’ll grant this one.
\(f_{C|BB'}\) is my estimate of the probability the couple will get along given they’re both attracted to one another.

Thus, 4k potential matches living in London. Combine this with the fact that people who are compatible are likely to more or less frequent the same places (not many teenagers on the shuffleboard court; not many octogenarians at punk rock shows), and the odds are not so bleak for the author.

Overall, I think this paper was fun novelty and I am glad Backus took the time to write it. Although I’ve offered some criticism of the work both in the podcast and here, my goal is not to knock down a cute paper, but rather to piggy back on it to illustrate a few more ideas that readers might benefit from. If nothing else, this work may get some readers thinking about probabilistic reasoning and introduce them to the Drake equation. Either way, that’s a win in my book.


The Girlfriend Equation

Linda: The data skeptic podcast is a weekly show featuring conversations about skepticism, critical thinking and data science.

Kyle: Welcome to another episode of the data skeptic podcast! I'm here with Linda as always for this mini-episode.

Linda: Hello I'm Linda.

Kyle: Linda I'm so excited because this episode kicks off a four week miniseries we're going to be doing all on the topic of love and data.

Linda: Whooh! I love love.

Kyle: I love data and love so do you want to expand yours maybe and include the other things that through it.

Linda: Ahaha, I'm just a normal simple folk haha so I just love love.

Kyle: Alriht. So yeah I'm excited because there's gonna be another episode we're gonna do on a similar theme and I have two great guests lined up. One is the senior data scientist from plenty of fish which is now my dating website, and the other is an author who wrote a great book about her experiences trying to both participate and manipulate the average that work on the show. So we're gonna do four week series here all about love and we're gonna kick it off with actually a usual suggestion. Linda you have looked at this right, The Girlfriend Equation?

Linda: Kyle sent me an email where there was a link to a pdf that had some guy in England or London I believe...

Kyle: That's right.

Linda: ...try to do some Math and calculate the probability that he could find a girlfriend by going out in London.

Kyle: That's right! And what do you think of this over-all?

Linda: So I look at it for 5 minutes or so and I never took a statistic class but I thought it was not accurate. The adds were too low.

Kyle: I'm gonna ask you why you think they were too low before we get into it let's walk through the process. This is essentially an "Oi mush" to the tricky question. Do you know what that is?

Linda: Nope.

Kyle: It's this really cool idea that Frank Drake put forward to trying to estimate the number of possible intelligent species that might be out in the rest of the universe trying to communicate with us so it's actually where our long episode or maybe long interview at some point so well skip that for now and say that he tried to follow the same logical process. So for example he starts with N which is the total population of the UK. Do you think that is a reasonable place to start?

Linda: Yeah. But you probably want to narrow down the population of London because he's just in London.

Kyle: We'll get there. The First term he multiplied by is the percentage of what you gonna call it, Great Britainers or UK-ers.

Linda: UK

Kyle: No, what's the term for one from that place?

Linda: Citizens.

Kyle: Alright, citizens in service to the queen are 51 percent female so he cuts that number down of about 60million to half of that. Then he cuts down that about 13 percent of that lives in London. So that seems reasonable so far?

Linda: Yeah, that's fine

Kyle: About twenty percent of that are women living in London between the ages of 24 and 34 because by the time he is writing this he was like 30 years old so he thought that was the reasonable age range. Do you have a problem with that?

Linda: That seems suspicious but let's assume he is correct.

Kyle: That's a reason because I mean you know there are people in a  happy relationships that aren't necessarily closed in age but generally speaking he gave a good range then he went on to say, "He presumed he would only end up being matched with someone who has a college degree of which 26 percent of people meet in the previous criteria are." How do you feel about that scaler?

Linda: 26 percent of people from the ages oblige

Kyle: Of female Londoners between 24 and 34, basically 1 and 4 such women have a college degree.

Linda: Really?

Kyle: Yeah

Linda: That seems low

Kyle: It seems low but it must be right

Linda: Especially in UK their tuition is incredibly cheap.

Kyle: Oh yeah?

Linda: Oh Yeah, they just...it's like a little tax or something each year or semester. I asked them. When I study abroad in England this is a long time though. That would be surprising.

Kyle: Yeah, that might be a little low but I'm gonna trust him on that just pronouncing maybe he got that right even let's say he doubled it 50 percent of Londoner female 24th to 34th have a college degree. That is just a factor of two so more or less I'll give now in. His next parameter, he multiplied them by the percentage of people whom he would be attracted to which he gives them 5 percent or 1 and 20.  Also I mean you won't narrow him we don't know what he's thinking but that sounds reasonable, right?

Linda: No, I think that's too low.

Kyle: Too low?

Linda: That is of the people on his age range

Kyle: Age range and college degree, so also sort of his social status.

Linda: Not so, he would just probably find more than 50 percent of women

Kyle: More than 50 come on, that would mean he has no taste at all. He will just like every other woman.

Linda: I don't think it take that much

Kyle: Oh come on, He have to chaste you for so long

Linda: Well that's a separate story

Kyle: I will maybe get to that one. But anyway if that's just to 5 percent, you'll not gonna date to anyone. So all that gets him down to 10,510 suitable matches for himÉ he then says, and this is where I think that's where his works is most to obvious.  ÒÓWell, 5 percent of those will be attracted back to me, 50 percent will be single and 10 percent will probably get along with me and me with them which get him down to like 26 possible candidates."

Linda: Yeah, now that's doubling too low. He think only 5 percent of those women will be attracted to him.

Kyle: yeah, right

Linda: Okay, I don't know what this person looks like. He is either the average, low average or above average so 5 percent to me sounds below average.

Kyle: You think so

Linda: Because I mean, most of the attraction is personality so if he is saying he's gonna have a good personality to attract someone. What is he saying? It's so odd

Kyle: So my main objection to that is that it's not necessarily his value but that he earlier multiplied by 5 percent of women he would be attracted to and then also tuck on 5 percent would be attracted back to him which I don't think is totally independent.

Linda: So, why would you not think their independent or you do think not independent.

Kyle: I do not. I think that given person A is attracted to Person B that is more likely that person B will also be attracted to Person A?

Linda: I don't knowÉ this is a great question for the maker of that act where is quite left and right

Kye: Thinder?

Linda: Thinder. How many people? What's the percentage of people whose life left and right on in each other?

Kyle: They'll hear in LA may be you'd get an engineer who thinder to come along as well

Linda: I would like to hear what they have to say.

Kyle: But he also says I object to this, that only 10 percent of all those math put into matches are those he get along with. That's again on top of only 5 percent he'll be attracted to and 5 percent will be attracted back to him. So that's kind a like saying out of a hundred people there's only 2.5 that will be mutually attracted.

"So quick editor's note here that is actually not right it's weak 2.5 percent so a quarter of a person or 1 & 400."

And that doesn't seem write to me. I think that it's not independent. And then, given we already say that we have two people mutually attracted, why would they will only have 1 and 10 percent of chances eating alone  with each other. What do you think?

Linda: I mean I don't know where this numbers are coming from. Because whether or not you get a long, some people, it's just you have to be able to change, be adaptable. I mean that's part of a relationship and so it seems odd that he goes well to date 10 women and 1 of 10 will stick as if it's he's almost saying it's random well it's not random.

Kyle: True. Yeah. Yeah.

Linda: Because I think if you went on a date they like each other the more likely to try even if it didn't work.

Kyle: Ah, good point. Maybe, he figured that in this  10 that even given effort only 1 in 10 will work out

Linda: I mean only 1 in 10 works out. I think there's something wrong with you.

Kyle: Haha

Linda: Well, I don't think this guy. He doesn't sound very likeable

Kyle: So in fairness...I don't know him personally the author of this is an economist, Peter Backus and also over all even though we're criticizing his work. I don't think he put this out there like it was the most rigorous proof. I think it's mostly for fun and you know, get some journalist to write about his other projects and stuff which is fine but I think my primary criticism is around how he doesn't consider the conditional probability that everything is independent but it sounds like you take most issues with this scaler values issues.

Linda: I don't know what the scaler, the numbers?

Kyle: Yeah, the numbers.

Linda: Yeah, I just don't know where they're coming from.

Kyle: In some cases I assume he went to like senses where he knew 51 percent.

Linda: Yeah but the like someone, like someone back, the chances how work out. I mean what his story? WhatÕs his statistic?

Kyle: Well, I think we need to get him in the show but I will agree that given the age group, given the college degree, given that his attracted to them, all these things put together is unlikely that it's only 5 percent would be attracted back and 10 percent of those would get along with him. So I think he's being pessimistic and all this. What do you think he's odds of É

Linda: Of meeting someone?!

Kyle: Yeah

Linda: So first of all he has to get out of his apartment probably.

Kyle: Okay, let's assume he does.

Linda: And I guess we assume the condition was that he asks someone on a date and she will say yes.

Kyle: Well I don't know I ask you once and you say, "No" and now we're married so I don't know

Linda: Yeah but that was separate, haha

Kyle: Okay

Linda: So we're saying lots of the odd that he'll ask someone out and they will say YES

Kyle: Aha

Linda: First of all he's gonna ask someone based he's not just gonna ask anyone so he's already self filtering

Kyle: That's true. Maybe even  self filtering strategically like you know someone I don't know, maybe top ranks, sern, senior scientist, he might not approach her because he's intimidated with them. He knows to land a better fish or something like that.

Linda: Yeah, I mean he's not if he sees someone married or has a partner, he's probably, "Oh I guess someone ask her out" so he is already setting up his chances of success by filtering them and then you know some people are open minded and will not and will say YES even they don't like you, as in they just don't know you.

Kyle: Well, IÕm hungry and broke so I guess this way can be.

Linda: No! Don't just agree to like have a conversation with them.

Kyle: Aha...

Linda: Coz I mean even you meet them in a bar, in a sidewalk or I don't really know what I think about you. I think people will be open-minded and say yes

Kyle: Oh yeah

Linda: For the first date

Kyle: Haha, His adds are a little better that he lets on in his paper especially when you're leaving in a densely populated area like London

Linda: Oh yeah, I mean I got in a subway during rush hour

Kyle: So if we gonna want to follow his adds, what do you think will gonna be the likelihood was in you know in a hindsight that you and I got married

Linda: At what point?

Kyle: Ah..

Linda: Are you assessing the adds?

Kyle: The vows I guess, I dont know.

Linda: Haha. I mean for all in the wedding day, the adds are pretty high that we'll say yes and get married as we planned the wedding.

Kyle: That's right, well maybe we should back off and add some colors and we just gonna talk about our meeting story.

Linda: Meeting??

Kyle: Yeah.

Linda: How we met?

Kyle: Yeah, what's your perspective on it?

Linda: Well, for viewers who don't know Kyle and I met at work at the company Christmas Party

Kyle: Aha

Linda: And Kyle was not dancing, haha.

Kyle: Right.

Linda: I didn't recognize him so I thought I would ask him a question.

Kyle: And we've never met before.

Linda: We've never met before and I was like, "Do you work here?" Because there are people who works there at the party and people who do not at our place of business. And he was like, "Ah yeah I work here" and I was like, "Are you lying?" and he was like, "huh". Because our workplace has 120 people at that time so I wasn't sure because you see you could remember 100 faces and I had not memorize his face so I was like, "I don't know you work there!" So that's how we met

Kyle: And seven days later wedding bells.

Linda: Nope.

Kyle: Haha.

Linda: Would you like to tell the rest?

Kyle: We didn't really start dating at first, we got to be really good friends, best friends even.

Linda: Yeah, we're good friendd for about 8months

Kyle: Aha.

Linda: Or eventually through 8months we became very good friends.

Kyle: Yeah, and then what happened?

Linda: Well I think this is your part to tell, haha, I think it sounds better coming from you.

Kyle: Well, I ask you if you wanted to go out and you said, "No".

Linda: That's what exactly happened.

Kyle: So the probability is zero for me.

Linda: Ahaha, so just a back track Kyle decide one day we're walking away from a bar which has a lot of coworkers, we went to a happy hours or something after work and Kyle is really excited and he seems like full of energy and he look at me and goes.. I can't remember exactly but I think you said to me like, "Will you go out with me on a date like I think it would be a good idea if we went out on a date?" And I was so kind of guard. It was like, "What?! No."

Kyle: Haha.

Linda: And Kyle take off his face and was like, "Oh Okay" and then you said something that how you didn't want to look weird and we continued. We able to continue being friends and then... several months later I developed a crush on him. And I think I have to climb on at first I said, "When my feelings changed I let you know" and so. And so I had a work of the nerve to tell him one day at work that I was like, "Well, maybe we can go on a date or something." I can't really remember my words maybe you can remember better.

Kyle: I just remember the unprofessional nature of it. Because when I do it, it wasn't on company time.

Linda: Haha. Yes, so then I was just like, "Yeah maybe we should go on a date". And we did, after that Kyle was still willing to go on a date with me.

Kyle: So thank goodness to that potion #9 I slept to the one that...

Linda: Well I always tell people it's not love at first sight.

Linda & Kyle: Ahahhaah.

Linda: Or even second, third, fourth, fifth or how many times we hang out off. You do not know who you're gonna marry sometimes.

Kyle: So maybe our associate through discussion Peter Backus who wrote "The Girlfriend Equation" will or perhaps has because it was an older paper but experienced similar fortunate circumstance for himself.

Linda: Yeah, and maybe he already knew the person that he was gonna marry later.

Kyle: Ah, so he's strategically turned down all these Londonites females?

Linda: Yeah, and when you'll gonna talk to him get a feel of what he's personality was, like who end up you marrying, are you married now and do you think your statistics are accurate?

Kyle: Yeah, maybe we'll get him into the show and we'll see what happened.

Linda: Because if he's married now, I don't think his adds were right. Haha.


Kyle: Well that's an interesting discussion that essentially the confirmation bias that will be the topic for another day.

Linda: I would like to hear this guy

Kyle: Well, thanks for joining me Linda.

Linda: Thank you for joining me.