Posted by: Sameer Agarwal | June 26, 2009

Should I ask her out? It’s all about Maths!

Warning: This blog post is a direct consequence of extreme joblessness (If you haven’t noticed, I am a graduate now!) and reading a bit too many articles on arbit topics from maths to game theory to love and  statistics on my super awesome electronic book reader \m/. First of all, the mathematical reasoning applied here may all be flawed,  at best, I am pretty much only a math enthusiast and if MA-101 is/was one of your favorite courses, please don’t bleed your eyes out by reading this further!

Well, now let us get back to the topic at hand. A couple of days back, I was cribbing as to “Why I will Never Have a Girlfriend” after reading this thought provoking article. The author used pure maths, statistics and proper census data to conclude that there are exactly 18,726 non-committed girls who are beautiful, intelligent and might also like me (yeah, the last point is debatable, but a safe 50% chance of liking me is taken into consideration… this is my blog… sue me). Though this number doesn’t appear to be very small,  statistically the search for 1 in 18,726 would take 67 years to complete. Hence the title of the article was well justified.

However, just when I was trying to overcome my disappointment I got to know that the beautiful girls are not in a much good position either!  However, the analysis isn’t as simple as the guys. It involves Game Theory, Nash Equilibrium and the famous NP-complete Interactive Decision Problem! (Yes, women are _really_ hard to understand!). To give a brief idea, let me remind you of the movie The Beautiful Mind. It dealt with a situation as follows:  Suppose you are hanging out in a bar with a couple of your friends and there is a group of beautiful girls all of whom are brunettes except one who is a blonde. Now, all the boys would like to approach the blonde first, however this doesnt quite seem to be a good strategy. Here is how Nash argued:

If we all go for the blonde, we block each other and not a single one of us is going to get her. So then we go for her friends, but they will all give us the cold shoulder because nobody likes to be second choice. But what if no one goes to the blonde? We don’t get in each other’s way and we don’ t insult the other girls. That’s the only way we win“.

Things are turning interesting now! This means that the prettiest girl in a group will always be available for potential guys to ask. But there is a catch! Consider a guy who is attracted to our beautiful blonde girl Carol and would like to talk to her. Realizing that Carol is shy, he is also a bit skeptical about his decision. But there are 3 possible outcomes as highlighted by Dr. José-Manuel Rey:

  1. He talks to Carol and she responds in a friendly manner and he gets her phone number. (‘a’ reward points)
  2. He does not approach Carol. Instead enjoys doing something else little less rewarding. (‘b’ reward points)
  3. He talks to Carol and she proves uninterested and he feels miserable for a week. (‘0’ reward points)

Let us associate reward points with each of these actions. First outcome has reward points ‘a’, second has b reward points and third has zero reward points. Clearly, a > b > 0. Now, if there are N guys taking independent decisions, we will have $2^ N$ possible set of actions (aah, NP-complete!). However, for a particular guy, success means that guy approaches Carol while all others donot. By symmetry, let us assume that the proabability of each guy approaching Carol is p, so for our man to approach Carol, the award associated with it is $a(1-p)^{N-1}.$ (He approaches Carol, while other N-1 guys don’t!). Compared to this, the reward he gets doing something else is b. So, for so called pareto optimal solution/ Nash Equilibrium,

\[ a(1-p)^{N-1}=b. \]

i.e. \[  p^\ast = 1- \left(\frac{b}{a}\right)^{\frac{1}{N-1}}. \]

blond girl

This result is pretty interesting. Some of the few points that it highlights is:

1. As N increases, p tends to zero! Which implies no one will go for the most beautiful girl around as everyone will be too afraid to ask her out. Waah!

2. If you are a beautiful girl (with a large value of N) reading this , don’t despair, make a move with any one out of those N guys

3. If you are a guy reading this, don’t despair if the girl has a large fan following! In fact, the more the better are your chances!

4. If you are neither, then stop pointing out Mathematical flaws in this article and get a life you loser! (yeah, look who is talking)

PS: On a serious note, Game Theory is a really really interesting topic! If anyone of you is interested, I recommend an excellent book by Roger B. Myerson titled Game Theory: Analysis of Conflict.

PS 2: The analysis is still incomplete due to an error pointed out by an ingenious personality (read point 4). I am still not working on it because I don’t give a damn!

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Responses

  1. Haha, true to your word, you did end up writing a blog on this! btw, you should give me credit for the ‘experiments’ we did for the sake of data collection yesterday in the Mall 😀

    • sure sir! and for this blonde pic art too!

  2. Lol max… all of this doesn’t matter.. and that is y u will never have a girlfriend 🙂
    You are now the president of the losers group with a free lifetime membership 😉

  3. Guess what now, Mr. Sameer Agarwal.. I was just wondering the consequences of the same theory, in a special case as of yours. Let me explain it if you will (ya as if I care for your permission),

    As for Sameer Agarwal, oops forgot to mention our honhaar PG Medalist, would have it the award/return in case of “He does not approach Carol. Instead enjoys doing something else” (like John Nash) i.e b >> a.

    Well I have left the job half done, and would leave the rest for the able IITians and otherwise who read this blog, including myself (read LOSERS) to comprehend.

  4. WoO! WHAT in the hell was that!! now, if we weren’t from a place called IIT, we could have figured ALL that out without any fuss! lol..

  5. Ah, so now I know why you guys don’t ask out pretty girls like us. I thought it was due to recession 😛

  6. Kya baat hai! This discussion is about a beautiful girl and no weightage being given to the relative looks of her N admirers! As always, this articles is male-biased. Agar mujhme tere jaisa dimaag hota to i would have come up with a more gender-balanced article..lol!

  7. hehe, what/who is your inspiration in writing this blog?

  8. Well, I love it! Amazing grasp of concepts displayed creatively 🙂

    BUT the N guys’ decision may not be independent of each other.. Guy 1 may decide not to approach Girl coz she put down/laughed at Guy 2 who’s hotter/sexier.. bla bla! And so the probability of each guy would not be p.. but hey! I’m not single now, look me up later 😉

    • Wow, you are smart 🙂

      Nice point! Situations in real life no doubt have complex dependence on them. That is why Nash equilibrium safely talks about “Independent Choices”. . Perhaps Avinash, the _actual_ game theory researcher may comment something on this.

  9. all these facts further complement IR 1 of our institute……kya tha yeh be 😛

    • hehe, extreme vellapa makes you do things like this 🙂

  10. Nerd.

    And you told me yesterday that XKCD’s lost it.

    Nerd and hypocrite. (Plus, incomplete math.)

    • That is the whole point of this blog post! When I understood that XKCD post, I couldn’t control laughing out loud and decided that this definitely deserves a blog post! XKCD is indeed becoming 0.2% humor, but I am trying my best to keep up 😉

      Regarding the maths, I am still working on it!

      • Nerd.

  11. 🙂

  12. Sameer, I landed on your blog while randomly browsing. One quick glance confirmed that the owner of the keyboard that was used to type this post is a guy with interesting tastes. And I couldn’t resist commenting!

    At the risk of being accused of point (4), I am unable to make sense of the expression a.(1-p)^(N-1). It does not sufficiently capture the probability of our poor guy actually proposing to Carol (which is ‘p’). Shouldn’t it go like a.p.(1-p)^(N-1)?

    Wonderful post otherwise. Are you still an active blogger?

    • Hey Vikram, thanks for reading through the blog entry and ‘daring’ to question my (non-existent) mathematical skills 😛 These days most of my time is spent in writing reviews research papers in networks and systems (including the awesome Portland paper written by your group) but I hope to write a couple more posts during December 🙂

      Regarding you question, I think the intention here is of taking into account the conditional probability in calculating the expectation value of the reward. Think of it more like P(none of the other guys approached carol | our friend approached her) which is a.p.(1-p)^(N-1)/p = a.(1-p)^(N-1). I agree that I did a pretty lame job at explaining things above but then I thought that the purpose of this post is mainly to serve as a confidence booster for geeks!

  13. Hey Sameer, thanks for the elaborating. Yes, the equation makes sense for conditional probability.

    By the way, am glad that you took time to read our PortLand paper. In fact, I should mention that I landed on your blog not entirely randomly. I read your comment on http://www.mosharaf.com/blog/2009/09/22/portland-a-scalable-fault-tolerant-layer-2-data-center-network-fabric/, and then followed you with a Google search!

    Now, would you like to take a stab at calculating the probability of my landing at your blog, given that I have landed at Mosharaf’s post? 😉

    -Vikram

    • Haha sure, here is my take on it, do let me know if I am wrong:

      Given p1 is the probability of you landing at Mosharaf’s website and p2 be the probability of you googling me, and N being the average number of websites you visit everyday, and assuming the power laws in the Internet still hold true, and further taking into account the standard heavy-tail probability distribution behavior of how humans access web, the final probability of you landing on this blog should be WHY THE HELL SHOULD WE CARE 😀

      • Ha ha. Nice getting to know you anyway!

        Vikram

  14. Same here 🙂


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