**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:

- He talks to Carol and she responds in a friendly manner and he gets her phone number. (‘a’ reward points)
- He does not approach Carol. Instead enjoys doing something else little less rewarding. (‘b’ reward points)
- 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 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 (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,

i.e.

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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