Tuesday, October 23, 2007

You Know Just Another Mathematical Proof !

Credits : Someone out there.

Suppose that you can go out with some number of guys, n. Assume that after going out with any number r (1 ≤ rn) of the men, you can rank them from most preferable (rank 1) to least preferable (rank r). At any stage, you can either stop and commit to one man, or go on to the next one. Further, assume that once a guy is rejected you can never go back.

For i = 1, …, n, let U(i) be the utility of selecting the guy with rank i among all n guys. We shall assume that U(1) ≥ U(2) ≥ … ≥ U(n). Let the random variable X denote the rank of the man that is selected. The goal is to find a rule with maximizes E(U(X)).

For a = 1, …, r and r = 1, …, n, let U*(a,r) denote the expected utility of the optimal continuation when r guys have been inspected and the rth guy has been found to have a rank a among the r. Also, let U0(a,r) denote the expected utility if the rth man is selected, and dating is terminated. Since we fixed an n,

U*(a,n) = U0(a,n) = U(a)
Now consider the probability than a man with rank a among the first r actually has rank b among all n men:
( b – 1 ) × ( nb ) / (

n )
a – 1 ra

The rank b must lie between the bounds ab ≤ (nr + a). Therefore,

U0(a,r) = U(b) ( b – 1 ) × ( nb ) / ( n )
a – 1 ra r

Clearly, after inspecting r guys, the expected utility of inspecting one more and continuing optimally is

1/(r+1) × U*(b, r+1)

Call this expression Z. From this, we can see that U*(a,r) = max(U0(a,r),Z). The optimal procedure is to continue if U*(a,r) > U0(a,r), and to commit when U*(a,r) = U0(a,r)

Now, consider the choice of utility function. Assume a spherical cow. Also, assume that U(1) = 1, and U(b) = 0 for b = 2, …, n. Then U0(1,r) = r/n, and U0(a,r) = 0 a = 2, …, r. Note that this is a fair approximation for the case of a soulmate. Then U*(1,r) = r/n, and should be continued if U*(1,r) > r/n.

It then follows that the optimal procedure is to go out with 1/e of the guys, and then select the first one thereafter which has rank 1.

Now, if n isn’t fixed, utility can be maximized by maximizing n. I’m a guy. QED.

An alternate proof can be constructed by assuming we’re both Bayesian reasoners, that disagreements about priors are irrational, and that my priors are rational. The proof is left as an exercise to the reader.

Sunday, October 14, 2007

NOTHING 8: Of Buses, Dogs and the Universe!

Ahh my groupies it’s that time of the year when you squeal with joy and basically have a good belly laugh, it is NOTHING time and as promised the brew is a quirky mix of Nerd meets Standup Comedian.

Anyways without me vexing eloquent and straining my unexercised Literary muscles it is time to introduce two new characters, now some of you would be thinking that the cast has been changing very erratically like them shows on Zee and Star Plus let me assure you that I am not pulling a Ekta K (K)apoor on you poor glued to your computer screen souls, in fact I am thinking of an episode where all the characters Party, so there.

Our first new character is none other than Comet the Cosmo Dog (No reference to that motivational magazine that is extensively used at IITs) and the Bus Girl here are a few quick snatches of our conversation:


Comet the Cosmo Dog: Ooh, man - you scared the crap out of me. See? There it is.

Me: But I was just trying to make you wear a sweater, you know winters are coming

Comet the Cosmo Dog: You want me to wear a sweater, why don’t you try wearing a tail for a while!

Me: Dude first you get up every day 10 min before me and behave like an Alarm gone wild with no snooze button and then this, How Rude !

Comet the Cosmo Dog: Do you know I am getting fat?

Me: Yea so!

Comet the Cosmo Dog: Genius what that means is that you aren’t getting enough exercise.

Me: What!

Comet the Cosmo Dog: On another note did you know that there is now a Facebook application especially for Pet Dogs, and you haven’t made me a profile till now!

Me: What, how, when and how exactly do you know about it!

Comet the Cosmo Dog: Do you think I am just lying there while you are busy playing your Poke wars with Ami buahahahaha !


Bus Girl: Do you like animals?

Me: Yes in fact I have a pet dog name Comet.

Bus Girl: Do you remember the meeting that you had missed!

Me: err yeah!

Bus Girl: Do you know what that means.

Me: That I was really engrossed in my work and missed a meeting and am very sorry about that.

Bus Girl: Yes but it also means that you are insensitive, rude...[1], mean…. [2], careless….. [3], also do you remember the first words that came out of yer mouth when you talked to me?

Me: How in my living memories can I forget that Faux Pas! (In my defense you are too hot to be flicking those lustrous hairs and staring so pensively at the traffic)

Bus Girl: It means that you are very abrupt…[4]

Me: Why don’t you tell me something good about me, I haven’t had a good day.

Bus Girl: Ummm Ahhh errrrr heeh ( No she isn’t pulling a Meg Ryan on me(You know the one in Harry met Sally( Brackets within brackets (Sue Me!))))

Me: 8 bit Processor

[1] Look into my eyes !

[2] Look into my eyes !

[3] Say nice things about me please.

[4] Shit I can't hypnotize to save my life :(.