*Originally posted by HuuBinh *

**I will mathematically prove that when a girl reduces her time being with a guy, the guy's interest level increases.**

Let say that the guy's interest level in the girl has a function:

IL = lnD

Where IL = interest level, a function of D, which is the number dating activities. This natural log function is concave which exibits diminishing marginal return of interest.

and he also derives happiness from other nondating activities:

O = lnN

Where O is other activities like sports, parties etc..., this function is also concave.

When we add these two types of utility (happinesss) together (dating & nondating) we have a utility function of:

U = lnD + lnN

This guy also faces a time constraint, which is also a function of dating & nondating activities (D&N), we have the function:

T = Dt + Nt

Where T = total amount of time devoted to dating and nondating.

For example: if he goes on 3 dates per week, each requires about 4 hours, then his dating time alone = 12hrs, excluding his nondating activities.

Individuals maximize their happiness subject to their constraints.

There, if we maximize U, and subject it to the time contraint T, to find the maximum number of dating activies and nondating activies, then we have:

MAX(D, N) s.t. (subject to) T

MAX(D,N) = lnD + lnN - (Dt + Nt - T)

then, i take the partial derivative with respect to D, and N.

dMAX/dD = 1/D - t = 0 ---> t = 1/D

dMAX/dN = 1/N - t = 0 ---> t = 1/N

divide: t/t = (1/D) / (1/N) ---> 1 = N/D

solve for D & N: we have a simple solution: D = N

The time constraint once again is: T = Dt + Nt,

I subsititute D in to this equation to solve for N, and the same for N to solve for D, to have:

N = T / (2t), this small t is the amount of time devoted to nondating activities and,

D = T / (2t), this small t is the amount of time devoted to dating activies;

Once again the utility function:

U = lnD + lnN,

Remember that, his romantic interest derives from dating alone which is the first part of the equation, lnD.

I then, substitute D & N into the ultiltiy function to have:

U = ln [ T / (2t) ] + ln [ T / (2t) ]

O.K. if the girl decides to reduce her time being with this guy, that means, the small t that indicates the amount of time he spends toward dating, must fall. If I isolate the interest level effect of this guy, i have:

IL = lnD

IL = ln [ T / (2t) ], if this (t) falls as a result of less time being with the girl, then according to the equation IL must RISE.

That means, he now has to shift some of his dating time toward his nondating activities. Lets use a numeric example.

In a given week, he has 20hrs to allocate between dating and nondating activities: T = 20

He goes on 2 dates per week each requires 5hrs, and 2 nondating activities each also requires 5hrs.

T = Dt + Nt ---> 2(5) + 2(5) = 20

Substitute these numbers into ultility equation, i have a total utility of:

U = lnD + lnN ---> ln [20 / (2*5) + ln [ 20 / (2*5) = 1.39 utils

*IL = lnD ---> 0.693

O = lnN ---> 0.693

Lets say now the girl still goes out with him 2 dates per week, but this time only 4 hours each instead of 5, thus dating time = 8 hrs.

Thus, he must now allocate that 2hrs extra toward his nondating activies.

Lets say that he still enjoys 2 nondating activities per week, but this time, 6hrs each.

We plug back into the equation to have.

T = Dt + Nt ---> 2(4) + 2(6) = 20hrs <--- still 20hrs per week

U = lnD + lnN ---> ln [20 / (2*4) + ln [ 20 / (2*6) = 1.43

If I isolate the the romantic interest effect alone, then:

IL = lnD ---> 0.916, this is higher than 0.693

This means, less time = more interest.

This result can also be generalized for guys, so SCARCE your time guys.

Comments are welcome!