A Simple Love Model

Don't let A go

Life is about meeting wonderful people. Some obscure force makes us like some people more than others. How much liking is enough to pursue a relationship with someone ?

Abstract models step into the dangerous territory of assigning numbers to everything. Let’s assume that when you meet someone, there’s a number to how much you like that person.

Denote by as a0a_0 how much you like a person that you meet today, at time t=0t = 0

Say the maximum you can like someone is A. Ideally, you’d want to meet that person today, and start a relationship with them forever. The value of meeting A today is the pleasure you receive from liking them today, plus tomorrow, plus the next, etc.

The value of accepting a0a_0 today is then

V(a0)=∑t=0∞at V(a_0) = \sum_{t=0}^\infty a_t

Not everyone is that lucky though. Let’s say every day you meet a new person, and you must either start a relationship with them or reject them forever. No pressure!

We’d like to capture the fact that we’d prefer to meet A today than tomorrow. We’re all somewhat impatient, don’t like uncertainty, and why delay the feeling of love to the future?

Let's use β\beta to capture that future love is less valuable. So if β=0.5\beta = 0.5, we will multiply the love of tomorrow by 0.50.5, and that of the day after by 0.52=0.250.5^2 = 0.25, and so on.

This simplifies the value of accepting a relationship today to

V(a0)=∑t=0∞βtat=a01−βV(a_0) = \sum_{t=0}^\infty \beta^t a_t = \frac{a_0}{1-\beta}

Good news: there must be some amount of liking someone, above which we will accept a relationship with them. It’s obvious we’ll accept A, but what about A - 1 ? or A - 5 ?

The level of liking that causes indifference between accepting or rejecting we call the reservation, or r, partner.

The pleasure from liking r from today to infinity equals that from finding someone you’ll like more in the future - adjusting for the cost of uncertainty and having to wait for that person’s love.

Formally this means that

r1−β=β[F(r)r1−β+∫rAat+11−βdF(at+1)]\frac{r}{1-\beta} = \beta \left[ F(r) \frac{r}{1-\beta} + \int_r^A \frac{a_{t+1}}{1-\beta} dF(a_{t+1}) \right]

There’s one last mystery - for the mathy ones, it’s the F(at)F(a_{t}). If the minimum I can like someone is 0 and the maximum is A, what’s the probability of meeting tomorrow person 0 or A or any of the values in between ?

Imagine today i meet an 8 - what’s the probability of meeting someone higher than that ? how many people are there above 8 ?

If we assume the probability of meeting an 8 is the same of meeting 0 or of meeting A then we have a uniform distribution.

This simplifies the previous to

r1−β=β1−β[r2A+A2−r22A]\frac{r}{1-\beta} = \frac{\beta}{1-\beta} \left[ \frac{r^2}{A} + \frac{A^2 - r^2}{2A} \right]

With a bit of algebra we get

r=1−1−β2βAr = \frac{1 - \sqrt{1 - \beta^2}}{\beta} A

Imagine I value twice the love i get today than that from tomorrow - or that I’m kinda scared about uncertainty.

Then β=0.5\beta = 0.5 so r=1−1−0.520.5Ar = \frac{1 - \sqrt{1 - 0.5^2}}{0.5} A and if A=10A = 10 then r=3r = 3

So if I like someone above r>3r > 3, I must accept it! No need to overthink! Maths gave us the answer!

The true beauty about A is that it requires no calculation. When you meet it you know, and don't be stupid to let it go.