'tis better to have loved and lost...

Today on the "solving optimisations for no good reason" show, we have the ever exciting topic of poems.

Now, I assume that the original author didn't consider the consequences of what he was writing, or the potential outcomes for all of human society in an age dominated by cliche of the words:

"Tis better to have loved and lost, than never to have loved at all."

Which goes on my list of things which don't make any sense; or rather, don't necessarily make any sense. The problem here is best analysed through some basic game-theory; convenient, really, as I don't know any other kind of game theory.



Assumptions:
Life is a two period model
There is one agent in the economy, entitled, "the love-struck fool" or, for simplicity, L.
All agents in the economy are rational.
In period one, the agent is in love.
In period two, the agent has 'lost'.

We expect our hero, L, to attempt to maximise his lifetime utility; that is, he wants the sum of his utilities to be as great as possible.

In period one, he has a utility function which can be derived from his level of consumption, his return on any capital invested previously, disutility from working and so on, and, of course, the utility associated with love (UL).

In period two, he has the same level of utility for working, capital income, and consumption. But whilst beforehand he was in the dizzy heights of love, he has now come to taste the bitter lows of a solitary existence, and so suffers from some disutility of (UM).

These functions can be normalised fairly easily, and I choose to assume that the two time periods are given equal weight by our maximising agent, so, in order for our condition that 'loving and losing is better than not loving at all', in our normalised equation;

UL - UM > 0
UL > UM

This seems not entirely unreasonable; the misery associated with a thing ending cannot possibly trump the thing existing in the first place...or can it?

First, there is a problem that the world is not a two period model. After whirlwind holiday romance, lasting, say, 2 week, L must return to his dreary job as a janitor (or whatever), knowing that he has lost his love;how long does it take for him to recover from this? 2 weeks certainly doesn't seem long enough, so perhaps there's a term of UM that lasts for a while, so that;

UM(t+1) = aUM(t) + e
(values in brackets denote subscripts here)

Obviously, this little model of mine lacks micro-foundations, but perhaps you could, given enough opportunity and evidence, calculate the exact nature of all of these things, at least on average?

The scary thought is, what happens if you are unable to reject the hypothesis on testing that a = 1 (or, more likely, that g=0, such that g = a-1); then misery becomes a random walk, and any shocks to your mood will be felt forever.

Anyway, have a nice day.