# Economic definition of true love

This page was originally posted in September 2011 with humorous intent. To my surprise it has now received almost 500 distinct hits, and almost 700 views. In the hope of appeasing my future audience, and just possibly getting a date out of it, I will now start adding actual content here.

## Current Availability

Ed is tentatively available for dating again at this time. Interested parties should (in no particular order):

• Be demonstrably female
• Be over the age of consent and subject the to standard rule: $\underline{Age} \ge \left(\frac{\overline{Age}}{2}\right)+7$
• Only use the work 'like' to express simile, agreeability or endearment
• Satify the requirements:
• $Height \ge \underline{Height}$
• $Weight \le \overline{Weight}$
• $|Heads| = 1\;$
• $|Tails| = 0\;$

Furthermore, it would be desireable if you:

• Do not actually need to demonstrate that you're female
• Are $\underline{Age}$
• Enjoying mocking the retards who misuse the word 'like'
• Are so tall that I consider imposing \overline{Height}
• Are not so thin that I consider imposing \underline{Weight}

It should be entirely unnecessary for me to suggest that you have at least one graduate degree with a basis in a mathematical discipline (i.e. math(s), econ, physics, engineering, etc.), as I would assume that you've stopped reading by now if you don't.

However, the above 'criteria' aside, if you genuinely believe:

$p\left(You \cap The\,One \ne \{\empty\}\,|\,First\,Glance\right) \gg 0$

## True Love

### Definition

Let $H$ denote the set of all entities (perhaps Humans, though we might also include dogs, cats and horses, according to historical precedent).

Let $T$ denote the set of pairs of individuals who have True Love, such that:

$\forall\{i,j\} \in T: \quad (i \succ_j h \quad \forall h \ne i) \and (j \succ_i h \quad \forall h \ne j), \quad h \in H \cap \{\emptyset\}$

Note that:

• The definition employs strict preferences. A polyamorous definition might allow weak preferences instead.
• The union with the empty set allows for people who would rather be alone (e.g. Liz Lemon/Tina Fey), provided that we allow a mild abuse of notation so that $\{\emptyset\} \succ_{i} h$.

### The Existence of True Love

Can we prove that $T \ne \{\emptyset\}$ ?

Rational preferences aren't sufficient to guarantee that $T \ne \{\emptyset\}$.

Proof:

Recall that a preference relation is rational if it is complete and transitive:

1. Completeness: $\forall x,y \in X: \quad x \succsim y \;\or\; y \succsim x$
2. Transitivity: $\forall x,y,z \in X: \quad \mbox{if}\; \; x \succsim y \;\and\; y \succsim x \;\mbox{then}\; x \succsim z$

Also recall the definition of the strict preference relation:

$x \succ y \quad \Leftrightarrow \quad x \succsim y \;\and\; y \not{\succsim} x$

Then suppose:

1. $\forall j \ne i \in H \quad i \succ_j h \quad \forall h\ne i \in H\quad\mbox{(Everyone loves Brad)}$
2. $\{\emptyset\} \succ_i h \quad \forall h \in H\quad\mbox{(Brad would rather be alone)}$

Then $T = \{\emptyset\}$ Q.E.D.

Adding the constraint that 'everybody loves somebody', or equivalently that:

$\forall i \in H \quad \exists h \in H \;\mbox{s.t. }\; h \succ_i \{\emptyset\}$

does not make rational preferences sufficient to guarantee that $T \ne \{\emptyset\}$.

Proof:

Suppose:

1. $\forall k \ne i,j \in H \quad i \succ_j h \quad \forall h\ne i,k \in H\quad\mbox{(Everyone, except Johnny, loves Brad)}$
2. $j \succ_i h \quad \forall h\ne j \in H\quad\mbox{(Brad loves Johnny)}$
3. $\exists h' \ne i,j \; \mbox{s.t.}\; h'\succ_j h \quad \forall h\ne h',i \in H\quad\mbox{(Johnny loves his wife)}$

Then $T = \{\emptyset\}$ Q.E.D.

Note: Objections to this proof on the grounds of the inclusion of Johnny Depp should be addressed to Matthew Rabin.