# Difference between revisions of "Economic definition of true love"

## 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:
• $Height \ge \underline{Height}$
• $Weight \le \overline{Weight}$
• $|Heads| = 1\;$
• $|Tails| = 0\;$
• Have at least one graduate degree with a basis in a mathematical discipline

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.