Economic definition of true love

This page was originally posted in September 2011 with humorous intent. To my surprise by mid-February 2012 it had received almost 500 distinct hits, and over 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. Though, most of it won't help you if you are actually interested in dating me. If that's the case, your best course of action is to email or reconsider.

Current Availability

Ed is tentatively available for dating again.

If you genuinely believe:

[math]p\left(You \,\cap\, The\;One \ne \{\empty\}\,|\,First\;Glance\right) \gg 0[/math]

then please stalk me at your earliest convenience.

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.

Rules of Dating

The Age Rule

I was unable to find a reference for the defacto standard age rule, but I believe it is a follows:

Denote two people [math]i\;[/math] and [math]j\;[/math] such that [math]Age_i \le Age_j[/math], then it is acceptable for them to date if and only if

[math]Age_i \ge \max \left\{\left(\frac{Age_j}{2}\right)+7\;,\;\underline{Age}\right\}[/math]

where [math]\underline{Age} = 18 \;\mbox{if}\; Age_j \ge 18[/math], except in Utah.

True Love

Definition

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

Let [math]T[/math] denote the set of pairs of individuals who have True Love, such that:

[math]\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\}[/math]

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 [math]\{\emptyset\} \succ_{i} h[/math].

The Existence of True Love

Can we prove that [math] T \ne \{\emptyset\}[/math] ?

The Brad Pitt Problem

Rational preferences aren't sufficient to guarantee that [math] T \ne \{\emptyset\}[/math].

Proof:

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

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

Also recall the definition of the strict preference relation:

[math]x \succ y \quad \Leftrightarrow \quad x \succsim y \;\and\; y \not{\succsim} x[/math]

Then suppose:

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

Then [math]T = \{\emptyset\}[/math] Q.E.D.

The Pitt-Depp Addendum

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

[math]\forall i \in H \quad \exists h \in H \;\mbox{s.t. }\; h \succ_i \{\emptyset\}[/math]

does not make rational preferences sufficient to guarantee that [math] T \ne \{\emptyset\}[/math].

Proof:

Suppose:

1. [math]\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)}[/math]
2. [math]j \succ_i h \quad \forall h\ne j \in H\quad\mbox{(Brad loves Johnny)}[/math]
3. [math]\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)}[/math]

Then [math]T = \{\emptyset\}[/math] Q.E.D.

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