Economic definition of true love

Preamble

I originally tried to write an economic definition of true love for Valentine's Day in 2009 on a page entitled "Dating Ed". It became one of the most popular pages on my website, receiving hundreds of thousands of views, and I maintained it across several different wikis. The version below no longer includes information about dating me, as I'm now married, but does bring back some other material that was deleted over the years.

Definition of True Love

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) \wedge (j \succ_i h \quad \forall h \ne j), \quad h \in H \cup \{\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 \;\lor\; y \succsim x[/math]
2. Transitivity: [math]\forall x,y,z \in X: \quad \mbox{if}\; \; x \succsim y \;\wedge\; y \succsim z \;\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 \;\wedge\; 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.

The Age Rule

The defacto standard age rule is as 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.

I finally found a source to attribute this to: XKCD predates my posting significantly with its 'Standard Creepiness Rule'.

Random Love

An amusing exploration of Random Love was recently posted as XKCD Blog article No. 9.