Economic definition of true love

Current Availability

Denoting Ed's availability [math]A[/math] at time [math]t[/math] as [math]A_t[/math],and defining [math]T[/math] as below, it appears that:

[math]\forall t \in \{now, \ldots, \infty\} \;\;A_t=0[/math]

because

[math]\{LB,EE\} \in T\; [/math]

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