Difference between revisions of "Economic definition of true love"
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*The union with the empty set allows for people who would rather be alone (e.g. Tiny Fey), provided that we allow a mild abuse of notation so that <math>i \succ_{\{\emptyset\}} h</math>. The inclusion of the empty set is not necessary with weak preferences as then we can allow <math> i \succsim_i i</math> without violating the definition of the preference relation. | *The union with the empty set allows for people who would rather be alone (e.g. Tiny Fey), provided that we allow a mild abuse of notation so that <math>i \succ_{\{\emptyset\}} h</math>. The inclusion of the empty set is not necessary with weak preferences as then we can allow <math> i \succsim_i i</math> without violating the definition of the preference relation. | ||
− | ===The | + | ===The Existence of True Love=== |
Can we prove that <math> T \ne \{\emptyset\}</math> ? | Can we prove that <math> T \ne \{\emptyset\}</math> ? |
Revision as of 16:53, 25 February 2012
Contents
Current Availability
I'm afraid that Ed is currently unavailable for dating at this time. Exceptions to this can be made if you have a Math(s) Ph.D.
That said, if you genuinely believe:
- [math]p\left(You \cap The\,One \ne \{\empty\}\,|\,First\,Glance\right) \gg 0[/math]
then please stop by my office (F533) at the Haas School of Business (map) at your earliest convenience.
Future Availability
Please check back for updates.
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,j) \and (j \succ_i h \quad \forall h \ne j,i), \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. Tiny Fey), provided that we allow a mild abuse of notation so that [math]i \succ_{\{\emptyset\}} h[/math]. The inclusion of the empty set is not necessary with weak preferences as then we can allow [math] i \succsim_i i[/math] without violating the definition of the preference relation.
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:
- Completeness: [math]\forall x,y \in X: \quad x \succsim y \;\or\; y \succsim x[/math]
- 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:
- [math]\forall j \ne i \in H \quad i \succ_j h \quad \forall h\ne i,j \in H\quad\mbox{(Everyone Loves Brad)}[/math]
- [math]\{\emptyset\} \succ_i h \quad \forall h\ne i \in H\quad\mbox{(Brad would rather be alone)}[/math]
Then [math]T = \{\emptyset\}[/math] Q.E.D.