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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 ([http://maps.google.com/maps?msid=218233511539606995594.0004adfa2636c2d290827&msa=0&ll=37.872008,-122.252512&spn=0.011501,0.015535&t=m&z=16&vpsrc=6 map]) at your earliest convenience. ==Future Availability== Please check back for updates. ==Definition of True Love== ===Definition===
===The Existence of True Love===
====The Brad Pitt Problem====
====The Pitt-Depp Addendum====
Economic definition of true love (view source)
Revision as of 20:00, 18 December 2020
, 20:00, 18 December 2020→The Pitt-Depp Addendum
==Current AvailabilityPreamble==
Ioriginally tried to write an [[economic definition of true love]] for Valentine'm afraid that s Day in 2009 on a page entitled "Dating Ed is '''currently unavailable''' for dating at this time". Exceptions to this can be made if you have a Math(s) PhIt became one of the most popular pages on my website, receiving hundreds of thousands of views, and I maintained it across several different wikis.DThe 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.
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 wedge (j \succ_i h \quad \forall h \ne j), \quad h \in H \cap 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 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.
Can we prove that <math> T \ne \{\emptyset\}</math> ?
Rational preferences aren't sufficient to guarantee that <math> T \ne \{\emptyset\}</math>.
Recall that a preference relation is rational if it is complete and transitive:
#Completeness: <math>\forall x,y \in X: \quad x \succsim y \;\orlor\; y \succsim x</math>#Transitivity: <math>\forall x,y,z \in X: \quad \mbox{if}\; \; x \succsim y \;\andwedge\; y \succsim x 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 \;\andwedge\; y \not{\succsim} x</math>
Then suppose:
Then <math>T = \{\emptyset\}</math> Q.E.D.
Adding the constraint that 'everybody loves somebody', or equivalently that:
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 [httphttps://elsascholar.berkeleyharvard.edu/~rabin/ capital-montana 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 [http://xkcd.com/314/ 'Standard Creepiness Rule']. ==Random Love== An amusing exploration of Random Love was recently posted as [http://what-if.xkcd.com/9/ XKCD Blog article No. 9].
