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That said, if you genuinely believe:==Definition of True Love==
then please stop by my office (F533) at the Haas School of Business ([httpThen suppose://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==#<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>#<math>\{\emptyset\} \succ_i h \quad \forall h \in H\quad\mbox{(Brad would rather be alone)}</math>
Please check back for updatesThen <math>T = \{\emptyset\}</math> Q.E.D.
===Definition===Adding the constraint that 'everybody loves somebody', or equivalently that:
Let :<math>\forall i \in H\quad \exists h \in H \;\mbox{s.t. }\; h \succ_i \{\emptyset\}</math> denote the set of all entities does not make rational preferences sufficient to guarantee that <math> T \ne \{\emptyset\}</math>. '''Proof''': Suppose:#<math>\forall k \ne i,j \in H \quad i \succ_j h \quad \forall h\ne i,k \in H\quad\mbox{(perhaps HumansEveryone, though we might also include dogsexcept Johnny, cats and horsesloves Brad)}</math>#<math>j \succ_i h \quad \forall h\ne j \in H\quad\mbox{(Brad loves Johnny)}</math>#<math>\exists h' \ne i,j \; \mbox{s.t.}\; h'\succ_j h \quad \forall h\ne h', according to historical precedenti \in H\quad\mbox{(Johnny loves his wife)}</math> Then <math>T = \{\emptyset\}</math> Q.E.D.
Let <math>T</math> denote Note: Objections to this proof on the set grounds of pairs the inclusion of individuals who have True Love, such thatJohnny Depp should be addressed to [https://scholar.harvard.edu/rabin/capital-montana Matthew Rabin].
:<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>==The Age Rule==
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 defacto standard age rule 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.follows:
===The Existance of True Love===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
Can we prove that :<math> T Age_i \ne ge \max \left\{\left(\frac{Age_j}{2}\right)+7\;,\;\underline{Age}\emptysetright\}</math> ?
Rational preferences arenI finally found a source to attribute this to: XKCD predates my posting significantly with its [http://xkcd.com/314/ 'Standard Creepiness Rule'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 \;\or\; y \succsim x</math>#Transitivity: <math>\forall x,y,z \in X: \quad \mbox{if}\; \quad x \succsim y \;\and\; y \succsim x \;\mbox{then}\; x \succsim z</math>==Random Love==
Also recall the definition An amusing exploration of the strict preference relationRandom Love was recently posted as [http::<math>x \succ y \quad \Leftrightarrow \quad \quad x \succsim y \;\and\; y \nsuccsim x</math>/what-if.xkcd.com/9/ XKCD Blog article No. 9].
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>p\leftforall\{i,j\} \in T: \quad (You i \cap 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>. ==TheExistence of True Love== Can we prove that <math> T \ne \,One {\emptyset\}</math> ? ===The Brad Pitt Problem=== Rational preferences aren't sufficient to guarantee that <math> T \ne \{\emptyemptyset\}</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 \;\lor\; y \succsim x</math>#Transitivity: <math>\forall x,Firsty,z \,Glancein 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 \right) not{\gg 0succsim} x</math>
==True Love=The Pitt-Depp Addendum===
where <math>\underline{Age} ====The Brad Pitt Problem====18 \;\mbox{if}\; Age_j \ge 18</math>, except in Utah.
