Changes

==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 (i \succ_j h \quad \forall h \ne i) \wedge (You 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\cap }</math> ? ===TheBrad 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,One y \in X: \quad x \succsim y \;\lor\; y \succsim x</math>#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: #<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>\{\emptyemptyset\} \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:#<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>#<math>j \succ_i h \quad \forall h\ne j \in H\quad\mbox{(Brad loves Johnny)}</math>#<math>\exists h' \ne i,Firstj \; \mbox{s.t.}\; h'\succ_j h \quad \forall h\ne h',Glancei \rightin H\quad\mbox{(Johnny loves his wife) }</math> Then <math>T = \gg 0{\emptyset\}</math> Q.E.D.

then please stop by my office (F533) at Note: Objections to this proof on the Haas School grounds of Business (the inclusion of Johnny Depp should be addressed to [httphttps://mapsscholar.googleharvard.comedu/maps?msid=218233511539606995594.0004adfa2636c2d290827&msa=0&ll=37.872008,rabin/capital-122.252512&spn=0.011501,0.015535&t=m&z=16&vpsrc=6 mapmontana Matthew Rabin]) at your earliest convenience.

==Future AvailabilityThe Age Rule==

Let where <math>H\underline{Age} = 18 \;\mbox{if}\; Age_j \ge 18</math> denote the set of all entities (perhaps Humans, though we might also include dogs, cats and horses, according to historical precedent)except in Utah.

Let <math>T<I finally found a source to attribute this to: XKCD predates my posting significantly with its [http://xkcd.com/math> denote the set of pairs of individuals who have True Love, such that:314/ 'Standard Creepiness Rule'].

Note thatAn amusing exploration of Random Love was recently posted as [http:*The definition employs strict preferences//what-if. A polyamorous definition might allow weak preferences insteadxkcd.*The union with the empty set allows for people who would rather be alone (ecom/9/ XKCD Blog article No.g. Tiny Fey). This is not necessary with weak preferences as then we can allow <math> i \succsim_i i</math> without violating the definition of the preference relation9].