Difference between revisions of "Economic definition of true love"
imported>Harsh m (→Definition) |
|||
(5 intermediate revisions by the same user not shown) | |||
Line 1: | Line 1: | ||
− | == | + | ==Preamble== |
− | + | I originally tried to write an [[economic definition of true love]] for Valentine's Day in 2009 on a page entitled "Dating Ed". It became one of the most popular pages on my website, receiving hundreds of thousands of views, and I maintained it across several different wikis. The 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. | |
− | + | ==Definition of True Love== | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | ==True Love | ||
− | |||
− | |||
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>H</math> denote the set of all entities (perhaps Humans, though we might also include dogs, cats and horses, according to historical precedent). | ||
Line 17: | Line 9: | ||
Let <math>T</math> denote the set of pairs of individuals who have True Love, such that: | 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) \ | + | :<math>\forall\{i,j\} \in T: \quad (i \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: | Note that: | ||
Line 23: | Line 15: | ||
*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 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> ? | 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>. | Rational preferences aren't sufficient to guarantee that <math> T \ne \{\emptyset\}</math>. | ||
Line 34: | Line 26: | ||
Recall that a preference relation is rational if it is complete and transitive: | Recall that a preference relation is rational if it is complete and transitive: | ||
− | #Completeness: <math>\forall x,y \in X: \quad x \succsim y \;\ | + | #Completeness: <math>\forall x,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 \;\ | + | #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: | Also recall the definition of the strict preference relation: | ||
− | :<math>x \succ y \quad \Leftrightarrow \quad x \succsim y \;\ | + | :<math>x \succ y \quad \Leftrightarrow \quad x \succsim y \;\wedge\; y \not{\succsim} x</math> |
Then suppose: | Then suppose: | ||
Line 47: | Line 39: | ||
Then <math>T = \{\emptyset\}</math> Q.E.D. | Then <math>T = \{\emptyset\}</math> Q.E.D. | ||
− | + | ===The Pitt-Depp Addendum=== | |
Adding the constraint that 'everybody loves somebody', or equivalently that: | Adding the constraint that 'everybody loves somebody', or equivalently that: | ||
Line 64: | Line 56: | ||
Then <math>T = \{\emptyset\}</math> Q.E.D. | 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 [ | + | Note: Objections to this proof on the grounds of the inclusion of Johnny Depp should be addressed to [https://scholar.harvard.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]. |
Revision as of 20:00, 18 December 2020
Contents
Preamble
I originally tried to write an economic definition of true love for Valentine's Day in 2009 on a page entitled "Dating Ed". It became one of the most popular pages on my website, receiving hundreds of thousands of views, and I maintained it across several different wikis. The 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.
Definition of True Love
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) \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].
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 \;\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]\{\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:
- [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,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.
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 'Standard Creepiness Rule'.
Random Love
An amusing exploration of Random Love was recently posted as XKCD Blog article No. 9.