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Economic definition of true love (view source)
Revision as of 21:21, 25 February 2012
, 21:21, 25 February 2012→The Existence of True Love
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.
Adding the constraint that 'everybody loves somebody', or equivalently that:
:<math>\forall i \in H \quad \exists h \ne i \in H \;\mbox{s.t. }\; h \succ_i \{\emptyset\}</math>
does not make rational preferences sufficient to guarantee that <math> T \ne \{\emptyset\}</math>.
Suppose:
#<math>\forall k \ne i,j \in H \quad i \succ_j h \quad \forall h\ne i,j,k \in H\quad\mbox{(Everyone, except Johnny, loves Brad)}</math>#<math>j \succ_i h \quad \forall h\ne i,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,j \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 [http://elsa.berkeley.edu/~rabin/ Matthew Rabin].
