Gibbard-Satterthwaite theorem
How can the consequence $ x \in X] be efficient under the preference pair [$ \succsim \in \mathscr{D}_I ?
$ y \succsim_i x \quad \forall i \in I
$ y \succ_j x \quad \exists j \in I
is that there is no consequence $ y that satisfies
In other words, if any one person can improve without making anyone worse off, that is not efficient.
social choice function $ f: \mathscr{D}_i \to X , if the consequence $ f(\succsim)] is efficient under the preference [$ \succsim] for all preference pairs [$ \in \mathscr{D}_I $ f is said to be efficient. ---
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