[1] Each voter reports his entire preference relation over the set of alternatives.
The set of reports is called a preference profile.
A social choice rule maps the preference profile to the selected alternative.
relative to each of the other alternatives either improves or stays the same as in
[2] Maskin monotonicity is a necessary condition for implementability in Nash equilibrium.
Moreover, any social choice rule that satisfies Maskin monotonicity and another property called "no veto power" can be implemented in Nash equilibrium form if there are three or more voters.