__Pareto superiority, inferiority, and optimality, and
the Pareto frontier__.

[For the sake of simplicity, the following definitions apply only to outcomes. We will also extend the definitions in obvious ways to apply to lotteries over outcomes, evaluated on the basis of their expected utility.] Where a group of agents have available individual choices that could result in possible outcomes O1, O2, O3, . . . , On:

(1) One possible outcome Oi is __Pareto superior__ to
another possible outcome Oj just in case at least one member of the group
prefers Oi to Oj (i.e., for at least one member of the group, the utility
of Oi is greater than the utility of Oj) and no member of the group prefers
Oj to Oi (i.e., for no member of the group is the utility of Oj less than
the utility of Oi).

(2) Oj is __Pareto inferior__ to Oi just in case Oi
is Pareto superior to Oj.

(3) An outcome Oi is __Pareto optimal__ just in case
no other possible outcome is Pareto superior to it.

(4) The __Pareto frontier__ is the set of all of the
possible outcomes that are Pareto optimal.