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.