# Pareto efficiency

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In game theory and economics, the concept of Pareto efficiency (or Pareto optimality) is a method to judge the efficiency of a set of decisions made by the participants. It was named after Vilfredo Pareto. Paretto efficiency is different from and should not be confused with Nash equilibrium.

A decision set (a combination of all decisions made by all participants) is called strong Pareto optimal if there is no other set in the entire decision space (all possible decision sets) in which at least one participant is strictly better off and no participant is worse off than he was as a result of the current decision set. A decision set is called weak Pareto optimal if there is no other set in the entire decision space in which every participant is strictly better off than he was as a result of the current decision set. While a strong Pareto optimal set is necessarily weak Pareto optimal, the converse is not necessarily true.

If there exist a decision set where at least one participant's outcome improves without anybody else's outcome worsening, the new decision set Pareto dominates the old set.

## When Pareto efficiency is not the equilibrium state - Prisoner's dilemna

 B A not confess confess not confess A: good / B: good A: bad / B: very good confess A: very good / B: bad A: OK / B: OK

In the one-stage game shown at the right side, prisoners A and B can concurrently choose between "not confess" and "confess". The result can either be "very good", "good", "OK", or "bad".

The green field ("not confess/not confess") is the Pareto optimal situation; the red field is the Nash equilibrium solution.

There exists no other decision set besides "not confess/not confess" that has an equal or better outcome for all participants. However, in "not confess/not confess" ("good" for A, "good" for B), A could switch to "confess". Assuming B does not change his decision, this improves A's result to "very good" while changing B's result to "bad". (There is no honor among thieves, so A is not concerned about B's welfare.) Knowing this, if B also changed his decision to "confess", B's result improves to "OK". However, A's result would then worsen, but only down to "OK" as well. Now at "confess/confess", if either A or B unilaterally changes to "not confess", his result worsens to "bad". Thus, "confess/confess", and not the Pareto optimum, would be the equilibrium outcome.

Generally in a game with finite steps, the equilibrium outcome may not necessarily be the Paretto efficient outcome.