Pareto Efficiency

Pareto Efficiency is a state where you cannot make one person better off without making another person worse off. We will define pareto efficiency when there are $n$ divisible items and $n$ agents.

def. Pareto Efficiency. Allocation $\overrightarrow{x_1},\dots ,\overrightarrow{x_n}$ is pareto efficiet where there exists no alternative allocation $\overrightarrow{y_1},\dots ,\overrightarrow{y_n}$ such that both:

$$\begin{array}{rlrl}& \exists i^{\ast }\text{ s.t. }& \overrightarrow{y_{i^{\ast }}}{⪰}_{i^{\ast }}\overrightarrow{x_{i^{\ast }}}& \text{ one agent benefits}\\ & \forall i,& \overrightarrow{y_i}{⪰}_i{x}_x& \text{others still not worse}\end{array}$$