1 Pareto Optimal
In welfare economics, a Pareto improvement formalizes the idea of outcomes being better in every possible way. An improvement is Pareto if at least one person in the system is better off without anybody else being worse off. A situation is called Pareto optimal (efficient) is all possible Pareto improvements have already been made, that is all win-win improvements have already been applied.
Formally, let \(X\) be the set of all feasible allocations and define utility functions \(u_{i}:X \to \mathbb{R}\) of agent \(i=1,\dots,n\). An allocation \(x^*\in X\) is Pareto optimal if there is no other allocation \(x \in X\) such that \(u_{i}(x)\geq u_{i}(x^*);\quad \forall i\) and \(u_{j}(x)>u_{j}(x^*)\) for at least one \(j\).
In other words, you cannot improve someones utility without lowering someone else’s.