Front Concept
Most lifelike problems involve several conflicting goals. For this reason, the concepts of Pareto fronts and archives have applications that range from economics to engineering. In Game Theory, we have these kinds of outcomes:
| Outcome | Description |
|---|---|
| Pareto efficient or Pareto optimal | No other outcome can increase the utility in one goal without decreasing the utility of any other goal |
| Pareto inefficient | There is another that can improve at least one goal without harming other goals |
| Pareto improvement over \(p\) | Better than the Pareto inefficient outcome \(p\) |
| Pareto dominated by \(p\) | Outcome \(p\) can improve at least one goal without harming other goals |
| Pareto dominated by \(p\) | Outcome \(p\) can improve at least one goal without harming other goals |
Although many outcomes can be Pareto optimal, no outcome dominates an outcome that is Pareto optimal. The set of all Pareto optimal outcomes is the Pareto front (also Pareto frontier, or Pareto set).
Example: Pareto front
This is a two-dimensional Pareto front. The region in gray is dominated by the front.
In this example, we consider lower values of \(f(x)\) to be a gain of utility
Formal Definition: Pareto front
The set \(P\) of all Pareto optimal outcomes, is defined as
where \(f_i(x)\) is the \(i\)-th goal in our problem
Every game has at least one outcome that is Pareto optimal.
The container pareto::front is an extension and an adapter of spatial containers for Pareto fronts. The container uses query predicates to find and erase any dominated solution whenever a new solution is inserted.