Pseudo code algorithm for beauty contestant question
You are asking for the Pareto-optimal front in 3D space.  This is a well-studied problem.

In 2D, there are $O(n \log n)$ time algorithms: see, e.g., What are some interesting applications of the skyline problem?, An online algorithm to find the Pareto frontier elements, minimum subset of dominating 2D points.

In 3D, life is more complicated, but apparently it is still possible to solve this in $O(n \log n)$ using fancy methods.  See the following paper:

On Finding the Maxima of a Set of Vectors.  H.T. Kung, F. Luccio, F.P.Preparata.  Journal of the ACM, vol 22 no 4, Oct 1975, pp.469--476.