We have also included a number of exercises and examples. These feature salient points connecting different topics and demonstrate the diverse mathematical techniques that arise in problems connected to the Shapley-Shubik power index. The references also include many citations of instructional materials and research materials that apply simple weighted-voting games to modeling of political institutions. In optimization problems, often students find a single extremum of a function that is assumed to be a maximum or a minimum. At best, saddle points are discarded when checking second order conditions en route to maxima or minima.
Game theory provides a setting where saddle points are the solution concept.
Paralleling how students learn to find extrema, we first consider finding saddle points on the interior of a set and then consider saddle-like points that appear on the boundary of compact sets. We conclude with a couple of examples from the game theory literature. The Microsoft Excel project for the Precalculus material is below. The Differential Equations project is a Mathematica file. It appears below, too.