Mathematics is being applied more and more frequently in the social and in the political sciences, mainly as a tool for making decisions. This often implies looking for optimal states and situations. In this chapter we explore some of the techniques used in optimalization.

The various units in this chapter are of different kinds. In section 5.1 we discuss zero-sum games, payoff matrices, and pure strategies at a level suited to grades 6-8. Mathematical models, in this case, help us to arrive at decisions.

In other sections mathematical analysis helps us to solve the problem completely: in section 5.2 written as a text for students, minimalization suggests a practical solution to a power-grid problem through graphs, spanning trees, and different minimizing algorithms; in section 5.3, the finding of maxima or minima without calculus, both at an elementary and an intermediate level, leads us also to a complete solution of the given problem.

Section 5.4 discusses the use of the least action principle for reflection and refraction in optics. The section on ‘least squares’ discusses ‘goodness of fit’, regression lines, and a geometric interpretation of the correlation coefficient. It ties in well with chapter 4 on statistics. We end this chapter with some remarks on mathematical models and decisions.

Games

We describe here an activity which we have found effective with children in grades 6-8. We start with a game between two players, Tom and Jerry. Each has a black checker and a red checker. At a signal each puts forward a closed hand containing a checker. Then they open their hands to show their choices.