We introduce the notion of a mathematical game. We give examples and classify them into various types, such as two-person games vs. n-person games (where n > 2), and zero-sum vs. constant-sum vs. variable-sum games. We carefully delineate the assumptions under which we operate in game theory. We illustrate how two-person games can be described by payoff matrices or by game trees. Using examples, including an analysis of the Battle of the Bismarck Sea from World War II, we develop the notions of a strategy, dominant strategy, and Nash equilibrium point of a game. Specializing to constant-sum games, we show the equivalence between Nash equilibrium and saddle point of a payoff matrix. We then consider games where the payoff matrix has no saddle point and develop the notion of a mixed strategy, after a quick review of some basic probability notions. Finally, we introduce the minimax theorem, which states that all constant-sum games have an optimal solution, and give a novel proof of the theorem in case the payoff matrix is 2 x 2.

This chapter provides a basic introduction to matrices, including the following: scaling, transposing, adding, and subtracting matrices; multiplying matrices and applications; finding determinants and inverses of 2 x 2 matrices; and solving systems of equations by matrix inversion. Applications of matrix algebra, including applications to cryptography and to Leontief economic models, are discussed.A method for finding inverses of larger matrices using elimination is developed in a series of exercises.

In this chapter, we introduce linear programming. We start with a simple algebra problem that can be modeled as a system of equations. However, that problem has no solution, so we modify the model to make it more realistic. The result is a standard form linear programming problem. This shows how linear programming arises naturally from the modeling process. Next, we show how to solve these problems in the case when there are two decision variables by the standad graphical technique, which we call graphing in the decision space. This leads naturally to a discussion of the corner point theorem. Finally, we exhibit an alternate graphical approach to solving these problems in the case when there are just two constraints but an arbitrary number of decision variables. We call this method graphing in the constraint space. This technique is a feature of this text; it is not covered in most texts. It is based on linear combinations of vectors in a plane and basic solutions to systems of linear equations, and it sets the stage for the simplex algorithm in Chapter 5.

In this chapter, we vary the assumptions under which we play a game, so the chapter can be regarded as a sort of sensitvity analysis of game theory. We illustrate reverse induction to solve games of perfect information if play is sequential rather than simultaneous. We also consider changes such as what happens if the players are allowed to communicate with each other or if your opponent is indifferent (such as nature) as opposed to a rational player. We consider ordinal games (where the outcomes are just ranked in order of preference rather than having numerical payoffs).Here is where we cover the famous dilemmas of game theory, such as the prisoner's dilemma, and discuss applications to politics and international relations (the arms race, the Cuban Missile Crisis, and federal government shutdowns due to budget gaps). We discuss the theory of moves, proposed by Brams in 1994 as a way of making ordinal game models more realistic, and offer our own small adjustment to this theory. We conclude the chapter with brief mention of n-person games and discuss games in characeristic function form. Examples include legislative voting systems, where we introduce power indices.