We continue the study of constant-sum games by illustrating how to solve them if the payoff matrix is larger than 2 x 2.We derive the method of equalizing expectation to solve such games, Williams's method of oddments, and finally, we show how to solve any constant-sum game using linear programming. This provides us with a full proof of the minimax theorem. Also, using linear programming, we can prove the square subgame theorem, which states that the solution to any constant-sum game is the same as a solution to one of its subgames that has a square payoff matrix. We then illustrate how to use Microsoft Excel or Wolfram Mathematica to solve such games. In the final section of the chapter, we study variable-sum games and introduce the notions of payoff polygon and Pareto efficiency of an outcome. We show that not every such game has a universally accepted solution, so there is no analog of the minimax theorem for such games. In the 2 x 2 case, we show how to find a Nash equilibrium using mixed strategies if necessary (Nash proved that any game has one). However, the equlibrium point so obtained may not be Pareto efficient so may not be a good "solution" to the game.

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.