Chapter summary

In the opening chapter of this book, we use the well-known game of chess to illustrate the notions of strategy and winning strategy. We then prove one of the first results in game theory, due to John von Neumann: in the game of chess either White (the first mover) has a winning strategy, or Black (the second mover) has a winning strategy, or each player has a strategy guaranteeing at least a draw. This is an important and nontrivial result, especially in view of the fact that to date, it is not known which of the above three alternatives holds, let alone what the winning strategy is, if one exists.

In later chapters of the book, this result takes a more general form and is applied to a large class of games.

We begin with an exposition of the elementary ideas in noncooperative game theory, by analyzing the game of chess. Although the theory that we will develop in this chapter relates to that specific game, in later chapters it will be developed to apply to much more general situations.

Schematic description of the game

The game of chess is played by two players, traditionally referred to as White and Black. At the start of a match, each player has sixteen pieces arranged on the chessboard. White is granted the opening move, following which each player in turn moves pieces on the board, according to a set of fixed rules.