The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy (Dover Books on Mathematics)

In this chapter we shall describe a method, called the pivot method, which is powerful enough to ferret out all solutions, and which is efficient enough to be practical; that is, it usually reaches the exact solution in a few steps. The method is more complicated—particularly to describe—than the methods discussed earlier, but we believe the carefu

The Theory of Games is a method of analyzing a conflict, according to the following abstraction: The conflict is a situation in which there are two sets of opposing interests; it may be regarded as a game between two players, each of whom represents one set of interests. Each player has a finite set of strategies from which he may, on any given pla

One should always look first for a saddle-point; the process is painless and concludes the work if there is one. Recall that we inspect each row to find its minimum, select the greatest of these, and then inspect the columns to find the maxima, selecting the least of these. If the two numbers (called, incidentally, the maxmin and minmax) are equal,

In zero-sum games the payoffs represent strictly an exchange of assets; one player wins the quantity that the other loses. We have compromised this principle somewhat in games played against Nature (used as examples here and there), where we have computed strategies for the personal player as if he were playing against an opponent who shared his va

However, one-person games (including Solitaire) may be regarded as a special kind of two-person game in which you are one of the players and Nature is the other.

This is one of the fundamental distinctions in Game Theory, namely, the number of persons—distinct sets of interests—that are present in the game. The form of analysis and the entire character of the situation depend on this number. There are three values, for the number of persons, which have special significance: one, two, and more-than-two.

The above two cases illustrate a fundamental distinction among games: It is important to know whether or not the sum of the payoffs, counting winnings as positive and losses as negative, to all players is zero. If it is, the game is known as a zero-sum game. If it is not, the game is known (mathematicians are not very imaginative at times) as a non

The average so found is called by mathematicians the expected value. It is evident that this is a use of language which requires special care in interpretation. We do not expect the value (in this case 6) to turn up when Blue and Red use the strategies which lead to this box—indeed, the payoff ‘6’ is actually impossible of occurrence in this box—bu

We have indicated that the number of persons involved is one of the important criteria for classifying and studying games, ‘person’ meaning a distinct set of interests. Another criterion has to do with the payoff: What happens at the end of the game?