Lectures on the Theory of Games

By Harold W. Kuhn | Go to book overview

Chapter Three
Extensive Games

3.1 Some Preliminary Restrictions

By now the reader will have suspected that matrix games must play an important role in the theory of zero-sum two-person games and that they are much more inclusive than they may have appeared at first sight. Indeed, it may be asserted that all finite zero-sum two-person games can be reduced to matrix games. The critical reader may justly complain that this statement is meaningless since wehave never defined what constitutes a game. Furthermore, although he understands vaguely what is meant by the word “game,” his ideas on the subject are nonmathematical and give no clue as to how to verify the connection between these and the mathematical concept of a matrix game. The heart of the problem is close to this confusion; in fact, our assertion above was a non-mathematical assertion. To verify that it is true, we must provide an axiom system with its interpretation for finite games in general and show that this coincides, as an interpreted system, with the notion of a matrix game for zero-sum two-person games. We will be concerned only with the coincidence of certain aspects, indeed, exactly the strategic aspects which formed the basis of our definition of a solution. We have already solved this problem in the special instance of Simplified Poker; although the payoff matrix contained none of the verbal apparatus of the game given by its rules, we were convinced that it carried enough of the strategic possibilities to find a solution.

Therefore, our main problem in this chapter will be to give an axiom system with its interpretation which formalizes our intuitive notion of a game. The strict purist will not be satisfied with these axioms since they are framed in geometric language which could be axiomatized in turn. However, this is a mere quibble; we will make constant use of the intuitive insight given by the geometric model.

Throughout this chapter, passages that serve as motivation, interpretation, or heuristic discussion will be placed in square brackets […]. This is done to emphasize the independence of the mathematical deductions of these sections.

3.2 The Axiom System

[In hunting for a geometrical scheme that will represent the combinatorial possibilities of an arbitrary game adequately, one might proceed by imagining the various

-59-

If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this book

This book has been saved
Highlights (0)
Some of your highlights are legacy items.
Citations (0)
Some of your citations are legacy items.
Notes (0)
Bookmarks (0)

You have no saved items from this book

Project items include:
• Saved book/article
• Highlights
• Quotes/citations
• Notes
• Bookmarks
Notes

Cited page

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited page

Lectures on the Theory of Games

• Title Page iii
• Contents v
• Author's Note vii
• Preface ix
• Chapter One - What is the Theory of Games? 1
• Chapter Two - Matrix Games 5
• Chapter Three - Extensive Games 59
• Chapter Four - Infinite Games 81
• Index 105
Settings

Settings

Typeface
Text size Reset View mode
Search within

Look up

Look up a word

• Dictionary
• Thesaurus
Please submit a word or phrase above.

Why can't I print more than one page at a time?

Full screen
/ 107

Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.