Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*2. Historical Considerations.* A history of general

decision-making is an impossibility, but histories of

important decisions in law, military operations, busi-

ness, etc., is another matter, though none of our con-

cern here. Games on the other hand, as far as both

their origin and development is concerned, as well as

their scientific analysis, have a long and varied history.

The roots of games go back deep into the animal

kingdom and to primitive society. Even the oldest

known games of Homo sapiens are abstract creations

“Homo ludens.” Games are present in all civilizations,

not only in great varieties of form, but they also appear

in disguises such as in ceremonies, liturgies, diplomatic

customs, or war, the latter being especially visible

during the time of maintenance of expensive private

mercenary armies. In Roman Imperial times public

games were a great burden on the state. In modern

ages the money transactions, say, in the United King-

dom from football pools, exceed those of some of the

largest corporations.

Since games have always occupied man in a very

real sense it is curious that it was so long before games

became a subject of scientific inquiry, especially in

view of the dominating role of uncertainty in games.

But finally the fundamental notion of *probability* arose

from a study of games of chance and is a creation of

the sixteenth century, developed by Girolamo Cardano

(cf. Ore, 1953) from which time Galileo, Blaise Pascal,

Christiaan Huygens, the Bernoullis, Pierre Simon de

Laplace and many others of equal distinction have

extended our understanding of this basic concept. It

is still the subject of searching mathematical analysis

without which it is impossible for modern science even

to attempt to describe the physical or social world.

Probability theory, not to be discussed further here

though to be used in an essential manner, deals in spite

of its complexity and high mathematical sophistication

with a simpler specialized game situation than that

encountered in those games in which true strategic

situations occur. These are characterized by the simul-

taneous appearance of several independent but inter-

acting human agents each pursuing his own goal.

Probability theory first explained chances in particular

games. But philosophical questions were raised, notably

by Laplace. The relationships between those games and

situations similar to them, but transcending them in

their human significance were subjected to analysis.

While some issues were clarified it immediately be-

came clear that buried under the obvious there were

further questions which awaited formulation and an-

swer, not all of them posed or given to this day. The

application of probability theory to physics, by then

an actively developing abstract mathematical disci-

pline, had to wait until the second half of the nine-

teenth century. Though it originated from the study

of a social phenomenon, i.e., from games of chance,

the application to social events—except for actuarial

purposes (J. Bernoulli)—lagged behind that made to

physics and astronomy.

The need for a *theory* of those games for whose

outcome probability alone is not decisive was clearly

seen, apparently for the first time, by Leibniz (1710)

who stated: “Games combining chance and skill give

the best representation of human life, particularly of

military affairs and of the practice of medicine which

necessarily depend partly on skill and partly on

chance.” Later, in his letter of July 29, 1715 to de

Montmort he said... *Il serait à souhaiter qu'on eut
un cours entier des jeux, traités mathématiquement*

(“... it would be desirable to have a complete study

made of games, treated mathematically”). Leibniz also

foresaw the possibility of

*simulation*of real life situa-

tions by indicating that naval problems could be stud-

ied by moving appropriate units representing ships on

maneuver boards. The similarity of chess to some real

life situations is obvious and was noted for example

as early as 1360 by Jacobus de Cessolis, or in 1404

by Dirk van Delft who saw in that game a microcosm

of society. The ancient Chinese game

*wei-ch'i,*better

known by its Japanese name of

*go*was always inter-

preted as a mirror of complex, primarily military,

operations. Later many authors have referred to the

“game of politics,” “the game of the market,” or of

the stock-exchange, etc. But it is one thing to observe

some similarity and quite another to establish a rigor-

ous and workable theory.

In 1713 when James de Waldegrave analyzed the

game “le Her,” as quoted in a letter from Pierre

Remond de Montmort to Nicholas Bernoulli (Baumol

and Goldfeld, 1968), a very different step was taken.

This remarkable study anticipated a specific case of

what is now known as the (optimal) minimax strategy

concept (see Section 7, below) applied to a matrix game

without a saddle point. However this matter was

entirely forgotten or perhaps never understood, and

has only been unearthed recently. Thus de Waldegrave

had no influence; also his solution would have remained

singular since the mathematics of his time would not

have made it possible to prove a generalization of his

specific result.

It is a moot question whether mathematics could

have developed rapidly in the direction which the

theory of games of strategy has taken. The interest of

mathematicians was then dominated by the study of

analysis, stimulated by the concomitant and inseparable

development of mechanics. It can be even argued that

it is at any rate largely an accident that the human

mind turned early towards the formal science of math-

ematics and not towards, say, the intriguing task of

formalizing law in a similarly rigorous manner.

There is no known record of any deeper scientific

concern with games of strategy for about 200 years,

though various authors, including C. F. Gauss and

others, have from time to time studied certain com-

binatorial problems arising in chess (e.g., Gauss deter-

mined the minimum number of queens needed to con-

trol the entire chess board). M. Reiss (1858), who even

given an extensive mathematical treatment of a game

that is not strictly a chance game. But his is a game

of “solitaire” and as such was not of great consequence.

It seems that this work too was forgotten and without

influence. Among others E. Zermelo (1912) and E.

Lasker (1918) advanced the understanding of chess

mathematically and philosophically. In 1924-27 É.

Borel published papers on a certain two-person game,

for which he found an optimal method of playing, but

he expressed belief that it would not be possible to

arrive at a general theorem. Confirming the well-

known danger of making negative statements in sci-

ence, John von Neumann in his important paper of

1928, “Zur Theorie der Gesellschaftsspiele” (

*Mathe-*

matische Annalen,100), proved precisely what Borel

matische Annalen,

had thought to be impossible: a general theorem which

guarantees that there is always an optimal strategy

available for a player: the now famous fundamental

and widely influential minimax theorem (cf. 7 below).

This paper, though decisive, was again neglected,

though in 1938 J. Ville gave a simplified and more

general version of the proof of the minimax theorem.

In 1944 appeared the

*Theory of Games and Economic*

Behaviorby John von Neumann and Oskar Morgen-

Behavior

stern, a large and comprehensive work, which defi-

nitely established the field. Since then an immense,

steadily growing literature on games and decision the-

ory has arisen in many countries. The theory developed

by von Neumann and Morgenstern has been extended,

applied and modified, but its basic structure and con-

cepts sustain the new developments. Decision theory

in the narrower, principally statistical, sense had de-

veloped due to the pioneering work of A. Wald (1950).

The minimax theorem is of crucial importance also.

The newest modifications and extensions of either game

theory or statistical decision theory are manifold, and

some brief indications are found in the text below. The

history of the theory of games of strategy to 1944 is

found in Morgenstern, 1972.

Dictionary of the History of Ideas | ||