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
of surprisingly high order, and justify the expression
“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
quoted Leibniz, is apparently the first author who has
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
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
Behavior by John von Neumann and Oskar Morgen-
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.
