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The Coming of Game Theory

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 GIANFRANCO GAMBARELLI and GUILLERMO OWEN

THE COMING OF GAME THEORY

We don’t control our lives the way a chess player controls his pieces, but life
is not roulette either.

(Ken Follett, 1985, p. 7)

ABSTRACT. This is a brief historical note on game theory. We cover its

historical roots (prior to its formal definition in 1944), and look at its
development until the late 1960’s.

KEY WORDS: Game theory, History

1. INTRODUCTION – THE FOUR PARADIGMS

We would like to think of game theory as the fourth in a line of

paradigms, as people learn to optimize results.
Originally, production was unnecessary. In Eden, Adam and
Eve received all that they needed without labor. It was only
after the fall that man had to earn his bread with the sweat of
his brow (Genesis, 3).

1.1. The First Paradigm

After the fall, as seen in the book of Genesis, there is a great
deal of productive activity. Cain is a farmer; Abel, a shepherd
(ch. 4). Noah builds a great ark (ch. 9). Jacob herds sheep for
his uncle Laban (ch. 29). Joseph builds great warehouses to
store the bounty of the seven fat years (ch. 41).
One thing we do not see in Genesis, however, is any attempt at
optimization. Cain and Abel do not use statistical analysis to
determine the best crops to plant or the best breeds to tend. Noah

Theory and Decision 56: 1–18, 2004.

2004 Kluwer Academic Publishers. Printed in the Netherlands.
2 GIANFRANCO GAMBARELLI AND GUILLERMO OWEN

does not attempt to determine an optimal size of ark (he builds

according to specifications given him by someone wiser). Joseph
does not use sophisticated inventory models for Pharaoh’s grain.
It could be that the author of Genesis is simply uninterested in
the mathematics of optimization; more likely, however, the idea
of optimization did not even occur to him and his people.

1.2. The Second Paradigm

A new phase arises as people realize that production – indeed,
most human activities – can be carried out more or less effi-
ciently. Some mathematics is needed for this: the development
of calculus was obviously the most important milestone for
optimization, followed closely (in importance) by linear pro-
gramming, but some attempts at optimization were made even
before this. (For example, statistics – numerical summaries of
data relating to matters of importance to the state – were col-
lected to determine the most productive activities, as an aid to
the state’s decision-makers.)

1.3. The Third Paradigm

Note that, in optimization, decision-makers generally thought of
themselves as working with (or against) a neutral nature. Even-
tually, however, they realized that their actions influenced the
decisions of others who were themselves trying to optimize
something – pleasure, utility, etc. Payoffs were a function of both
actions. As an example, the demand curve was studied: tradesmen
set their prices so as to maximize profits, taking into account both
their own costs and potential customers’ willingness to pay. As
another, police departments choose optimal levels of patrolling,
so as to minimize some combination of their own costs and the
damages caused by speeding motorists – knowing that motorists’
propensity to speed will depend on the level of patrolling.
In this third paradigm, it is necessary to study other indi-
viduals’ desires (utility). For example, polling data is of great
importance for marketing purposes.
Note, however, that the decision-maker does not (in this
paradigm) consider the possibility of strategic action by the
 THE COMING OF GAME THEORY 3

others. Thus, the monopolist does not consider the possibility

that consumers might form a buyers’ cooperative to bargain for
lower prices or to boycott him. The highway patrol does not
consider the possibility that motorists might band together
against traffic enforcers, e.g., by signalling the patrolmen’s
locations to each other.

1.4. The Fourth Paradigm

Finally, man realizes that he is interacting, and in competition,
with other individuals who are, themselves, aware of this. Thus
he must
(a) outsmart others,
(b) learn from others’ behaviour,
(c) cooperate with others,
(d) bargain with others.

2. FOREWORD

The modern Game Theory as Interactive Decision-Making is

universally recognized as originating in 1944, with the publi-
cation of von Neumann and Morgenstern’s The Theory of
Games and Economic Behavior. Earlier studies (even those by
von Neumann) had not been introduced in the context of a
precise science, which the publication of the above book cre-
ated. Nevertheless, there were indeed earlier studies. We will
thus rather arbitrarily look at the foundations of game theory
by studying three general periods: Prehistory – anything up to
1900. Ancient history – 1900 to 1944. Early history – 1944 to
1969. Anything since 1970 will be considered recent history,
and thus will not be treated here (though we reserve the right to
continue with a subsequent article).

2.1. Prehistory
The minimax problem (i.e., the existence of equilibrium strat-
egies minimizing the maximum expected loss for each player)
dates back to the early 18th Century. At that time, the concept
4 GIANFRANCO GAMBARELLI AND GUILLERMO OWEN

was used by James Waldegrave and Pierre-Rémond de

Montmort to analyze the card game Le Her (see (Baumol and
Goldfeld, 1968)). Waldegrave noticed that the game could
essentially be reduced to a 2 · 2 matrix (all other strategies
were dominated), noticed a lack of equilibrium here, and
talked about the probability that either player would make the
correct guess, but did not actually give a solution in terms of
mixed strategies. Later Nicolas and Bernoulli, while studying
Le Her, introduced the concept of expected utility and dem-
onstrated its potential applications in Economics (1738). Still
in the 18th Century, a preliminary to cooperative game theory
appeared in the form of the first known power index, which
was developed by Luther Martin in the 1780’s (see Riker (1986)
for this).
In the 19th Century, we see the general idea of equilibrium
strategies emerging, especially in the work of Cournot (1838),
who studied the equilibria of an oligopolistic production game.
Only pure strategies were considered, but because of the form
of the profit function, such equilibria existed in many cases.
Edgeworth (1881) gave a strong impetus to the development
of bargaining theory with his introduction of the contract
curve. Essentially, he was introducing, in a two-person context,
the two important ideas of individual rationality and Pareto-
optimality.
In a less academic setting, the general use of interactive
strategies can be found in the work of Poe (1845) and Doyle
(1891). These two giants of literature both considered a duel of
wits between the protagonist (a detective) and an intelligent
antagonist, though they lacked the concept of mixed strategies
as solutions to the games they considered.

2.2. Ancient History

As the 19th Century turned into the 20th, parlor games gave
further impulse to the theory. Bertrand (1924), for example, in
1899 studied a simplified form of baccarat in which he
emphasized psychological as well as mathematical aspects of
the problem. Zermelo proved in 1913 that finite two-person
 THE COMING OF GAME THEORY 5

zero-sum games of perfect information (such as chess, checkers,

etc.) are strictly determined. Steinhaus studied in 1925 a class of
problems, ranging from card games to limited-move chess
games, to pursuit. Although significant results were not
achieved (due to the considerable combinatorial analysis and
differential calculus problems encountered), the work on pur-
suit was an important precursor of dynamic games.
In the early 20th Century, new concepts were developed.
Émile Borel is credited with the first modern formulation of a
mixed strategy, in 1921, and with finding the minimax solution
for some classes of two-person games in 1924 and in 1927.
However, the correct formulation and proof of the general
theorem is attributed to von Neumann in his work of (1928). A
simpler proof was developed by von Neumann in (1937) using
the Brouwer fixed point theorem. Later proofs were based on
separation theorems. Among the first of these was one by
Borel’s student, Ville, in 1938; another was by Weyl, in 1950,
with a precise method based on one of his previous works on
convex polyhedra (1935).
Numerous other works could be cited, e.g. some military
strategy from past centuries.

2.3. Early History: Princeton

In the 1930’s and 1940’s, Princeton had become a living science
museum. Here grand old men met, just as in a gentlemen’s club,
leaving youth to conquer the world. But the minds of the old
scientists resonated and the fundamentals that would rule our
century were born: new theories in physics, computer science,
mathematics, economics. In this greenhouse, ideas germinated,
exploiting the synergies of experts from different fields, and
from this, modern Game Theory emerged. The Theory of
Games and Economic Behavior was in fact the collaboration of
the mathematician John von Neumann and the economist
Morgenstern (1944). Morgenstern recalled nostalgically in 1976
how they occasionally spent evenings together in the company
of men such as Einstein, Bohr, and Weyl, and how the ideas for
their book matured.
6 GIANFRANCO GAMBARELLI AND GUILLERMO OWEN

3. ORIGINS AND FIRST APPLICATIONS TO OTHER SCIENCES

The real beginnings of Modern Game Theory can be dated

from 1944 for two reasons: first, previous works were frag-
mentary and lacked organization; second, these works did not
attract much attention. With the publication of von Neumann
and Morgenstern’s book, the Theory of Games had its own
concrete organization of fundamental topics at both competi-
tive and cooperative levels. Furthermore, the reputation of the
two authors attracted the attention of both mathematicians and
economists.
It is fashionable nowadays to say that 1944’s The Theory of
Games and Economic Behavior was merely a recompilation of
previous work and contained no new results. In fact, while no
new, profound theorems are proved in this book, there are some
very important developments. First, the introduction of infor-
mation sets led to a formal definition of strategy, and thus al-
lowed for the reduction from the extensive to the normal form of
the game. In dealing with cooperative games, the treatment of
coalitions introduced the characteristic form (admittedly a
misnomer) and gave a formal definition to the very important
concept of imputation. The introduction of von Neumann–
Morgenstern solutions led down a blind alley, but nevertheless
included the idea of domination, which would eventually lead to
the important concept of the core. The second edition of TGEB
(1947) gave a strict development of utility theory.
Thus game theory can really be said to have started – as a
formal science – in 1944. The same is true for the computer, which
officially appeared in 1944, although the MARK1 was a per-
fected version of Howard Aiken’s ASCC, which dated back to
the previous year but was more or less unknown to the general
public, because of military secrecy. Here, too, there are numerous
precedents from Hollerith to Pascal and so on, back to the first
abacus made from ‘‘calculi’’, i.e., stones aligned on the ground.
The coincidence between the beginnings of computers and
modern game theory is not due to pure chance, as the genius of
von Neumann was a determining factor in the invention of the
computer, too. Perhaps if he had not met the English logician,
 THE COMING OF GAME THEORY 7

Alan Turing (a meeting which is even today shrouded in mys-

tery due to war department secrecy), the first computer would
have been developed in Germany by Friedrich Zuse, and the
course of history would have been changed.
The links between games and computer science were very
close at first, especially since both were used for military pur-
poses. Proof exists of the American use of games in wartime by
the Operations Evaluation Group of the U.S. Navy and by the
Statistical Research Group of the U.S. Air Force (see (Rees
1980)) and in the post-war period by the Rand Project. Subse-
quently, games and computer science continued to develop to-
gether, giving rise to very important synergies. In fact, the first
studies in linear programming were developed by George
Dantzig in order to solve problems in two-person game theory
(see (Albers and Alexanderson, 1985)). Later, games required
the application of general mathematics, and this gave the
stimulus for research in other fields: fixed points and convex sets
for extending the minimax theorem; duality and combinatorics
for mathematical programming related to matrix games; and in
general topology, probability, statistics, and theory of sets.
The next section deals with the major contributions of Game
Theory to the field of economics, which did not occur until much
later. Although there were many such well-founded contribu-
tions already in the founders’ book, economists were slow to
appreciate their importance. We will therefore only examine
Morgenstern’s principal new contributions by quoting Andrew
Schotter (who is to be considered Morgenstern’s last student):
Three major things are accomplished. First, the problem for economic sci-
ence is shifted from a neoclassical world composed of myriad individual
Robinson Crusoes existing in isolation and facing fixed parameters against
which to maximize, to one of a society of many individuals, each of whose
decisions matters. The problem is not how Robinson Crusoe acts when he is
shipwrecked, but rather how he acts once Friday arrives. This change of
metaphor was a totally new departure for economics, one not appreciated
for many years.

Second, the entire issue of cardinal utility is discussed … An attempt is

made to keep the axioms as close as possible to those needed to prove the
existence of an ordinal utility function under certainty …
8 GIANFRANCO GAMBARELLI AND GUILLERMO OWEN

Finally, the entire process of modeling exchange as an n-person cooperative

game and searching for a ‘‘solution’’ is described … While the neoclassical
theory of price formation was calculus-based and relied on first-order
conditions to define equilibrium, game theory, especially cooperative game
theory, relied more on solving systems of inequalities … While the neo-
classical solution would often be included within the set of cooperative
solutions … the theory of games offered new and appealing other solutions
to the problem of exchange.

(Andrew Schotter, 1992, p. 98)

Thus, beyond the military applications, which did nothing but

hamper the early development of the theory (as seen in (Mi-
rowski, 1991)), early interest came from mathematics (where
experts were able to find new problems and new mathematical
applications) and from economics (where the new models
seemed set to completely revolutionize current theory). This
caused controversy between the ‘‘classicists’’ and the modern
school of thought (see Theocharis (1983) and Arrow and In-
triligator (1981)).
Furthermore, there was an important change in mathemat-
ical theory. Existing quantitative models in economics had links
to physics (do not forget that applied mathematical principles
at that time belonged to the world of physics, and economists
interested in quantitative methods naturally relied on them).
Even Leon Walras and Vilfredo Pareto had engineering back-
grounds. This bond was not, however, without its drawbacks,
as Giorgio Szegö claimed:
About 50 years ago it was finally recognized that economic phenomena had
certain characteristics that were totally different from those of the physical
world, which made them in certain cases completely unsuitable for
descriptions of a mechanical nature. This is because a single economic agent
behaves not only according to past and present values of certain variables
but, contrary to what happens in the physical world, also according to his or
her own (possibly not rationally justifiable) expectations about the future
values of these quantities. … Contrary to the situation in mechanics, no
invariant law of conservation or variational principle seems to hold for
economic systems.

(Giorgio Szegö, 1982, p. 3)

 THE COMING OF GAME THEORY 9

The new approach by von Neumann and Morgenstern, there-

fore, opened up new horizons. But, contrary to expectations,
the first decade of these new models was difficult to digest for
the economists, who were unable at that time to understand
them in depth. There was therefore no great revolution in the
years following 1944 and the classical economists dismissed this
new science as nothing more than a passing fad.

4. THE MATHEMATICAL REVOLUTION OF THE 1950’S AND

1960’S

At the middle of the 20th Century, then, the science of game

theory had been founded. It was, however, a science still in its
infancy. The minimax theorem told us that optimal mixed
strategies existed, but, except for 2 · 2 games and a few other
special cases, no efficient methods existed for their computa-
tion. Non-zero-sum games were effectively ignored. For
n-person games, only a very strange solution concept existed.
Further, von Neumann and Morgenstern had made two very
strong assumptions, namely, transferable utility (linear side
payments) and complete information (full knowledge of the
rules and of other players’ utility), which, while useful in sim-
plifying the problems studied, narrowed the scope of the the-
ory. Moreover, for all but very small (extensive form) games,
the normal form of the game tended to be enormous. (As an
example of this, the normal form for such a simple game as tic-
tac-toe, even after all symmetries are used to reduce size, is a
matrix with over 1000 rows and columns.)
In the first of a series of volumes of papers, Contributions to
the Theory of Games, Kuhn and Tucker (1950) listed 14 out-
standing problems. These included:

(1)1 To find a computational technique of general applicability

for finite zero- sum two-person games with large numbers
of pure strategies …
(7)1 To establish the existence of a solution (in the sense of von
Neumann–Morgenstern) of an arbitrary n-person game …
10 GIANFRANCO GAMBARELLI AND GUILLERMO OWEN

(10) To ascribe a formal ‘‘value’’ to an arbitrary n-person

game …
(12) To develop a comprehensive theory of games in extensive
form with which to analyze the role of information …
(13) To develop a dynamic theory of games …
(14) How does one characterize and find the solution of games
in which each player wishes to maximize some non-linear
utility function of the payoff.

As it happened, the first of these problems was soon solved,

with two different approaches. On the one hand, Robinson
(1951) proved the convergence of the method of fictitious play.
On the other, the close relationship between two-person zero-
sum games and linear programming meant that the very pow-
erful simplex method developed by George Dantzig (see, on this
matter, Koopmans et al. (1951)) could be used for computation
of optimal strategies.
As the 1950’s continued, a major change in game theory,
principally theoretical in nature, took place. The young
Princeton mathematicians played a leading role. Foremost
among these were Nash and Shapley, though many others
contributed.
Nash wrote in (1950a) and (1951) an in-depth general treat-
ment on the notions of equilibrium for non-cooperative games.
In this way he took the theory beyond the limits of constant-sum
games (on which his predecessors had concentrated) and gen-
eralized both the minimax concept (to non-zero sum games) and
the results obtained in (1838) by Cournot (to general n-person
games). His work on the minimax concept provided a new
interpretation of the problem of players’ expectations that led to
a far-reaching discussion on refinements of equilibria which is no
less interesting even today. Nash (1950b) also developed the
basic methodology for the analysis of bargaining, and in 1953 a
no less important concept for the solution of cooperative games
with non-transferable utility.
The core was developed by Donald Gillies in his Ph.D. thesis
at the Department of Mathematics (1953). This formed the
basis for fundamental developments in economics, being as it
 THE COMING OF GAME THEORY 11

was a generalization of Edgeworth’s (1881) contract curve for

the n-person case.
The value concept was introduced by Shapley (1953a) to
solve two fundamental problems that until then had effectively
stopped the development of the n-person cooperative models:
existence and uniqueness. Up until then there had been fruitless
attempts to prove a general existence theorem for von Neu-
mann–Morgenstern stable sets for cooperative n-person games.
It would be another 20 years before Lucas’s counter-examples
of (1968 and 1969). Furthermore, the non-uniqueness of the
solution had already been noticed by von Neumann and
Morgenstern in (1944). Moreover, the core, the set of undom-
inated imputations, could be empty or contain many imputa-
tions. Shapley set quite reasonable axioms and determined that
there was a unique function (the value) satisfying these axioms
for all n-person cooperative games with transferable utility. In
(1954) Shapley, collaborating with Shubik, applied this same
value to voting games and showed that it could also be thought
of as an index of voting power.
Shapley (1953b) also made an important stride in the
development of dynamic games by introducing stochastic
games, in which the game passes from position to position
according to probability distributions influenced by both
players. Shapley proved the existence of a value for these
games, though they are formally infinite, and found ways of
computing optimal strategies.
Shapley’s value article was published in the second volume
(1953) of the series started at Princeton in 1950 by Kuhn and
Tucker. These two mathematicians not only studied con-
strained optimization, but also made important contributions
to the development of Game Theory. Tucker worked with
Dantzig on the development of linear programming for the
military application of matrix games. Kuhn (1950) analyzed
extensive games and in 1952 proposed the first complete pre-
sentation and discussion on the proofs of the minimax theorem
in existence at that time. Kuhn was also interested in the
applications of convex sets to games, and especially in signaling
strategies.
12 GIANFRANCO GAMBARELLI AND GUILLERMO OWEN

Signaling strategies were introduced for the first time in or-

ganic form by Gerald Thompson in 1953, and formed the basis
of further studies on games with various types of information,
which are still of great interest today.
In 1957, Everett generalized Shapley’s results on stochastic
games by introducing recursive games (the difference being that
a stochastic game will terminate with probability 1, whereas a
recursive game has positive probability of continuing without
end).
At this point we should mention some of the many other
intellectuals who were involved in Game Theory at Prince-
ton in the early 1950s: Richard Bellman, David Gale, John
Isbell, Samuel Karlin, John Kemeny, John Mayberry, John
McCarthy, Harlan and William Mills, Marvin Minsky, How-
ard Raiffa, Norman Shapiro, Martin Shubik, Laurie Snell and
David Yarmish. Later these were joined by Robert Aumann,
Ralph Gomory, William Lucas, John Milnor, and Herbert
Scarf.
The early 1950’s saw the first textbook on game theory
(McKinsey, 1952) (this was still a little complex for non-
mathematicians), and the first popular editions appeared:
McDonald (1950), Riker (1953) and Williams (1954).
In the meantime, the language used by economists was
changing, due in good part to the influence of game theory:
If one looks back to the 1930s from the present and reads in the major
economic journals and examines the major treatises, one is struck by a sense
of ‘the foreign’. … If, on the other hand, we read economics journal articles
published in the 1950s, we are on comfortable terrain: the land is familiar,
the language seems sensible and appropriate. Something happened in the
decade of the 1940s; during those years economics was transformed from a
‘historical’ discipline to a ‘mathematical’ one.

(Roy Weintraub, 1992, p. 3)

As the 1950’s ended, Aumann began to study n-person games

without transferable utility. (Nash (1953) had only considered
two-person games.) Aumann and Peleg (1960) generalized
much of the von Neumann–Morgenstern theory to these
games.
 THE COMING OF GAME THEORY 13

In 1962, Bondareva neatly characterized games with a non-

empty core, in terms of balanced collections of subsets. More or
less simultaneously (and independently), Shapley (1965)
obtained essentially the same results. These results were gen-
eralized to games without transferable utility by Scarf in 1967,
using a clever combinatorial argument due to Lemke (1965).
Aumann also collaborated with Maschler (1964) to develop
a new solution concept for n-person games: the bargaining set.
Maschler in turn, working first with Davis (1965), and then
with Maschler and Peleg (1966), developed the important
concept of the kernel. Along these same lines, Schmeidler in
(1969) developed the nucleolus.
Aumann and Maschler (1963, see also 1995) had been among
the first to study the problem of incomplete information in
games (i.e., lack of information as to other players‘ interests).
In a series of three important articles, Harsanyi (1967, 1968a, b)
developed a sound theory which is still the basis for work done
today.
The important developments made since the 1960’s cannot
be adequately described in a single paragraph. We will, there-
fore, give a brief outline.
The theory spread from Princeton to the rest of the United
States and experienced its first concrete applications. Luce and
Raiffa’s book (1957), although in a mathematical vein, was
more successful than the previous one by McKinsey (1952), as
non-mathematicians had by now understood how important it
was to make an effort to further their knowledge of the subject.
The same can be said for Schelling’s publication in 1960 as well
as that of Owen (1968), which had some influence in spreading
the theory worldwide as these were translated into numerous
languages.

ACKNOWLEDGEMENTS

This paper is a new version of an article published in 1994 in the

European Journal of Business Education (4, 1, pp. 30–45), which
appeared in Italian translation in the book ‘Giochi competitivi e
14 GIANFRANCO GAMBARELLI AND GUILLERMO OWEN

cooperativi ’ ( I ed., Cedam, Padova, 1997; II ed., Giappichelli,

Torino, 2003). The authors are most grateful to Dunia Milanesi
for her invaluable help in updating this paper.

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Addresses for correspondence: Gianfranco Gambarelli, Department of

Mathematics, Statistics, Computer Science and Applications, University of
Bergamo, Via dei Caniana 2, Bergamo 24127, Italy. E-mail: gambarex@-
unibg.it

Guillermo Owen, Naval Postgraduate School – Mathematics Code MA/

ON, Naval Postgraduate School, Monterey 93943, CA, USA. E-mail:
gowen@nps.navy.mil

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