University of Virginia Library

GAME THEORY

1. Decisions and Games. Human life is an unbroken
sequence of decisions made by the conscious individual.
He is continuously confronted with the need for mak-
ing choices, some of them of narrow, others of very
wide scope. In some cases he commands much infor-
mation about consequences of a particular choice, in
most he is quite uncertain. Some affect the immediate
present, others commit him for a distant future. Some
decisions are entirely his own—whether to have an-
other cup of tea, to go for a walk; some involve other
persons—whether to marry X. Many decisions are
made with respect to nature: what planting to choose,
what weather to expect. Many decisions are made by
groups of individuals, and the group decision can be
arrived at by a great variety of processes.

Some decisions arise from a logical structure as in
law. There are also mathematical and logical decisions;
whether π is a transcendental number, or whether to
accept a particular proof of the existence of God. Even
in mathematics there may be uncertainty, as Gödel has
shown.

Decisions must also be made when an individual
plays a game. (First, however, he must make the deci-
sion to play, which is normally though not necessarily
a voluntary one.) In playing the game, the individual
follows rules which, together with the decisions made,
usually determine the winner. In all cases the desire
for optimality (maximum rewards) will arise since
clearly a decision is a choice among alternatives and
the “best” decision will be preferred over all others.
Many different kinds of decisions occur in connection
with games of various types, the number of different
games being indeterminate since always new games
can be invented. In order to describe the behavior of
individuals and to evaluate their choices, criteria have
to be known or must be established.

Clearly a comprehensive theory of decision-making
would encompass virtually all of voluntary human
activity and as such would be an absurd undertaking,
given the infinity of human situations. A more reasona-
ble approach is to develop a science, or sciences, deal-
ing with the principles, so as to govern decision-making
in well-defined settings. In what follows the structure
of that theory will be laid bare as far as this is possible
without going into the use of the underlying mathe-
matics.

Game theory represents a rigorous, mathematical
approach towards providing concepts and methods for
making reasonable decisions in a great variety of
human situations. Thus decision theory becomes part
of game theory. The basic features of the theory are
described in Section 7, below.

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.

3. Utility. Before discussing games of strategy
proper and developing the essence of the theory, a
clarification of the medium in which the payoff is made
is needed. When a game is played for money then the
winnings in money can be taken as the criterion for
the outcome, be it a game of chance or of strategy.
But when the score is not set in ready-made numerical
terms, or even by a simple “win” or “lose” declaration,
the matter is more difficult. While it would be possible
merely to postulate the existence of a number, it is
desirable to show how a numerical criterion of a speci-
fiable character can be established. This was accom-
plished (von Neumann and Morgenstern, 1944) by
showing that “utility” can be defined as a number up
to a positive linear transformation without fixing a unit
or a zero. In these terms payoffs will be expressed. The
utility concept takes prior rank even over money units,
though they be available. Utility thus defined is what
the individual will fundamentally aim for when select-
ing his strategy. The above-mentioned numerical ex-
pression is obtained from a small set of plausible axioms
by combining probability and an individual's com-
pletely ordered set of preferences (fulfilling the
Archimedean order property), showing that the indi-
vidual will think in terms of expected utility. It is
proved that these axioms define “utility” and make it
numerical in the desired manner. It is an additional
step to assume that the individual will endeavor to
maximize this utility.

The new utility theory also has given rise to a large
literature. Though modifications of the original version
have been proposed (e.g., the use of subjective,
Bayesian probabilities instead of the frequency con-
cept, etc.) the theory has entered virtually all writings
on decision-making and the more modern treatments
of economics. The theory has its antecedents in D.
Bernoulli's famous treatment of the “St. Petersburg
Paradox” (1738; Menger, 1934 and 1967) in which he
introduced the notion of moral expectation, i.e., a value
concept, in order to account for the fact that in spite
of an infinitely large mathematical expectation in that
game a person will not risk his entire possessions as
a stake in order to be allowed to play this game, even
if it could be offered. The second step in the direction
of von Neumann-Morgenstern utility theory was taken
by F. P. Ramsey in his “Truth and Probability” (1926;
1931); but this paper was only rediscovered after the
expected utility theory in the von Neumann-
Morgenstern formulation was developed and had be-
come dominant. The use of subjective probability does
not invalidate the theory (Pfanzagl, 1962; 1967) as was
already noted on the occasion of the original formula-
tion in 1944. The new theory of numerical utility is
not identical with theories of “cardinal” or “ordinal”
utility of the older and neo-classical economists, nor
has it a basis in philosophical or political utilitarianism.

In order to establish further the empirical validity
and power of the new theory a great number of exper-
iments have been made—a novum in this field. These
experiments attempt to test the validity of the under-
lying axioms, and to clarify the question of how indi-
viduals behave typically in situations involving risk.
This behavior is clearly a phenomenon that any theory
of decision-making has to take into account, given the
glaring fact of the prevalence of chance in human
affairs.

The development of the new theory of utility defi-
nitely advances our ability to analyze decisions (Fish-

burn, 1970) and raises important philosophical issues
(Martin, 1963).

4. Games as Models of Human Actions. Games can
be classified into two broad but sharply different cate-
gories: (a) games of chance and (b) games of strategy,
which contain chance games as a simple, special case.
In (a) the outcome is totally independent of the action
of the playing individual as, e.g., in roulette. There
may nevertheless be different manners of betting on
the outcome, e.g., the player must decide whether to
place a given stake in one throw, or to distribute it
over several places (numbers, colors), or over several
plays. These questions lead to important exercises in
probability theory but they do not alter the funda-
mental simple chance character of the game. In (b)
the outcome is controlled neither by chance alone, nor
by the individual player alone, but by each player to
some extent. Chance may (as in poker) or may not (as
in chess) be present.

It is the entirety of the actions of the players—and
of chance if nature intervenes—which determines the
outcome and the equilibria (which the theory is to
determine). In the course of a play the interests of the
players are sometimes opposed to each other, some-
times parallel.

The significance of game theory is that besides ex-
plaining games proper, suitable games can be identified
strictly with important other human actions which they
therefore model. This is to be understood in the precise
manner in which models are used in science, as when
the planets are considered to be mere mass points and
a theory of the solar system is built on that basis. In
the same manner military, political, economic, and
other processes can be identically represented by cer-
tain games of strategy. If a theory of such games can
be established then a theory for the modeled processes
is obtained. Such a theory would necessarily have to
be mathematical. Its structure turns out to be quite
different from that of classical mechanics and, a fortiori,
from the differential and integral calculus. This is due
to the essentially combinatorial character of the prob-
lems encountered and to the wide divergence of the
underlying phenomena from physical phenomena.
Among molecules or stars there is no cooperation, no
opposition of interest, no information processing or
withholding, no bluffing, no discrimination, no exploi-
tation. Matter may collide, coalesce, explode, etc., but
there is no conscious activity.

It was to be expected that the widespread attempts
to use the concepts and techniques that had originated
in the natural sciences must ultimately fail when ap-
plied to social phenomena. But the acceptance of new
approaches is slow and difficult in any field and the
impact of natural science thinking is hard to break.

The world of social phenomena is embedded in that
of natural phenomena. But the two are different and
as a consequence the structure of the sciences dealing
with them will differ too. All sciences must, of course,
have elements in common such as are dealt with in
the theory of knowledge. However, the connections
must not be overrated. Montaigne spoke of the need
for separate scientific languages and this need has now
become quite evident. It can be demonstrated that
ultimately different fields of inquiry will generate even
their own “logic.” For example, the logic of quantum
mechanics is best described by a projective geometry
in which the distributive law—which in algebra means
that a(b+c) = ab + ac—does not hold (Birkhoff and
von Neumann). It is to be expected that a calculus as
germane to the social sciences may someday be devel-
oped (or discovered?) as differential calculus is to me-
chanics. Other parts of the natural sciences show signs
of producing their own mathematical disciplines and
structures, and this process may repeat itself. One
important aspect of game theory is that it has already
given rise to considerable, purely mathematical activ-
ity. This process is only in its beginning, but proves
once more that the development of mathematics is
ultimately dependent on the mathematician being in-
volved with empirical problems. Mathematics cannot
proceed solely on the basis of purely formalistic and
possibly aesthetic grounds. Thus the creation of game
theory may be of a significance transcending in that
respect its material content.

The essential justification for taking games of strat-
egy as models for large classes of human behavior was
already stated in the first paragraph of this section:
our acts are interdependent in very complex manners
and it is the precise form of this interdependence that
has to be established. Interdependence has, of course,
been recognized, but even where neo-classical eco-
nomics of the Walras-Pareto type tried to describe this
interdependence, the attempt failed because there was
no rigorous method to account for interaction which
is evident especially when the number of agents is
small, as in oligopoly (few sellers). Instead large num-
bers of participants were introduced (under the mis-
nomer of “free competition”) such that asymptotically
none had any perceptible influence on any other par-
ticipant and consequently not on the outcome, each
merely facing fixed conditions. Thus the individual's
alleged task was only to maximize his profit or utility
rather than to account for the activities of the “others.”
Instead of solving the empirically given economic
problem, it was disputed away; but reality does not
disappear. In international politics there are clearly
never more than a few states, in parliaments a few
parties, in military operations a few armies, divisions,

ships, etc. So effective decision units tend to remain
small. The interaction of decisions remains more obvi-
ous and rigorous theory is wanting.

5. Rational Behavior. A purpose of social science,
of law, of philosophy has been for a long time to give
meaning to the notion of “rational behavior,” to ac-
count for “irrationality,” to discover, for example in
criminal cases, whether a given individual could be
considered as having acted rationally or not. In general
there appears to exist an intuitive notion of what “ra-
tional” must mean. Frequently this notion would be
based on experience; but experience varies with each
individual, and whether any person has an intuitively
clear idea of “rationality” is doubtful. In the simple
case in which an individual wishes to maximize a
certain quantity, say utility, and provided he controls
all factors or variables on which his utility depends,
then we shall not hesitate to say that he acts rationally
if he makes decisions such that he actually obtains this
maximum, or at least moves stepwise in its direction.
Thus, rationality is predicated on two things: (a) the
identification of a goal in the form of preferences
formed, possibly stated numerically, and (b) control
over all the variables that determine the attainment
of the goal.

The first condition requires that the individual have
a clear notion of what he wants and that he possess
sufficient information which will identify the goal he
wishes to reach. The second condition requires that
the individual be able to determine first the variables,
and second the consequences of the changes he may
make in setting their values for reaching the intended
goal, and finally that he actually can set the values
of the variables as it may appear proper to him. The
amount of foresight demanded (especially if the goal
should be distant) is considerable but this point shall
not be considered further. The control factor, however,
is of primary concern: if nature intervenes in his in-
tended behavior, the individual can control an in-
different nature by means of statistical adjustment; the
farmer, for example, can arrange his planting so that
on the average neither a very dry nor a very wet
summer will hurt him. Whether nature is always in-
different is another question (Morgenstern, 1967). But
it is an entirely different matter if among the variables
there are some that are controlled by other individuals
having opposite aims. This lack of complete control
is clearly the case in zero-sum (winnings compensate
losses exactly) two-person games of strategy, but also
in business, in military combat, in political struggles
and the like. It is then not possible simply, and in fact,
to maximize whatever it may be the individual would
like to maximize, for the simple reason that no such
maximum exists. It is then not clear intuitively which
course of action is “better” than another for the indi-
vidual, let alone which one is optimal.

To determine optimal, or “rational” behavior is pre-
cisely the task of the mathematical theory of games.
Rational behavior is not an assumption of that theory;
rather, its identification is one of its outcomes. What
is assumed is that the individual prefers a larger advan-
tage for himself to a smaller one. If these advantages
can be described and measured and are understood by
the individual, and if he chooses not to pursue the
required course, then there is a limited definition of
nonrational behavior for such situations. It is assumed
that the demonstration (if the theory succeeds) of the
optimal course of action is as convincing to the indi-
vidual as a mathematical proof is in the case of a
mathematical problem. But the theory allows that a
participant may deviate from his optimal course, in
which case an advantage accrues to the others who
maintain their optimal strategies.

Prior to the advent of game theory the term “ra-
tional” had been used loosely as referring to both of
the two conceptually different situations set forth
above, as if there were no difference. The transfer of
the notion of rationality from the completely control-
lable maximizing condition to one in which there is
no exclusive control over the variables is inadmissible.
This has been the cause of innumerable difficulties
permeating much of philosophical, political, and eco-
nomic writing. No side conditions, however compli-
cated, which may be imposed or exist when one is
confronted with a clear maximum problem changes the
situation conceptually. In the case where full control
exists side conditions merely make the task of reaching
the maximum more difficult—perhaps even impossible,
for example, because it may computationally be out
of reach. But even in its most complicated form it is
conceptually different—and vastly simpler—than the
problem faced by, say, a chess player or a poker player,
and consequently by any one whose activities have to
be modeled by games of strategy. The conceptual
difference does not lie in numbers of variables or in
computational difficulties; but we note that the solution
of games becomes extremely difficult both when the
number of strategies is large (even with as few players
as in chess) and also when the number of participants
increases, though each may have only a few strategies.

When there are, say, 100 variables of which one
individual controls 99 the other the remaining one, this
appears to be a different situation from that when there
are only 2 variables and each player controls one. Yet
conceptually the two are identical. No practical con-
siderations, such as possibly assigning weights to varia-
bles, and the like, in an effort to reduce difficulties of
action, will work. The fundamental conceptual differ-

ence and difficulty remains and has to be resolved by
the theory.

6. Normative or Descriptive Theory. The purpose
of a theory of decision-making, or specifically of game
theory, is to advise a person how to behave by choosing
optimally from the set of his available strategies, in
situations subject to the theory. If he decides knowingly
to deviate from the indicated course he has either
substituted another goal, or dislikes the means (for
moral and other reasons). It is then a matter of termi-
nology whether he is still considered to be a rational
actor. The theory at any rate can take such deviations
into consideration. Clearly, some technically available
strategies may be inadmissible in legal, moral, and
other respects. In some cases these questions do not
arise: chess is played equally whether the opponents
are rich, poor, Catholics, Muhammadans, communists
or capitalists. But business or political deals are affected
by such circumstances. Advice can be given with or
without constraints which involve morals or religion.

The theory is also descriptive, as it must be if it is
to be used as a model. It might be argued that the
theory cannot describe past events since before it was
created individuals could not have followed the
optimal strategy which only the theory could discover!
The answer is that for some situations the individual
can find it by trial and error and a tradition of empirical
knowledge could develop after repeated trials. How-
ever, the same objection applies regarding ordinary
maximum problems (provided they are even given):
the identification and computation of the maximum
was (and in many cases still is) out of reach even for
very large organizations; yet they behave as if they
could find it and they try to work in that direction.

Theories thus can be viewed as being both descrip-
tive and normative. In the natural sciences a similar
apparent conflict shows up in interpreting phenomena
as either causally or teleologically related, while in fact
this distinction may resolve into merely a matter of
how the differential equations are written.

Thus the theory is capable of being used in both
manners. Future descriptions of reality will be im-
proved if the concepts of the theory are available and
future actions of those who use the theory in order
to improve their own rationality will be superior (ex-
cept in the simplest cases where the correct answer
is also intuitively accessible). Clearly, if more and more
players act rationally, using the theory, there will be
shifts in actual behavior and in real events to be de-
scribed. This is an interesting phenomenon worth
pointing out. It has philosophical significance: progress
in the natural sciences does not affect natural phenom-
ena, but the spread of knowledge of the workable social
sciences changes social phenomena via changed indi
vidual behavior from which fact there may be a feed-
back into the social sciences (Morgenstern, 1935).

7. Basic Concepts: Game Theory and Social Struc-
ture. The description of a game of strategy involves
a number of new concepts. Obviously, games are first
classified by virtue of the number of players or partici-
pants: 1, 2,..., n. Second, when the winnings of some
are compensated exactly by the losses of others, the
game is zero-sum. The sum can also be positive (when
all gain), negative (when all lose), constant, or variable.
Games are “essential,” when there is an advantage in
forming coalitions, which can happen even in zero-sum
games, but only when n ≧ 3. This expresses advan-
tages in cooperation; it can develop even when there
are only two players, but then the game has to be
non-zero sum. Games are “inessential” when there is
no such advantage, in which case each player proceeds
independently for himself. Note, however, that he still
does not control the outcome for himself by his actions
alone; the “others” are always present, and sometimes
also nature is present as an agent.

Games are played according to rules which are
immutable and must be known to the players. A rule
cannot be violated since then the game would cease;
it would be abandoned or go over into another
game—if that is possible. A tacit assumption is that
players agree to play. They do this without doubt when
playing for pleasure. When games are used as models,
it may however happen that one's participation in the
modeled situation is not voluntary. For example, a
country may be forced into a military conflict; or, in
order to survive and to earn a living a person may
have to engage in certain economic activities. Games
come to an end; the rules provide for this. Again in
the modeled situation one play of a certain game—a
play being the concrete, historical occurrence of a
game—may follow another play of the same game, or
the play of one game may follow that of another game
and so on. Sometimes it is possible to view such se-
quences as supergames and to treat them as an entity.
In some games as in chess the players are perfectly
informed about all previous moves, in others they have
only partial information about them. Sometimes the
players are not even fully informed about themselves,
as e.g., in bridge, which is a two-player game, but each
player (e.g., North and South) plays through two rep-
resentatives. In this case information about oneself and
to oneself is only disclosed by the manner of playing.
In addition chance enters, since the cards are dealt at
random. This example gives a first indication of the
great complexity that confronts any attempt at theory
even under simple conditions. In poker, bluffing is
added, as the pretense by some players of having cer-
tain sets of cards can become an element in the play.

Decisions have to be made, when to bluff in the face
of possible bluffs by the other players, how to surmise
bluffs by others, and many more such factors.

The rules normally specify sequences of moves,
countermoves and tell when the game has terminated.
It is possible to view games described in this “extensive”
form strictly equivalently by introducing the notion of
strategies, which are the complete plans made up by
each player for such series of moves. Games are then
described in the “normalized” form and it is thus that
they shall be treated in what follows. In choosing a
pure strategy, i.e., by specifying the precise complete
course of action, the player may or may not be at a
disadvantage in expected values if he has to make his
choice openly before the other player makes his choice.
If there is no disadvantage, then the game has a saddle
point in the payoff matrix, for if the first player chooses
his optimal strategy, then no matter what the second
player may do, he cannot depress the first one's ex-
pected payoff below a certain value which is the value
of the saddle point. Exactly the same is then true
conversely for the second player. Games having saddle
points in pure strategies are strictly determined. In
these cases there is no value of information flowing,
voluntarily or involuntarily, from one player to the
other. Each behaves rationally if in pursuit of his in-
tended maximum benefit he chooses his pure strategy
so that he is guaranteed at least as much as corresponds
to the value of the saddle point. If the other player
deviates from his optimal strategy, i.e., behaves “irra-
tionally,” the first one can only gain.

However, games usually have no saddle points in
pure strategies. A player forced to disclose his pure
strategy would then be at a disadvantage and the
question arises whether there is at all an optimal way
of playing. The attempt of opponents to outguess each
other by the chain of thought: I think that he thinks
that I think he thinks... will never lead to a resolution
of the dilemma exemplified by the Sherlock Holmes—
Professor Moriarty pursuit case (Morgenstern, 1928),
which corresponds exactly to a qualified game of
matching pennies. How then shall one proceed?

John von Neumann proved in 1928 that for these
games which are not strictly determined a saddle point
always exists if the players resort to proper so-called
mixed strategies: the now famous minimax theorem.
A mixed strategy means that instead of selecting a
particular pure strategy from the whole set of all
available pure strategies, the player must assign a
specific probability to each one of them such that at
least one will be played. A properly chosen chance
device will then determine the strategy actually
chosen. The player himself will not know which strat-
egy he will actually play; hence he cannot be found
out by his adversary and he cannot even accidentally
disclose his choice, which if he did would be disastrous.

The fundamental “Minimax Theorem” assures that
the player, using mixed strategies, can always find a
correctly computed optimal mixed strategy to protect
himself (minimizing the worst in expected values that
can happen to him) precisely as in strictly determined
games he can identify, and even announce, his optimal
pure strategy. The original proof of this theorem in-
volved very advanced methods of topology and func-
tional analysis. The theorem is of outstanding impor-
tance and has had wide ramifications: the original
theory of games for any number of players rests on
it. Though the implications of the theorem have often
been found uncomfortable (and were termed “pessi-
mistic”), it stands unchallenged. As is often the case
in mathematics, other simpler proofs have later been
offered, by von Neumann himself as well as by others,
such as using concepts from the theory of convex
bodies, a theory which in turn has greatly benefited
from these developments.

It is necessary to examine the significance of the use
of mixed strategies since they involve probabilities in
situations in which “rational” behavior is looked for.
It seems difficult, at first, to accept the idea that
“rationality”—which appears to demand a clear, defi-
nite plan, a deterministic resolution—should be
achieved by the use of probabilistic devices. Yet pre-
cisely such is the case.

In games of chance the task is to determine and then
to evaluate probabilities inherent in the game; in games
of strategy we introduce probability in order to obtain
the optimal choice of strategy. This is philosophically
of some interest. For example, the French mathe-
matician É. Borel asserted that the human mind cannot
produce random sequences of anything; humans need
to invent devices which will do this for them. Borel
did not and could not give a mathematical proof be-
cause his assertion is not a mathematical one. It is
noteworthy, incidentally, that recent studies of the
brain seem to indicate, however, that some uncertainty
and randomness in its operation are essential for its
proper functioning.

The identification of the correct probabilities with
which to use each pure strategy is a mathematical
task—sometimes computationally formidable—and is
accomplished by use of rigorous theory. Putting these
probabilities to use requires then a suitable physical
generating device which always can be constructed.
In practice players may merely approximate such
devices where these would tend to be very compli-
cated. In some cases they will produce them exactly,
as in matching pennies. In this game, on matching
either heads or tails, one unit will be paid to the first

matching player; when not matching, one unit by the
first to the second. This game, clearly zero-sum and
of complete antagonism between the two players, is
not strictly determined. Hence each will protect him-
self against being found out. As is well known the
optimally correct way of playing is for both players
to toss his coin simultaneously with the other player,
which is equivalent to choosing each of the only two
available strategies with probabilities 1/2, 1/2. The coin
itself when tossed will either show heads or tails pre-
cisely with the required probabilities.

The manner in which this game is played makes
it appear to be a game of chance, but in reality it is
one of strategy. This incidentally illustrates a grave
difficulty of giving correct descriptions of social events!
The probabilities of 1/2, 1/2, have to be changed if there
should be a premium, say, on matching on heads over
matching on tails. The new probabilities that secure
the saddle point can no longer be guessed at or be
found intuitively; they have to be computed from the
theory, so quickly does the true, mathematical analysis
which requires the full use of the complex theory have
to be invoked. When the number of strategies goes
beyond two the computational difficulties increase at
any rate; the computations may become impossible
even when the game is strictly determined, as in chess,
where there are about 10120 strategies. The existence
proofs of optimal strategies are valid nevertheless.

The problem now arises how a social equilibrium
can be described when there are more than two deci-
sion makers. Here only the most basic concepts can
be indicated as a full description would require much
space and intricate mathematical analysis. The struc-
ture is this: when in a zero-sum game n ≧ 3, then the
possibility of cooperation among players arises, and
they will form coalitions wherever possible. In order
to be considered for inclusion in a coalition a player
may offer side payments to other players; some may
be admitted under less favorable terms (when n > 3)
than those set by the initial members of the coalition
and the like. When a coalition wins, the proceeds have
to be divided among the partners and these then find
themselves in the same kind of conflict situation which
arises for the players of a zero-sum two-person game.

The totality of all payments to all players is an
“imputation.” In order to determine an equilibrium it
appears to be necessary to find a particular imputation
that is “better,” that is, more acceptable, from among
all possible ones than any other. Such an imputation
then “dominates” all other imputations. But that would
be the case in inessential games. Only for those is there
a unique social optimum, a division of the proceeds
of the game played by society which cannot be im-
proved upon and which therefore is imposed or im
poses itself upon society as the best stable arrangement.
But since cooperation is a basic feature of human
organization these games are of little interest. No such
single imputation exists for essential n-person games.
Domination is then not transitive, thus reflecting a
well-known condition of social arrangements in which
circularity often occurs (as, for instance, in the relative
values of teams in sports).

Thus the hope of finding a uniquely best solution
for human affairs is in vain: there is no stability for
such arrangements. Political, social, and economic
schemes have been proposed under the tacit, but fre-
quently even open, assumption that this is possible
when men organize themselves freely. Only the iso-
lated individual or a fully centralized (usually dicta-
torial) society can produce a scheme that it considers
better than any other and that it hopes to be able to
enforce.

Thus there is, in general, no “best” all dominating
scheme of distribution or imputation; but there may
be a number of imputations which do not dominate
each other and which among them dominate everything
else. Such imputations, therefore, must be considered
by society. They form a special “stable set,” originally
called the “solution set.” Any one of the imputations
belonging to this stable set is a possible, acceptable
social arrangement.

A stable set is precisely a set S of imputations, no
one of which dominates any other, and such that every
other possible imputation not in S is dominated by
some imputation in S. (Technically, the imputations
belonging to each stable or solution set are not even
partially ordered and, a fortiori, the elements of this
set are not comparable with one another.)

The stability that such a set possesses is unlike the
more familiar stability of physical equilibria. For no
single imputation can be stable by itself; it can always
be disturbed, not by “forces” (as a physical equilibrium
could be), but by the proposal of a different arrange-
ment by which it is dominated. Such a proposal must
necessarily lie outside of S. But for every such proposal,
there is always a counter-proposal which dominates
the proposal, and which lies in S. Thus a peculiar,
delicate but effective equilibrium results which has
nothing to do with the usual equilibria of physics; the
process of proposal and counter-proposal always leads
to an imputation in S. Indeed the present notion differs
so profoundly from the usual ideas of stability and
equilibrium that one would prefer to avoid even the
use of the words. But no better ones have yet been
found.

There may exist, even simultaneously, different,
conflicting solution sets or standards of behavior, each
one with any number of different imputations, always

more than one, sometimes even infinitely many. But
those within the respective solution sets are merely
alternative to each other; they are not in fundamental
conflict as are the different standards.

Clearly, it is difficult to identify solutions, i.e., sets
of imputations with the required properties, even from
the whole set of all possible imputations. In 1968 W. F.
Lucas made the important discovery of a game of
10 players that has no solution (in the so-called charac-
teristic function form). The question is open whether
this is a rare case and what modifications in concepts
and methods may be necessary to assure solvability.
In all other cases so far investigated solutions have been
found.

These admittedly difficult notions emerge from the
rigorous mathematical theory whose empirical basis is
formed by facts that are not questioned even by current
social and economic theory, though these theories have
not rendered a successful account of the nature of
decision-making. The lack of identification of a single
settlement or imputation is not a deficiency of game
theory. Rather there is herein revealed a fundamental
characteristic of social, human organization which
cannot be described adequately by other means.

In the light of these considerations one of the stand-
ard concepts currently used in describing a social op-
timum, the so-called Pareto optimum (formulated by
v. Pareto, 1909) appears at best to be an oversimplifi-
cation. That notion says that the optimal point is
reached when no one can improve his position without
deteriorating that of others. What is lacking in that
formulation, among other things, is to account for
nonuniqueness, uncertainty, deceit, etc., hence a more
comprehensive frame within which individuals make
decisions that guarantee a precisely defined but differ-
ent stability (Morgenstern, 1965).

The appearance of novel and complicated notions
is due to a mathematical analysis that is germane to
the subject matter and has nothing to do with any
ideological or other conception of society. The mathe-
matical analysis unravels implications of some gener-
ally accepted facts and observations, axiomatically
stated, and then leads via the fundamental minimax
theorem to the discovery of relationships in the empir-
ically given social world which without the aid of the
new theory have either escaped notice altogether or
were at best only vaguely and qualitatively described.
Since inventions are possible in the social world this
process is an unending one, which means that new
concepts and theorems have arisen and more are bound
to arise. For example, new concepts of solution struc-
ture have emerged. It may even happen that social
organizations are proposed that have no stable sets;
and that only work in a manner that is quite different
from the original intentions, even though these may
have involved sound philosophical and ideological
principles.

Physics studies given physical facts and is not con-
fronted with this type of creation; it faces in this sense
a static world (though it may be expanding!) as far as
we can tell. Not all given physical facts are known;
new effects are constantly being discovered but it is
doubtful that they are currently being created, while
it is certain that novel forms of social organization are
being and will be invented. We know that the life
sciences are also, and in fact more clearly, confronted
with the evolutionary creation of new phenomena, not
only with their discovery, as in the case of physics.
But on the other hand, the time spans which are neces-
sary for genetic change are so great as to make this
concern with the creation of new phenomena (other
than breeding of new plants and animals) to have as
yet no practical importance in this context.

This goes to show that the intellectual situation in
the social sciences is disquieting even when one ab-
stracts from the further complication presented by the
existence of frequently changing  ideologies .

There is thus no hope to penetrate into the intricate
web of social interdependencies by means of concepts
derived from the physical sciences, although thinking
along such lines still dominates. This is partly due to
the immense success of physics and the slow develop-
ment even of any proper description of the social
world. Where this description has used abstract con-
cepts these were mainly taken from the physical sci-
ences. Thus a recasting of the records of past social
events is necessary. The two movements of description
and theory formation are as inseparably interrelated
as they were in physics and astronomy where the
analysis of simple processes, for instance, that of a
freely falling body, led to mechanics and to the dis-
covery of the appropriate tool of the differential cal-
culus. Fate will not be easier for the social sciences
and in this methodological situation lies the deep phil-
osophical significance of game theory, i.e., of the new
analyses of human decision-making and the interlocking
of such decisions.

To give but one illustration: a formal system of
society may be fully symmetric, i.e., give each member
exactly the same possibility, such as laissez-faire, and
thereby have provisions of complete freedom and
equality. But the possibility of cooperation via coali-
tions, agreements, and the like produces nonsymmetric
arrangements so that the intent of the law-maker can-
not be maintained without forbidding coalitions which
then would run afoul of the principle of freedom.
While this asymmetry is sometimes not very hard to
discover there are other, more elusive cases; but in

order to be accepted the mathematical theory must
first yield results which are also obtainable from com-
mon sense experience. However, theory must in addi-
tion be able to predict the emerging structures and
show how the inner nature of social processes works.

8. Applications. Application is the final test of
theories but may be hard to come by. Decision theory
and game theory have a potentially wide range of uses.
Those already made are limited partly because of the
newness of the field, because of computational difficul-
ties, and partly because the theories are in a state of
active development which produces new concepts and
theorems. The distance in time and difficulty from an
abstract theory to application is always large when a
fundamentally novel development occurs. This period
may stretch over generations. Some directions of ap-
plication are becoming clear, however. Decision theory
is basic for, and indeed inseparable from, modern sta-
tistics. The use of the minimax theorem has given rise
to a new turn in that science (primarily due to A. Wald)
and produced a large literature. Noteworthy is a study
by J. Milnor (1954) on games against nature in which
various possible criteria, due certain authors such as
Laplace, A. Wald, L. J. Savage, and L. Hurwicz, were
investigated regarding their compatibility. Milnor
showed that no criteria satisfy all of a reasonable set
of axioms and it is an open problem whether new ideas
can be evolved to resolve this impasse. Since this is
a game against nature, then our incomplete and
changing knowledge of nature's laws also has to be
taken into account—a further complication not spe-
cifically considered by Milnor or others. Nature may
be infinitely complex and therefore can never be
“found out” completely.

Game theory has a profound bearing on economics.
Many special problems have been attacked such as
oligopoly (markets with few sellers) which could never
be adequately treated by conventional methods. Par-
ticularly noteworthy is the work by Shapley and Shubik
(1965 to date). The penetration to other areas such as
bargaining, auctions, bidding processes, general equi-
librium, etc., is slow but steady. The very structure
of existing theory is threatened once it is recognized
that there is no determinism and that no one, not even
the state, controls all variables, as was explained above.
But recognition of this indeterminism demands the
scrapping of more than can be immediately replaced,
and this causes a profoundly disturbing situation: one
shows the logical inadequacy of existing theories but
cannot offer a specific immediate and detailed replace-
ment. Also recall that false theories often have had
significant workability (Ptolemy) and therefore, though
doomed, could live together with their ultimate re-
placement (Copernicus) for a considerable time.

Sociology, with a less advanced theory than eco-
nomics will undoubtedly become a fertile field for
applications once the connections are seen. In particu-
lar the distinction between the rules of games and the
standards of behavior (which depend on previously
formulated rules but are the consequences rather than
antecedents of games) offer wide areas for sociological
investigations.

In political science there are increasingly many ap-
plications. Going back to Condorcet's voting paradox
(1785), which is the possibility of an inconsistent col-
lective choice, even when individual choices are con-
sistent, great strides have been made in illuminating
voting procedures (Farquharson, 1969), many of these
steps resting on the theory of weighted majority games.
In addition political power play, with favors granted,
side payments made, bluffs, promises kept and broken,
is as ideal and fetile a field for the new concepts as
one could wish, but the path is thorny, especially
because of the preliminary, difficult quantification of
matters such as “political advantage” and the like. Of
particular significance is the illumination of the bar-
gaining and negotiation process. A considerable litera-
ture has emerged which is of great practical value
though it is highly technical. One question, for exam-
ple, is how the contracting parties should deal with
disclosure of their own utility functions in the process
of negotiating. Another is the proof, given by von
Neumann and Morgenstern (1944), that of two bar-
gaining parties the one will get the upper hand which
has the finer utility scale, a better discernment of ad-
vantages. Negotiation is always possible except when
there is full antagonism, which exists only in a zero-sum
two person game. In all other cases negotiations are
possible, whether the game be zero-sum or not.

The application to military matters is obvious and
some possibilities have been explored extensively in
many countries. The idea of a “strategy” has after all
since ancient times been embedded in military activi-
ties, but it is noteworthy that the modern theory did
not take its inspiration from the military field but from
social games as a far more general and fruitful area
from which it could radiate.

Combat and conflict, however, are as deeply rooted
in human nature as is cooperation, so that the combi-
nation of both, emerging with singular clarity in
military affairs, makes this field naturally attractive for
study. As a consequence there is now a game theoretic
literature concerning combat, deployment, attrition,
deterrence, pursuit, and the like. Also the insight that
in war—especially in nuclear war—both parties may
lose (“Pyrrhic victories”) has found precision in the
formulation of games with negative payoffs to all. In
most cases it is only in the 1960's that all these notions

have become precise and were in part successfully
applied in a concrete and computational form.

Game theory has also been used in ethics, biology,
physics, and even engineering. This spread of appli-
cations is two-fold. First, in ethics the problems of
decision-making are essential, and it may appear that
they consist primarily in imposing constraints on the
individual or on society (Braithwaite, 1955). This view
would exclude technically feasible strategies for moral
reasons (though permitted within the rules of the
game). This exclusion of strategies shows how ethical
decisions involve other persons, positively or nega-
tively, directly or indirectly, singly or in groups, as well
as compromises and commitment. An ethics that con-
siders only a normative system of possible ideals (which
can never be fully explicit in view of the infinity of
situations that may be encountered), or single decisions
by single, isolated individuals is unable to deal with
crucial issues of that field. The mere exclusion of a
feasible strategy on moral grounds implies that the
consequences of its use are known and can be disap-
proved. But the consequences depend also on the strat-
egies chosen by the others and prediction of this type
may be impossible. The moral code may forbid murder
but accept killing on command in war, and then try
to qualify what kind of commands are valid and which
are not. This goes clearly beyond the mere establish-
ment of an abstract normative system, not considered
in action. Analysis taking into account the above points
leads to a probabilistic ethics if only because the not
strictly determined games demand the use of mixed
strategies. These ideas are now only in the first state
of development. They are fundamentally different from
previous abortive applications of mathematics to eth-
ics, such as by Spinoza.

Second, in the other areas game theory appears as
a mathematical technique rather than as a model.
Certain processes, say in engineering, can be inter-
preted as if they were games because of a formal
correspondence. This then makes the use of the exten-
sive mathematical apparatus of game theory possible.
Illustrations would necessarily be of a rather special-
ized character and are therefore omitted here, though
the large field of linear programming with its many
variants (of great practical importance) must be men-
tioned. Game theory and programming theory are
closely related by virtue of the well known duality
theorem for linear programming.

Biologists (Lewontin, 1961; Slobodkin, 1964) have
interpreted evolution in game theoretic terms, in spite
of the difficulty for a nonteleological biology to use
the purposeful orientation of game theory. By means
of appropriate reinterpretation, including that of util-
ity, it is shown that game theory can give answers to
problems of evolution not provided for by the theory
of population genetics. It is possible to identify an
optimal strategy for survival of populations in dif-
ferent environments.

Some of these developments involve game theory
strictly as a technology (not as a model) and in some
it is still doubtful whether a true model character can
be accepted (as possibly in biology). There are here
transitional phases of high interest and it is impossible
to foresee the development of these tendencies.

9. Philosophical Aspects. The appraisal of the phil-
osophical significance of a new field of science, or of
a fundamental turn in its treatment, or of the appear-
ance of a new scientific language expressing new con-
cepts, is an extremely delicate matter. Hence little shall
be said here as it may be premature to do so. But if
we attribute philosophical meaning to the fact that the
study of decision-making under a wide set of circum-
stances has not only affected significantly sciences like
statistics, but is spreading to other fields as a new
mathematical discipline—game theory—and is influ-
encing even pure mathematics, then we are justified
in speaking of a philosophically relevant development.

While raising no claims of equal importance, the
development of game theory has created a shift of
standpoints in viewing the social world and human
behavior, just as relativity theory and quantum me-
chanics have provided a new outlook on physical real-
ity. It is too early to be very specific: in those other
two areas it took years before the strange new concepts
of space curvature, of an infinite but bounded space,
of the Heisenberg uncertainty relationship, and of
Bohr's principle of complementarity (to name only a
few) were properly incorporated into philosophy, and
it is doubtful whether this process has already come
to an end. Consequently it will likewise be many years
before the philosophical discussion of the new outlook
due to game theory will have crystallized.

In statements about the philosophic significance of
a scientific area it would help if it were unambiguously
clear what is meant by “philosophy.” Philosophy has
a difficult but fairly well defined scope when it comes
to analyzing problems of knowledge, of verification,
of the meaning of truth. But to determine the philo-
sophical meaning of a new scientific development is
almost impossible while that change is rapidly pro-
gressing. Therefore only some tentative remarks shall
be made in which there is no attempt to order them
according to their significance or to be exhaustive. Nor
can one be sure that the principal philosophical mean-
ing does not lie elsewhere.

(a) We are confronted with a new development
concerning our understanding of reason and rationality
as the previous sections have indicated. Both being

possible human attributes we are now in possession of
precise concepts that were lacking or undefined for-
merly. We have a mathematical theory that is largely
combinatorial in character and whatever ultimate
crises mathematics itself may be afflicted with there
has never been any doubt cast on the final character
of combinatorics. The new light thrown on the problem
of rational behavior has shown that there is here not
one problem but many, that they inevitably lead to
formulations requiring mathematical analysis, that one
is now capable of providing such analysis at consid-
erable depth and that actual computations are possible,
though limited by physical processes such as speed and
memory of the computers.

Mathematics thus has encroached on another field
of human activity in a decisive manner, and it is certain
that it will never be dislodged from it again. We also
note that axiomatics, so far the ultimate formal expres-
sion we are capable of giving to theories, has now for
the first time firmly established itself in the social
sciences.

(b) A further step has been taken in the behavioral
sciences by the replacement of determinism by the
new, extended, role which has been assigned to proba-
bility though the indeterminacy introduced is not in
all respects that of a probabilistic nature (as is shown,
e.g., by the uncertainty regarding which imputation
in a solution set in an n-person game will be chosen).
This also affects the ideas held concerning prediction:
neither deterministic nor probabilistic approaches need
to work, as uncertainty of a different kind appears to
prevail in many social setups and decision situations.

(c) Modern decision theory has thrown new light on
the nature and role of information, its flow from indi-
vidual to individual and on the value and cost of
obtaining it or preventing it from spreading. In the
same spirit mention must be made of the fact that one
has gained control—no doubt in an initial manner
only—of the troublesome notion of utility by tying it
firmly to expectations and various forms of probability.

(d) The immense complexity of social actions and
their interplay has been laid bare. It is seen that it
is greater by several orders of magnitude over what
earlier writers in the social sciences had contemplated,
and it has been shown—though only in part and so
far mainly indirectly—how and why the classical
formalistic approaches must fail. It is probably no
exaggeration to state that social science will prove to
be far more difficult than physics and that it will re-
quire (as indicated earlier) the development of new
mathematical disciplines.

There is, in particular, one philosophical conse-
quence that must be stressed because it seems to have
escaped proper attention thus far: it was emphasized
above that certain formal system of society will of
necessity work in a manner different from the inten-
tions of the designers. More generally we state that no
complete formalization of society is possible: if a
formalization is made, it is either incomplete or self-
contradictory. Hence the attempt can only be to for-
malize as much as possible and to supplement the
formalism by new formalistic decisions in those con-
crete situations where it fails. Every social theory must
therefore be dynamic, proceeding from one formalism
to another. The axiomatization of games conforms to
this fact, since the axioms require neither categoricity
nor completeness because new games can always be
invented and these can serve as prototypes for new
social arrangements.

The theory of finding optimal strategies in decision-
making has thus produced a new paradigm for the
social and behavioral sciences. It will take considerable
time before the full impact of this development is felt.
But one philosophical meaning cannot be missed even
now: the push towards a more general theory firmly
based on combinatorial mathematical concepts and
procedures.

However, before philosophy reaches its ultimate
state of becoming the most general abstract science,
in the sense of Leibniz' Mathesis universalis, philo-
sophical activity may itself be viewed as a game. This
only appears to be a heretic idea. Plato in Parmenides
did speak of philosophy as a game and the Sophists
engaged openly in philosophical contests. Philosophical
schools have always competed with each other, as is
the case in all sciences in different stages of their
development. The same applies to art; it suffices to
recall the contests between Leonardo da Vinci and
Michelangelo. With this remark we return to the
opening observation in this paper which showed the
deep roots of games in human affairs to be such that
we may speak rightly of man as Homo ludens.

BIBLIOGRAPHY

R. J. Aumann and M. Maschler, eds. Recent Advances
in Game Theory (Princeton, 1962). W. J. Baumol and S. M.
Goldfeld, eds., Percursors in Mathematical Economics: An
Anthology (London, 1968), Introduction by H. Kuhn, pp.
1-9. Claude Berge, Théorie générale des jeux à n-personnes
(Paris, 1957). R. B. Braithwaite, Theory of Games as a Tool
for the Moral Philosopher (Cambridge, 1955). M. Dresher,
L. S. Shapley, and A. W. Tucker, eds., Advances in Game
Theory, Annals of Mathematics Studies, No. 52 (1964). R.
Farquharson, The Theory of Voting (New Haven, 1969).
M. Fréchet and J. v. Neumann, “Commentary on the Three
Notes of Émile Borel,” Econometrica, 21 (1953), 118-27. P. C.
Fishburn, Utility Theory for Decision Making (New York,
1970). K. P. Heiss, “Game Theory and Human Conflicts,”

Methods of Operations Research, 5 (1968), 182-204. Johan
Huizinga, homo Ludens (Leyden, 1938). G. Klaus, Spiel-
theorie in Philosophischer Sicht (Berlin, 1968). W. Krelle,
Präferenz und Entscheidungstheorie (Tübingen, 1968). H. W.
Kuhn, A. W. Tucker, et al., eds., Contributions to the Theory
of Games, 4 vols. (Princeton, 1950-59); Bibliography in
Vol. IV (1959), compiled by Dorothea M. Thompson and
Gerald L. Thompson. G. W. Leibniz, “Annotatio de qui-
busdam ludis,” Miscellanea Berolinensia (Berlin, 1710), p.
22. R. C. Lewontin, “Evolution and the Theory of Games,”
Journal of Theoretical Biology, 1 (1961), 382-403. W. F.
Lucas, “A Game with No Solutions,” Bulletin of the Ameri-
can Mathematical Society, 74 (1968), 237-39; idem, “Some
Recent Developments in n-Person Game Theory,” SIAM
Review, 13 (1971). R. D. Luce and H. Raiffa, Games and
Decision (New York, 1957). Richard M. Martin, Intension
and Decision (New York, 1963). K. Menger, “Das
Unsicherheitsmoment in der Wertlehre,” Zeitschrift für
Nationalökonomie, 5 (1934), 459-85; trans. Wolfgang
Schoellkopf, with the assistance of W. Giles Mellon, as “The
Role of Uncertainty in Economics,” in M. Shubik, ed., Essays
in Mathematical Economics in Honor of Oskar Morgenstern
(Princeton, 1967), pp. 211-31. J. W. Milnor, “Games Against
Nature,” in R. M. Thrall, C. H. Coombs, and R. L. Davis,
eds., Decision Processes (New York, 1954). O. Morgenstern,
Wirtschaftsprognose, eine Untersuchung ihrer Voraussetzun-
gen und Möglichkeiten (Vienna, 1928); idem, “Vollkommene
Voraussicht und Wirtschaftliches Gleichgewicht,” Zeit-
schrift für Nationalökonomie, 6 (1935), 337-57; idem, “Pareto
Optimum and Economic Organization,” Systeme und
Methoden in den Wirtschafts und Sozialwissenschaften, ed.
N. Kloten (Tübingen, 1964), pp. 573-86; idem, in prepara-
tion, “History of Game Theory to 1944,” International Jour-
nal of Game Theory, 1 (1972). Oskar Morgenstern and John
von Neumann, Theory of Games and Economic Behavior
(1944); 3rd ed. (Princeton, 1953). John von Neumann, “Zur
Theorie der Gesellschaftsspiele,” Mathematische Annalen,
100 (1928), 295-320; trans. Sonya Borgmann, in Kuhn, et
al., IV, 13-42. O. Ore, Cardano, The Gambling Scholar
(Princeton, 1953). V. Pareto, Manuel d'économie politique
(Paris, 1909). J. Pfanzagl, “Subjective Probability derived
from the Morgenstern-von Neumann Utility Concept,”
in M. Shubik, ed., Essays in Mathematical Economics
... (Princeton, 1967), pp. 237-51. F. P. Ramsey, “Truth
and Probability” (1926), reprinted in The Foundations of
Mathematics and Other Logical Essays (New York, 1931).
M. Reiss, “Beiträge zur Theorie des Solitärspiels,” Crelle's
Journal, 54 (1858), 344-79. W. Riker, The Theory of Political
Coalitions (New Haven, 1962). L. S. Shapley and M. Shubik,
Competition, Welfare and the Theory of Games, Vol. I, in
preparation. M. Shubik, ed., Game Theory and Related
Approaches to Social Behavior (New York, 1964). L. B.
Slobodkin, “The Strategy of Evolution,” American Scientist,
52, No. 3 (1964), 342-56. P. Suppes, “The Philosophical
Relevance of Decision Theory,” Journal of Philosophy, 18
(1961), 605-14. J. Ville, “Sur la théorie générale des jeux
où intervient l'habilité des joueurs,” in Émile Borel, ed.,
Traité du calcul des probabilités et de ses applications, Vol.
4: Applications diverses et conclusion (Paris, 1938), 104-13.
A. Wald, Statistical Decision Functions (New York, 1950).
James Waldegrave, Excerpt from a Letter (1713); see Baumol
and Goldfeld, above. E. Zermelo, “Über eine Anwendung
der Mengenlehre auf die Theorie des Schachspiels,” Proceed-
ings of the Fifth International Congress of Mathematicians
(Cambridge, 1912), 2, 501-04.

OSKAR MORGENSTERN

[See also Art and Play; Axiomatization; Chance; Indeter-
minacy in Physics; Probability; Social Welfare; Utility.]