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Dictionary of the History of Ideas | ||

#### 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

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

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

in disguises such as in ceremonies, liturgies, diplomatic

customs, or war, the latter being especially visible

during the time of maintenance of expensive private

mercenary armies. In Roman Imperial times public

games were a great burden on the state. In modern

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

dom from football pools, exceed those of some of the

largest corporations.

Since games have always occupied man in a very

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

became a subject of scientific inquiry, especially in

view of the dominating role of uncertainty in games.

But finally the fundamental notion of *probability* arose

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

the sixteenth century, developed by Girolamo Cardano

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

Christiaan Huygens, the Bernoullis, Pierre Simon de

Laplace and many others of equal distinction have

extended our understanding of this basic concept. It

is still the subject of searching mathematical analysis

without which it is impossible for modern science even

to attempt to describe the physical or social world.

Probability theory, not to be discussed further here

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

of its complexity and high mathematical sophistication

with a simpler specialized game situation than that

encountered in those games in which true strategic

situations occur. These are characterized by the simul-

taneous appearance of several independent but inter-

acting human agents each pursuing his own goal.

Probability theory first explained chances in particular

games. But philosophical questions were raised, notably

by Laplace. The relationships between those games and

situations similar to them, but transcending them in

their human significance were subjected to analysis.

While some issues were clarified it immediately be-

came clear that buried under the obvious there were

further questions which awaited formulation and an-

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

application of probability theory to physics, by then

an actively developing abstract mathematical disci-

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

teenth century. Though it originated from the study

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

the application to social events—except for actuarial

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

physics and astronomy.

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

outcome probability alone is not decisive was clearly

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

who stated: “Games combining chance and skill give

the best representation of human life, particularly of

military affairs and of the practice of medicine which

necessarily depend partly on skill and partly on

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

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

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

made of games, treated mathematically”). Leibniz also

foresaw the possibility of

*simulation*of real life situa-

tions by indicating that naval problems could be stud-

ied by moving appropriate units representing ships on

maneuver boards. The similarity of chess to some real

life situations is obvious and was noted for example

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

by Dirk van Delft who saw in that game a microcosm

of society. The ancient Chinese game

*wei-ch'i,*better

known by its Japanese name of

*go*was always inter-

preted as a mirror of complex, primarily military,

operations. Later many authors have referred to the

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

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

some similarity and quite another to establish a rigor-

ous and workable theory.

In 1713 when James de Waldegrave analyzed the

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

Remond de Montmort to Nicholas Bernoulli (Baumol

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

This remarkable study anticipated a specific case of

what is now known as the (optimal) minimax strategy

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

without a saddle point. However this matter was

entirely forgotten or perhaps never understood, and

has only been unearthed recently. Thus de Waldegrave

had no influence; also his solution would have remained

singular since the mathematics of his time would not

have made it possible to prove a generalization of his

specific result.

It is a moot question whether mathematics could

have developed rapidly in the direction which the

theory of games of strategy has taken. The interest of

mathematicians was then dominated by the study of

analysis, stimulated by the concomitant and inseparable

development of mechanics. It can be even argued that

it is at any rate largely an accident that the human

mind turned early towards the formal science of math-

ematics and not towards, say, the intriguing task of

formalizing law in a similarly rigorous manner.

There is no known record of any deeper scientific

concern with games of strategy for about 200 years,

though various authors, including C. F. Gauss and

others, have from time to time studied certain com-

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

mined the minimum number of queens needed to con-

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

given an extensive mathematical treatment of a game

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

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

It seems that this work too was forgotten and without

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

Lasker (1918) advanced the understanding of chess

mathematically and philosophically. In 1924-27 É.

Borel published papers on a certain two-person game,

for which he found an optimal method of playing, but

he expressed belief that it would not be possible to

arrive at a general theorem. Confirming the well-

known danger of making negative statements in sci-

ence, John von Neumann in his important paper of

1928, “Zur Theorie der Gesellschaftsspiele” (

*Mathe-*

matische Annalen,100), proved precisely what Borel

matische Annalen,

had thought to be impossible: a general theorem which

guarantees that there is always an optimal strategy

available for a player: the now famous fundamental

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

This paper, though decisive, was again neglected,

though in 1938 J. Ville gave a simplified and more

general version of the proof of the minimax theorem.

In 1944 appeared the

*Theory of Games and Economic*

Behaviorby John von Neumann and Oskar Morgen-

Behavior

stern, a large and comprehensive work, which defi-

nitely established the field. Since then an immense,

steadily growing literature on games and decision the-

ory has arisen in many countries. The theory developed

by von Neumann and Morgenstern has been extended,

applied and modified, but its basic structure and con-

cepts sustain the new developments. Decision theory

in the narrower, principally statistical, sense had de-

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

The minimax theorem is of crucial importance also.

The newest modifications and extensions of either game

theory or statistical decision theory are manifold, and

some brief indications are found in the text below. The

history of the theory of games of strategy to 1944 is

found in Morgenstern, 1972.

*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-

(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,

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

*non*rational 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-

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.

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

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

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

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

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

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.

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*Théorie générale des jeux à n-personnes*

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*Theory of Games as a Tool*

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OSKAR MORGENSTERN

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

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