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.
