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.
