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).
